SEVENTEENEQUATIONS

THATCHANGEDTHEWORLD

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SEVENTEENEQUATIONSTHATCHANGEDTHEWORLD

IANSTEWART

FirstpublishedinGreatBritainin2012byPROFILEBOOKSLTD3AExmouthHouse

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ContentsWhyEquations?

1Thesquawonthehippopotamus

Pythagoras’sTheorem

2Shorteningtheproceedings

Logarithms

3Ghostsofdepartedquantities

Calculus

4Thesystemoftheworld

Newton’sLawofGravity

5Portentoftheidealworld

TheSquareRootofMinusOne

6Muchadoaboutknotting

Euler’sFormulaforPolyhedra

7Patternsofchance

NormalDistribution

8Goodvibrations

WaveEquation

9Ripplesandblips

FourierTransform

10Theascentofhumanity

Navier–StokesEquation

11Wavesintheether

Maxwell’sEquations

12Lawanddisorder

SecondLawofThermodynamics

13Onethingisabsolute

Relativity

14Quantumweirdness

Schrödinger’sEquation

15Codes,communications,andcomputers

InformationTheory

16Theimbalanceofnature

ChaosTheory

17TheMidasformula

Black–ScholesEquation

WhereNext?NotesIllustrationCreditsIndex

Toavoidethetediouserepetitionofthesewoordes:isequalleto:IwillsetteasIdoeofteninwoorkeuse,apaireofparalleles,orgemowelinesofonelengthe: ,bicausenoe.2.thynges,canbemoareequalle.

RobertRecorde,TheWhetstoneofWitte,1557

WhyEquations?Equationsarethelifebloodofmathematics,science,andtechnology.Withoutthem,ourworldwouldnotexistinitspresentform.However,equationshaveareputationforbeingscary:StephenHawking’spublisherstoldhimthateveryequationwouldhalvethesalesofABriefHistoryofTime,butthentheyignoredtheirownadviceandallowedhimtoincludeE=mc2whencuttingitoutwouldallegedlyhavesoldanother10millioncopies.I’monHawking’sside.Equationsaretooimportanttobehiddenaway.Buthispublishershadapointtoo:equationsareformalandaustere,theylookcomplicated,andeventhoseofuswholoveequationscanbeputoffifwearebombardedwiththem.

Inthisbook,Ihaveanexcuse.Sinceit’saboutequations,IcannomoreavoidincludingthemthanIcouldwriteabookaboutmountaineeringwithoutusingtheword‘mountain’.Iwanttoconvinceyouthatequationshaveplayedavitalpartincreatingtoday’sworld,frommapmakingtosatnav,frommusictotelevision,fromdiscoveringAmericatoexploringthemoonsofJupiter.Fortunately,youdon’tneedtobearocketscientisttoappreciatethepoetryandbeautyofagood,significantequation.

Therearetwokindsofequationsinmathematics,whichonthesurfacelookverysimilar.Onekindpresentsrelationsbetweenvariousmathematicalquantities:thetaskistoprovetheequationistrue.Theotherkindprovidesinformationaboutanunknownquantity,andthemathematician’staskistosolveit–tomaketheunknownknown.Thedistinctionisnotclear-cut,becausesometimesthesameequationcanbeusedinbothways,butit’sausefulguideline.Youwillfindbothkindshere.

Equationsinpuremathematicsaregenerallyofthefirstkind:theyrevealdeepandbeautifulpatternsandregularities.Theyarevalidbecause,givenourbasicassumptionsaboutthelogicalstructureofmathematics,thereisnoalternative.Pythagoras’stheorem,whichisanequationexpressedinthelanguageofgeometry,isanexample.IfyouacceptEuclid’sbasicassumptionsaboutgeometry,thenPythagoras’stheoremistrue.

Equationsinappliedmathematicsandmathematicalphysicsareusuallyofthesecondkind.Theyencodeinformationabouttherealworld;theyexpresspropertiesoftheuniversethatcouldinprinciplehavebeenverydifferent.Newton’slawofgravityisagoodexample.Ittellsushowtheattractiveforcebetweentwobodiesdependsontheirmasses,andhowfaraparttheyare.Solving

theresultingequationstellsushowtheplanetsorbittheSun,orhowtodesignatrajectoryforaspaceprobe.ButNewton’slawisn’tamathematicaltheorem;it’strueforphysicalreasons,itfitsobservations.Thelawofgravitymighthavebeendifferent.Indeed,itisdifferent:Einstein’sgeneraltheoryofrelativityimprovesonNewtonbyfittingsomeobservationsbetter,whilenotmessingupthosewherewealreadyknowNewton’slawdoesagoodjob.

Thecourseofhumanhistoryhasbeenredirected,timeandtimeagain,byanequation.Equationshavehiddenpowers.Theyrevealtheinnermostsecretsofnature.Thisisnotthetraditionalwayforhistorianstoorganisetheriseandfallofcivilisations.Kingsandqueensandwarsandnaturaldisastersaboundinthehistorybooks,butequationsarethinontheground.Thisisunfair.InVictoriantimes,MichaelFaradaywasdemonstratingconnectionsbetweenmagnetismandelectricitytoaudiencesattheRoyalInstitutioninLondon.Allegedly,PrimeMinisterWilliamGladstoneaskedwhetheranythingofpracticalconsequencewouldcomefromit.Itissaid(onthebasisofverylittleactualevidence,butwhyruinanicestory?)thatFaradayreplied:‘Yes,sir.Onedayyouwilltaxit.’Ifhedidsaythat,hewasright.JamesClerkMaxwelltransformedearlyexperimentalobservationsandempiricallawsaboutmagnetismandelectricityintoasystemofequationsforelectromagnetism.Amongthemanyconsequenceswereradio,radar,andtelevision.

Anequationderivesitspowerfromasimplesource.Ittellsusthattwocalculations,whichappeardifferent,havethesameanswer.Thekeysymbolistheequalssign,=.Theoriginsofmostmathematicalsymbolsareeitherlostinthemistsofantiquity,oraresorecentthatthereisnodoubtwheretheycamefrom.Theequalssignisunusualbecauseitdatesbackmorethan450years,yetwenotonlyknowwhoinventedit,weevenknowwhy.TheinventorwasRobertRecorde,in1557,inTheWhetstoneofWitte.Heusedtwoparallellines(heusedanobsoletewordgemowe,meaning‘twin’)toavoidtediousrepetitionofthewords‘isequalto’.Hechosethatsymbolbecause‘notwothingscanbemoreequal’.Recordechosewell.Hissymbolhasremainedinusefor450years.

Thepowerofequationsliesinthephilosophicallydifficultcorrespondencebetweenmathematics,acollectivecreationofhumanminds,andanexternalphysicalreality.Equationsmodeldeeppatternsintheoutsideworld.Bylearningtovalueequations,andtoreadthestoriestheytell,wecanuncovervitalfeaturesoftheworldaroundus.Inprinciple,theremightbeotherwaystoachievethesameresult.Manypeoplepreferwordstosymbols;language,too,givesuspoweroveroursurroundings.Buttheverdictofscienceandtechnologyisthatwordsaretooimprecise,andtoolimited,toprovideaneffectiveroutetothe

deeperaspectsofreality.Theyaretoocolouredbyhuman-levelassumptions.Wordsalonecan’tprovidetheessentialinsights.

Equationscan.Theyhavebeenaprimemoverinhumancivilisationforthousandsofyears.Throughouthistory,equationshavebeenpullingthestringsofsociety.Tuckedawaybehindthescenes,tobesure–buttheinfluencewasthere,whetheritwasnoticedornot.Thisisthestoryoftheascentofhumanity,toldthrough17equations.

1Thesquawonthehippopotamus

Pythagoras’sTheorem

Whatdoesittellus?

Howthethreesidesofaright-angledtrianglearerelated.

Whyisthatimportant?

Itprovidesavitallinkbetweengeometryandalgebra,allowingustocalculatedistancesintermsofcoordinates.Italsoinspiredtrigonometry.

Whatdiditleadto?

Surveying,navigation,andmorerecentlyspecialandgeneralrelativity–thebestcurrenttheoriesofspace,time,andgravity.

Askanyschoolstudenttonameafamousmathematician,and,assumingtheycanthinkofone,moreoftenthannottheywilloptforPythagoras.Ifnot,Archimedesmightspringtomind.EventheillustriousIsaacNewtonhastoplaythirdfiddletothesetwosuperstarsoftheancientworld.Archimedeswasanintellectualgiant,andPythagorasprobablywasn’t,buthedeservesmorecreditthanheisoftengiven.Notforwhatheachieved,butforwhathesetinmotion.

PythagoraswasbornontheGreekislandofSamos,intheeasternAegean,around570BC.Hewasaphilosopherandageometer.Whatlittleweknowabouthislifecomesfrommuchlaterwritersanditshistoricalaccuracyisquestionable,butthekeyeventsareprobablycorrect.Around530BChemovedtoCroton,aGreekcolonyinwhatisnowItaly.Therehefoundedaphilosophico-religiouscult,thePythagoreans,whobelievedthattheuniverseisbasedonnumber.Theirfounder’spresent-dayfamerestsonthetheoremthatbearshisname.Ithasbeentaughtformorethan2000years,andhasenteredpopularculture.The1958movieMerryAndrew,starringDannyKaye,includesasongwhoselyricsbegin:

Thesquareonthehypotenuseofarighttriangleisequaltothesumofthesquaresonthetwoadjacentsides.

Thesonggoesonwithsomedoubleentendreaboutnotlettingyourparticipledangle,andassociatesEinstein,Newton,andtheWrightbrotherswiththefamoustheorem.Thefirsttwoexclaim‘Eureka!’;no,thatwasArchimedes.Youwilldeducethatthelyricsarenothotonhistoricalaccuracy,butthat’sHollywoodforyou.However,inChapter13wewillseethatthelyricist(JohnnyMercer)wasspotonwithEinstein,probablymoresothanherealised.

Pythagoras’stheoremappearsinawell-knownjoke,withterriblepunsaboutthesquawonthehippopotamus.Thejokecanbefoundallovertheinternet,butit’smuchhardertodiscoverwhereitcamefrom.1TherearePythagorascartoons,T-shirts,andaGreekstamp,Figure1.

Fig1GreekstampshowingPythagoras’stheorem.

Allthisfussnotwithstanding,wehavenoideawhetherPythagorasactuallyprovedhistheorem.Infact,wedon’tknowwhetheritwashistheorematall.ItcouldwellhavebeendiscoveredbyoneofPythagoras’sminions,orsomeBabylonianorSumerianscribe.ButPythagorasgotthecredit,andhisnamestuck.Whateveritsorigins,thetheoremanditsconsequenceshavehadagiganticimpactonhumanhistory.Theyliterallyopenedupourworld.

TheGreeksdidnotexpressPythagoras’stheoremasanequationinthemodernsymbolicsense.Thatcamelaterwiththedevelopmentofalgebra.Inancienttimes,thetheoremwasexpressedverballyandgeometrically.Itattaineditsmostpolishedform,anditsfirstrecordedproof,inthewritingsofEuclidofAlexandria.Around250BCEuclidbecamethefirstmodernmathematicianwhenhewrotehisfamousElements,themostinfluentialmathematicaltextbookever.Euclidturnedgeometryintologicbymakinghisbasicassumptionsexplicitandinvokingthemtogivesystematicproofsforallofhistheorems.Hebuiltaconceptualtowerwhosefoundationswerepoints,lines,andcircles,andwhosepinnaclewastheexistenceofpreciselyfiveregularsolids.

OneofthejewelsinEuclid’scrownwaswhatwenowcallPythagoras’stheorem:Proposition47ofBookIoftheElements.InthefamoustranslationbySirThomasHeaththispropositionreads:‘Inright-angledtrianglesthesquareonthesidesubtendingtherightangleisequaltothesquaresonthesidescontainingtherightangle.’

Nohippopotamus,then.Nohypotenuse.Notevenanexplicit‘sum’or‘add’.Justthatfunnyword‘subtend’,whichbasicallymeans‘beoppositeto’.

However,Pythagoras’stheoremclearlyexpressesanequation,becauseitcontainsthatvitalword:equal.

Forthepurposesofhighermathematics,theGreeksworkedwithlinesandareasinsteadofnumbers.SoPythagorasandhisGreeksuccessorswoulddecodethetheoremasanequalityofareas:‘Theareaofasquareconstructedusingthelongestsideofaright-angledtriangleisthesumoftheareasofthesquaresformedfromtheothertwosides.’Thelongestsideisthefamoushypotenuse,whichmeans‘tostretchunder’,whichitdoesifyoudrawthediagramintheappropriateorientation,asinFigure2(left).

Withinamere2000years,Pythagoras’stheoremhadbeenrecastasthealgebraicequation

a2+b2=c2

wherecisthelengthofthehypotenuse,aandbarethelengthsoftheothertwosides,andthesmallraised2means‘square’.Algebraically,thesquareofanynumberisthatnumbermultipliedbyitself,andweallknowthattheareaofanysquareisthesquareofthelengthofitsside.SoPythagoras’sequation,asIshallrenameit,saysthesamethingthatEuclidsaid–exceptforvariouspsychologicalbaggagetodowithhowtheancientsthoughtaboutbasicmathematicalconceptslikenumbersandareas,whichIwon’tgointo.

Pythagoras’sequationhasmanyusesandimplications.Mostdirectly,itletsyoucalculatethelengthofthehypotenuse,giventheothertwosides.Forinstance,supposethata=3andb=4.Thenc2=a2+b2=32+42=9+16=25.Thereforec=5.Thisisthefamous3–4–5triangle,ubiquitousinschoolmathematics,andthesimplestexampleofaPythagoreantriple:alistofthreewholenumbersthatsatisfiesPythagoras’sequation.Thenextsimplest,otherthanscaledversionssuchas6–8–10,isthe5–12–13triangle.Thereareinfinitelymanysuchtriples,andtheGreeksknewhowtoconstructthemall.Theystillretainsomeinterestinnumbertheory,andeveninthelastdecadenewfeatureshavebeendiscovered.

Insteadofusingaandbtoworkoutc,youcanproceedindirectly,andsolvetheequationtoobtainaprovidedthatyouknowbandc.Youcanalsoanswermoresubtlequestions,aswewillshortlysee.

Fig2ConstructionlinesforEuclid’sproofofPythagoras.Middleandright:Alternativeproofofthetheorem.Theoutersquareshaveequalareas,andtheshadedtrianglesallhaveequalareas.Thereforethetiltedwhitesquarehasthesameareaastheothertwowhitesquarescombined.

Whyisthetheoremtrue?Euclid’sproofisquitecomplicated,anditinvolvesdrawingfiveextralinesonthediagram,Figure2(left),andappealingtoseveralpreviouslyprovedtheorems.Victorianschoolboys(fewgirlsdidgeometryinthosedays)referredtoitirreverentlyasPythagoras’spants.Astraightforwardandintuitiveproof,thoughnotthemostelegant,usesfourcopiesofthetriangletorelatetwosolutionsofthesamemathematicaljigsawpuzzle,Figure2(right).Thepictureiscompelling,butfillinginthelogicaldetailsrequiressomethought.Forinstance:howdoweknowthatthetiltedwhiteregioninthemiddlepictureisasquare?

ThereistantalisingevidencethatPythagoras’stheoremwasknownlongbeforePythagoras.ABabylonianclaytablet2intheBritishMuseumcontains,incuneiformscript,amathematicalproblemandanswerthatcanbeparaphrasedas:

4isthelengthand5thediagonal.Whatisthebreadth?4times4is16.5times5is25.Take16from25toobtain9.WhattimeswhatmustItaketoget9?3times3is9.Therefore3isthebreadth.

SotheBabylonianscertainlyknewaboutthe3–4–5triangle,athousandyearsbeforePythagoras.

Anothertablet,YBC7289fromtheBabyloniancollectionofYaleUniversity,isshowninFigure3(left).Itshowsadiagramofasquareofside30,whosediagonalismarkedwithtwolistsofnumbers:1,24,51,10and42,25,35.TheBabyloniansemployedbase-60notationfornumbers,sothefirstlistactuallyrefersto1+24/60+51/602+10/603,whichindecimalsis1.4142129.Thesquarerootof2is1.4142135.Thesecondlistis30timesthis.SotheBabyloniansknewthatthediagonalofasquareisitssidemultipliedbythesquarerootof2.Since12+12=2= ,thistooisaninstanceofPythagoras’stheorem.

Fig3YBC7289.Right:Plimpton322.

Evenmoreremarkable,thoughmoreenigmatic,isthetabletPlimpton322fromGeorgeArthurPlimpton’scollectionatColumbiaUniversity,Figure3(right).Itisatableofnumbers,withfourcolumnsand15rows.Thefinalcolumnjustliststherownumber,from1to15.In1945historiansofscienceOttoNeugebauerandAbrahamSachs3noticedthatineachrow,thesquareofthenumber(sayc)inthethirdcolumn,minusthesquareofthenumber(sayb)inthesecondcolumn,isitselfasquare(saya).Itfollowsthata2+b2=c2,sothetableappearstorecordPythagoreantriples.Atleast,thisisthecaseprovidedfourapparenterrorsarecorrected.However,itisnotabsolutelycertainthatPlimpton322hasanythingtodowithPythagoreantriples,andevenifitdoes,itmightjusthavebeenaconvenientlistoftriangleswhoseareaswereeasytocalculate.Thesecouldthenbeassembledtoyieldgoodapproximationstoothertrianglesandothershapes,perhapsforlandmeasurement.

AnothericonicancientcivilisationisthatofEgypt.ThereissomeevidencethatPythagorasmayhavevisitedEgyptasayoungman,andsomehave

conjecturedthatthisiswherehelearnedhistheorem.ThesurvivingrecordsofEgyptianmathematicsofferscantsupportforthisidea,buttheyarefewandspecialised.Itisoftenstated,typicallyinthecontextofpyramids,thattheEgyptianslaidoutrightanglesusinga3–4–5triangle,formedfromalengthofstringwithknotsat12equalintervals,andthatarchaeologistshavefoundstringsofthatkind.However,neitherclaimmakesmuchsense.Suchatechniquewouldnotbeveryreliable,becausestringscanstretchandtheknotswouldhavetobeveryaccuratelyspaced.TheprecisionwithwhichthepyramidsatGizaarebuiltissuperiortoanythingthatcouldbeachievedwithsuchastring.Farmorepracticaltools,similartoacarpenter’ssetsquare,havebeenfound.EgyptologistsspecialisinginancientEgyptianmathematicsknowofnorecordsofstringbeingemployedtoforma3–4–5triangle,andnoexamplesofsuchstringsexist.Sothisstory,charmingthoughitmaybe,isalmostcertainlyamyth.

IfPythagorascouldbetransplantedintotoday’sworldhewouldnoticemanydifferences.Inhisday,medicalknowledgewasrudimentary,lightingcamefromcandlesandburningtorches,andthefastestformsofcommunicationwereamessengeronhorsebackoralightedbeacononahilltop.TheknownworldencompassedmuchofEurope,Asia,andAfrica–butnottheAmericas,Australia,theArctic,ortheAntarctic.Manyculturesconsideredtheworldtobeflat:acirculardiscorevenasquarealignedwiththefourcardinalpoints.DespitethediscoveriesofclassicalGreecethisbeliefwasstillwidespreadinmedievaltimes,intheformoforbisterraemaps,Figure4.

Whofirstrealisedtheworldisround?AccordingtoDiogenesLaertius,athird-centuryGreekbiographer,itwasPythagoras.InhisbookLivesandOpinionsofEminentPhilosophers,acollectionofsayingsandbiographicalnotesthatisoneofourmainhistoricalsourcesfortheprivatelivesofthephilosophersofancientGreece,hewrote:‘PythagoraswasthefirstwhocalledtheEarthround,thoughTheophrastusattributesthistoParmenidesandZenotoHesiod.’TheancientGreeksoftenclaimedthatmajordiscoverieshadbeenmadebytheirfamousforebears,irrespectiveofhistoricalfact,sowecan’ttakethestatementatfacevalue,butitisnotindisputethatfromthefifthcenturyBCallreputableGreekphilosophersandmathematiciansconsideredtheEarthtoberound.TheideadoesseemtohaveoriginatedaroundthetimeofPythagoras,anditmighthavecomefromoneofhisfollowers.Oritmighthavebeencommoncurrency,basedonevidencesuchastheroundshadowoftheEarthontheMoonduringaneclipse,ortheanalogywithanobviouslyroundMoon.

Fig4Mapoftheworldmadearound1100bytheMoroccancartographeral-IdrisiforKingRogerofSicily.

EvenfortheGreeks,though,theEarthwasthecentreoftheuniverseandeverythingelserevolvedaroundit.Navigationwascarriedoutbydeadreckoning:lookingatthestarsandfollowingthecoastline.Pythagoras’sequationchangedallthat.Itsethumanityonthepathtotoday’sunderstandingofthegeographyofourplanetanditsplaceintheSolarSystem.Itwasavitalfirststeptowardsthegeometrictechniquesneededformapmaking,navigation,andsurveying.Italsoprovidedthekeytoavitallyimportantrelationbetweengeometryandalgebra.Thislineofdevelopmentleadsfromancienttimesrightthroughtogeneralrelativityandmoderncosmology,seeChapter13.Pythagoras’sequationopenedupentirelynewdirectionsforhumanexploration,bothmetaphoricallyandliterally.Itrevealedtheshapeofourworldanditsplaceintheuniverse.

Manyofthetrianglesencounteredinreallifearenotright-angled,sotheequation’sdirectapplicationsmayseemlimited.However,anytrianglecanbecutintotworight-angledonesasinFigure6(page11),andanypolygonalshapecanbecutintotriangles.Soright-angledtrianglesarethekey:theyprovethatthereisausefulrelationbetweentheshapeofatriangleandthelengthsofitssides.Thesubjectthatdevelopedfromthisinsightistrigonometry:‘trianglemeasurement’.

Theright-angledtriangleisfundamentaltotrigonometry,andinparticularitdeterminesthebasictrigonometricfunctions:sine,cosine,andtangent.ThenamesareArabicinorigin,andthehistoryofthesefunctionsandtheirmanypredecessorsshowsthecomplicatedroutebywhichtoday’sversionofthetopic

arose.I’llcuttothechaseandexplaintheeventualoutcome.Aright-angledtrianglehas,ofcourse,arightangle,butitsothertwoanglesarearbitrary,apartfromaddingto90°.Associatedwithanyanglearethreefunctions,thatis,rulesforcalculatinganassociatednumber.FortheanglemarkedAinFigure5,usingthetraditionala,b,cforthethreesides,wedefinethesine(sin),cosine(cos),andtangent(tan)likethis:

sinA=a/ccosA=b/ctanA=a/b

ThesequantitiesdependonlyontheangleA,becauseallright-angledtriangleswithagivenangleAareidenticalexceptforscale.

Fig5Trigonometryisbasedonaright-angletriangle.

Inconsequence,itispossibletodrawupatableofthevaluesofsin,cos,andtan,forarangeofangles,andthenusethemtocalculatefeaturesofright-angledtriangles.Atypicalapplication,whichgoesbacktoancienttimes,istocalculatetheheightofatallcolumnusingonlymeasurementsmadeontheground.Supposethat,fromadistanceof100metres,theangletothetopofthecolumnis22°.TakeA=22°inFigure5,sothataistheheightofthecolumn.Thenthedefinitionofthetangentfunctiontellsusthat

tan22°=a/100

sothat

a=100tan22°.

Sincetan22°is0.404,tothreedecimalplaces,wededucethata=40.4metres.

Fig6Splittingatriangleintotwowithrightangles.

Onceinpossessionoftrigonometricfunctions,itisstraightforwardtoextendPythagoras’sequationtotrianglesthatdonothavearightangle.Figure6showsatrianglewithanangleCandsidesa,b,c.Splitthetriangleintotworight-angledonesasshown.ThentwoapplicationsofPythagorasandsomealgebra4provethat

a2+b2−2abcosC=c2

whichissimilartoPythagoras’sequation,exceptfortheextraterm−2abcosC.This‘cosinerule’doesthesamejobasPythagoras,relatingctoaandb,butnowwehavetoincludeinformationabouttheangleC.

Thecosineruleisoneofthemainstaysoftrigonometry.Ifweknowtwosidesofatriangleandtheanglebetweenthem,wecanuseittocalculatethethirdside.Otherequationsthentellustheremainingangles.Alloftheseequationscanultimatelybetracedbacktoright-angledtriangles.

Armedwithtrigonometricequationsandsuitablemeasuringapparatus,wecancarryoutsurveysandmakeaccuratemaps.Thisisnotanewidea.ItappearsintheRhindPapyrus,acollectionofancientEgyptianmathematicaltechniquesdatingfrom1650BC.TheGreekphilosopherThalesusedthegeometryoftrianglestoestimatetheheightsoftheGizapyramidsinabout600BC.HeroofAlexandriadescribedthesametechniquein50AD.Around240BCGreekmathematician,Eratosthenes,calculatedthesizeoftheEarthbyobservingtheangleoftheSunatnoonintwodifferentplaces:AlexandriaandSyene(nowAswan)inEgypt.AsuccessionofArabianscholarspreservedanddevelopedthesemethods,applyingtheminparticulartoastronomicalmeasurementssuchasthesizeoftheEarth.

Surveyingbegantotakeoffin1533whentheDutchmapmakerGemmaFrisiusexplainedhowtousetrigonometrytoproduceaccuratemaps,inLibellusdeLocorumDescribendorumRatione(‘BookletConcerningaWayofDescribingPlaces’).WordofthemethodspreadacrossEurope,reachingtheearsoftheDanishnoblemanandastronomerTychoBrahe.In1579TychousedittomakeanaccuratemapofHven,theislandwherehisobservatorywaslocated.By1615theDutchmathematicianWillebrordSnellius(SnelvanRoyen)haddevelopedthemethodintoessentiallyitsmodernform:triangulation.Theareabeingsurveyediscoveredwithanetworkoftriangles.Bymeasuringoneinitiallengthverycarefully,andmanyangles,thelocationsofthecornersofthetriangle,andhenceanyinterestingfeatureswithinthem,canbecalculated.SnelliusworkedoutthedistancebetweentwoDutchtowns,AlkmaarandBergenopZoom,usinganetworkof33triangles.Hechosethesetownsbecausetheylayonthesamelineoflongitudeandwereexactlyonedegreeofarcapart.Knowingthedistancebetweenthem,hecouldworkoutthesizeoftheEarth,whichhepublishedinhisEratosthenesBatavus(‘TheDutchEratosthenes’)in1617.Hisresultisaccuratetowithin4%.HealsomodifiedtheequationsoftrigonometrytoreflectthesphericalnatureoftheEarth’ssurface,animportantsteptowardseffectivenavigation.

Triangulationisanindirectmethodforcalculatingdistancesusingangles.Whensurveyingastretchofland,beitabuildingsiteoracountry,themainpracticalconsiderationisthatitismucheasiertomeasureanglesthanitistomeasuredistances.Triangulationletsusmeasureafewdistancesandlotsofangles;theneverythingelsefollowsfromthetrigonometricequations.Themethodbeginsbysettingoutonelinebetweentwopoints,calledthebaseline,andmeasuringitslengthdirectlytoveryhighaccuracy.Thenwechooseaprominentpointinthelandscapethatisvisiblefrombothendsofthebaseline,andmeasuretheanglefromthebaselinetothatpoint,atbothendsofthebaseline.Nowwehaveatriangle,andweknowonesideofitandtwoangles,whichfixitsshapeandsize.Wecanthenusetrigonometrytoworkouttheothertwosides.

Ineffect,wenowhavetwomorebaselines:thenewlycalculatedsidesofthetriangle.Fromthose,wecanmeasureanglestoother,moredistantpoints.Continuethisprocesstocreateanetworkoftrianglesthatcoverstheareabeingsurveyed.Withineachtriangle,observetheanglestoallnoteworthyfeatures–churchtowers,crossroads,andsoon.Thesametrigonometrictrickpinpointstheirpreciselocations.Asafinaltwist,theaccuracyoftheentiresurveycanbecheckedbymeasuringoneofthefinalsidesdirectly.

Bythelateeighteenthcentury,triangulationwasbeingemployedroutinelyinsurveys.TheOrdnanceSurveyofGreatBritainbeganin1783,taking70yearstocompletethetask.TheGreatTrigonometricSurveyofIndia,whichamongotherthingsmappedtheHimalayasanddeterminedtheheightofMountEverest,startedin1801.Inthetwenty-firstcentury,mostlarge-scalesurveyingisdoneusingsatellitephotographsandGPS(theGlobalPositioningSystem).Explicittriangulationisnolongeremployed.Butitisstillthere,behindthescenes,inthemethodsusedtodeducelocationsfromthesatellitedata.

Pythagoras’stheoremwasalsovitaltotheinventionofcoordinategeometry.Thisisawaytorepresentgeometricfiguresintermsofnumbers,usingasystemoflines,knownasaxes,labelledwithnumbers.ThemostfamiliarversionisknownasCartesiancoordinatesintheplane,inhonouroftheFrenchmathematicianandphilosopherRenéDescartes,whowasoneofthegreatpioneersinthisarea–thoughnotthefirst.Drawtwolines:ahorizontalonelabelledxandaverticalonelabelledy.Theselinesareknownasaxes(pluralofaxis),andtheycrossatapointcalledtheorigin.Markpointsalongthesetwoaxesaccordingtotheirdistancefromtheorigin,likethemarkingsonaruler:positivenumberstotherightandup,negativetotheleftanddown.Nowwecandetermineanypointintheplaneintermsoftwonumbersxandy,itscoordinates,byconnectingthepointtothetwoaxesasinFigure7.Thepairofnumbers(x,y)completelyspecifiesthelocationofthepoint.

Fig7Thetwoaxesandthecoordinatesofapoint.

Thegreatmathematiciansofseventeenth-centuryEuroperealisedthat,inthiscontext,alineorcurveintheplanecorrespondstothesetofsolutions(x,y)ofsomeequationinxandy.Forinstance,y=xdeterminesadiagonallineslopingfromlowerlefttotopright,because(x,y)liesonthatlineifandonlyify=x.Ingeneral,alinearequation–oftheformax+by=cforconstantsa,b,c–correspondstoastraightline,andviceversa.

Whatequationcorrespondstoacircle?ThisiswherePythagoras’sequationcomesin.Itimpliesthatthedistancerfromtheorigintothepoint(x,y)satisfies

r2=x2+y2

andwecansolvethisforrtoobtain

Sincethesetofallpointsthatlieatdistancerfromtheoriginisacircleofradiusr,whosecentreistheorigin,sothesameequationdefinesacircle.Moregenerally,thecircleofradiusrwithcentreat(a,b)correspondstotheequation

(x−a)2+(y−b)2=r2

andthesameequationdeterminesthedistancerbetweenthetwopoints(a,b)and(x,y).SoPythagoras’stheoremtellsustwovitalthings:whichequationsyieldcircles,andhowtocalculatedistancesfromcoordinates.

Pythagoras’stheorem,then,isimportantinitsownright,butitexertsevenmoreinfluencethroughitsgeneralisations.HereIwillpursuejustonestrandoftheselaterdevelopmentstobringouttheconnectionwithrelativity,towhichwereturninChapter13.

TheproofofPythagoras’stheoreminEuclid’sElementsplacesthetheoremfirmlywithintherealmofEuclideangeometry.Therewasatimewhenthatphrasecouldhavebeenreplacedbyjust‘geometry’,becauseitwasgenerallyassumedthatEuclid’sgeometrywasthetruegeometryofphysicalspace.Itwasobvious.Likemostthingsassumedtobeobvious,itturnedouttobefalse.

Euclidderivedallofhistheoremsfromasmallnumberofbasicassumptions,whichheclassifiedasdefinitions,axioms,andcommonnotions.Hisset-upwas

elegant,intuitive,andconcise,withoneglaringexception,hisfifthaxiom:‘Ifastraightlinefallingontwostraightlinesmakestheinterioranglesonthesamesidelessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthatsideonwhicharetheangleslessthanthetworightangles.’It’sabitofamouthful:Figure8mayhelp.

Fig8Euclid’sparallelaxiom.

Forwelloverathousandyears,mathematicianstriedtorepairwhattheysawasaflaw.Theyweren’tjustlookingforsomethingsimplerandmoreintuitivethatwouldachievethesameend,althoughseveralofthemfoundsuchthings.Theywantedtogetridoftheawkwardaxiomaltogether,byprovingit.Afterseveralcenturies,mathematiciansfinallyrealisedthattherewerealternative‘non-Euclidean’geometries,implyingthatnosuchproofexisted.ThesenewgeometrieswerejustaslogicallyconsistentasEuclid’s,andtheyobeyedallofhisaxiomsexcepttheparallelaxiom.Theycouldbeinterpretedasthegeometryofgeodesics–shortestpaths–oncurvedsurfaces,Figure9.Thisfocusedattentiononthemeaningofcurvature.

Fig9Curvatureofasurface.zerocurvature.Middle:positivecurvature.negative

curvature.

TheplaneofEuclidisflat,curvaturezero.Aspherehasthesamecurvatureeverywhere,anditispositive:nearanypointitlookslikeadome.(Asatechnicalfinepoint:greatcirclesmeetintwopoints,notoneasEuclid’saxiomsrequire,sosphericalgeometryismodifiedbyidentifyingantipodalpointsonthesphere–consideringthemtobeidentical.Thesurfacebecomesaso-calledprojectiveplaneandthegeometryiscalledelliptic.)Asurfaceofconstantnegativecurvaturealsoexists:nearanypoint,itlookslikeasaddle.Thissurfaceiscalledthehyperbolicplane,anditcanberepresentedinseveralentirelyprosaicways.Perhapsthesimplestistoconsideritastheinteriorofacirculardisc,andtodefine‘line’asanarcofacirclemeetingtheedgeofthediscatrightangles(Figure10).

Fig10Discmodelofthehyperbolicplane.AllthreelinesthroughPfailtomeetlineL.

Itmightseemthat,whileplanegeometrymightbenon-Euclidean,thismustbeimpossibleforthegeometryofspace.Youcanbendasurfacebypushingitintoathirddimension,butyoucan’tbendspacebecausethere’snoroomforanextradimensionalongwhichtopushit.However,thisisarathernaiveview.Forexample,wecanmodelthree-dimensionalhyperbolicspaceusingtheinteriorofasphere.Linesaremodelledasarcsofcirclesthatmeettheboundaryatrightangles,andplanesaremodelledaspartsofspheresthatmeettheboundaryatrightangles.Thisgeometryisthree-dimensional,satisfiesallofEuclid’saxiomsexcepttheFifth,andinasensethatcanbepinneddownitdefinesacurvedthree-dimensionalspace.Butit’snotcurvedroundanything,orinanynewdirection.

It’sjustcurved.

Withallthesenewgeometriesavailable,anewpointofviewbegantooccupycentrestage–butasphysics,notmathematics.Sincespacedoesn’thavetobeEuclidean,whatshapeisit?Scientistsrealisedthattheydidn’tactuallyknow.In1813,Gauss,knowingthatinacurvedspacetheanglesofatriangledonotaddto180°,measuredtheanglesofatriangleformedbythreemountains–theBrocken,theHohehagen,andtheInselberg.Heobtainedasum15secondsofarcgreaterthan180°.Ifcorrect,thisindicatedthatspace(inthatregion,atleast)waspositivelycurved.Butyou’dneedafarlargertriangle,andfarmoreaccuratemeasurements,toeliminateobservationalerrors.SoGauss’sobservationswereinconclusive.SpacemightbeEuclidean,andthenagain,itmightnotbe.

Myremarkthatthree-dimensionalhyperbolicspaceis‘justcurved’dependsonanewpointofviewaboutcurvature,whichalsogoesbacktoGauss.Thespherehasconstantpositivecurvature,andthehyperbolicplanehasconstantnegativecurvature.Butthecurvatureofasurfacedoesn’thavetobeconstant.Itmightbesharplycurvedinsomeplaces,lesssharplycurvedinothers.Indeed,itmightbepositiveinsomeregionsbutnegativeinothers.Thecurvaturecouldvarycontinuouslyfromplacetoplace.Ifthesurfacelookslikeadog’sbone,thentheblobsattheendsarepositivelycurvedbutthepartthatjoinsthemisnegativelycurved.

Gausssearchedforaformulatocharacterisethecurvatureofasurfaceatanypoint.Whenheeventuallyfoundit,andpublisheditinhisDisquisitionesGeneralesCircaSuperficiesCurva(‘GeneralResearchonCurvedSurfaces’)of1828,henameditthe‘remarkabletheorem’.Whatwassoremarkable?Gausshadstartedfromthenaiveviewofcurvature:embedthesurfaceinthree-dimensionalspaceandcalculatehowbentitis.Buttheanswertoldhimthatthissurroundingspacedidn’tmatter.Itdidn’tenterintotheformula.Hewrote:‘Theformula…leadsitselftotheremarkabletheorem:Ifacurvedsurfaceisdevelopeduponanyothersurfacewhatever,themeasureofcurvatureineachpointremainsunchanged.’By‘developed’hemeant‘wrappedround’.

Takeaflatsheetofpaper,zerocurvature.Nowwrapitroundabottle.Ifthebottleiscylindricalthepaperfitsperfectly,withoutbeingfolded,stretched,ortorn.Itisbentasfarasvisualappearancegoes,butit’satrivialkindofbending,becauseithasn’tchangedgeometryonthepaperinanyway.It’sjustchangedhowthepaperrelatestothesurroundingspace.Drawaright-angledtriangleontheflatpaper,measureitssides,checkPythagoras.Nowwrapthediagramroundabottle.Thelengthsofsides,measuredalongthepaper,don’tchange.

Pythagorasisstilltrue.Thesurfaceofasphere,however,hasnonzerocurvature.Soitisnotpossible

towrapasheetofpapersothatitfitssnuglyagainstasphere,withoutfoldingit,stretchingit,ortearingit.Geometryonasphereisintrinsicallydifferentfromgeometryonaplane.Forexample,theEarth’sequatorandthelinesoflongitudefor0°and90°toitsnorthdetermineatrianglethathasthreerightanglesandthreeequalsides(assumingtheEarthtobeasphere).SoPythagoras’sequationisfalse.

Todaywecallcurvatureinitsintrinsicsense‘Gaussiancurvature’.Gaussexplainedwhyitisimportantusingavividanalogy,stillcurrent.Imagineanantconfinedtothesurface.Howcanitworkoutwhetherthesurfaceiscurved?Itcan’tstepoutsidethesurfacetoseewhetheritlooksbent.ButitcanuseGauss’sformulabymakingsuitablemeasurementspurelywithinthesurface.Weareinthesamepositionastheantwhenwetrytofigureoutthetruegeometryofourspace.Wecan’tstepoutsideit.Beforewecanemulatetheantbytakingmeasurements,however,weneedaformulaforthecurvatureofaspaceofthreedimensions.Gaussdidn’thaveone.Butoneofhisstudents,inafitofrecklessness,claimedthathedid.

ThestudentwasGeorgBernhardRiemann,andhewastryingtoachievewhatGermanuniversitiescallHabilitation,thenextstepafteraPhD.InRiemann’sdaythismeantthatyoucouldchargestudentsafeeforyourlectures.Thenandnow,gainingHabilitationrequirespresentingyourresearchinapubliclecturethatisalsoanexamination.Thecandidateoffersseveraltopics,andtheexaminer,whichinRiemann’scasewasGauss,choosesone.Riemann,abrilliantmathematicaltalent,listedseveralorthodoxtopicsthatheknewbackwards,butinarushofbloodtothebrainhealsosuggested‘Onthehypotheseswhichlieatthefoundationofgeometry’.Gausshadlongbeeninterestedinjustthat,andhenaturallyselecteditforRiemann’sexamination.

Riemanninstantlyregrettedofferingsomethingsochallenging.Hehadaheartydislikeofpublicspeaking,andhehadn’tthoughtthemathematicsthroughindetail.Hejusthadsomevague,thoughfascinating,ideasaboutcurvedspace.Inanynumberofdimensions.WhatGausshaddonefortwodimensions,withhisremarkabletheorem,Riemannwantedtodoinasmanydimensionsasyoulike.Nowhehadtoperform,andfast.Thelecturewaslooming.Thepressurenearlygavehimanervousbreakdown,andhisdayjobhelpingGauss’scollaboratorWilhelmWeberwithexperimentsinelectricitydidn’thelp.Well,maybeitdid,becausewhileRiemannwasthinkingabouttherelationbetween

electricalandmagneticforcesinthedayjob,herealisedthatforcecanberelatedtocurvature.Workingbackwards,hecouldusethemathematicsofforcestodefinecurvature,asrequiredforhisexamination.

In1854Riemanndeliveredhislecture,whichwaswarmlyreceived,andnowonder.Hebeganbydefiningwhathecalleda‘manifold’,inthesenseofmany-foldedness.Formally,a‘manifold’,isspecifiedbyasystemofmanycoordinates,togetherwithaformulaforthedistancebetweennearbypoints,nowcalledaRiemannianmetric.Informally,amanifoldisamultidimensionalspaceinallitsglory.TheclimaxofRiemann’slecturewasaformulathatgeneralisedGauss’sremarkabletheorem:itdefinedthecurvatureofthemanifoldsolelyintermsofitsmetric.AnditisherethatthetalecomesfullcirclelikethesnakeOrobourosandswallowsitsowntail,becausethemetriccontainsvisibleremnantsofPythagoras.

Suppose,forexample,thatthemanifoldhasthreedimensions.Letthecoordinatesofapointbe(x,y,z),andlet(x+dx,y+dy,z+dz)beanearbypoint,wherethedmeans‘alittlebitof’.IfthespaceisEuclidean,withzerocurvature,thedistancedsbetweenthesetwopointssatisfiestheequation

ds2=dx2+dy2+dz2

andthisisjustPythagoras,restrictedtopointsthatareclosetogether.Ifthespaceiscurved,withvariablecurvaturefrompointtopoint,theanalogousformula,themetric,lookslikethis:

ds2=Xdx2+Ydy2+Zdz2+2Udxdy+2Vdxdz+2Wdydz

HereX,Y,Z,U,V,Wcandependonx,yandz.Itmayseemabitofamouthful,butlikePythagoras’sequationitinvolvessumsofsquares(andcloselyrelatedproductsoftwoquantitieslikedxdy)plusafewbellsandwhistles.The2soccurbecausetheformulacanbepackagedasa3×3table,ormatrix:

whereX,Y,Zappearonce,butU,V,Wappeartwice.Thetableissymmetricaboutitsdiagonal;inthelanguageofdifferentialgeometryitisasymmetrictensor.Riemann’sgeneralisationofGauss’sremarkabletheoremisaformulaforthecurvatureofthemanifold,atanygivenpoint,intermsofthistensor.Inthe

specialcasewhenPythagorasapplies,thecurvatureturnsouttobezero.SothevalidityofPythagoras’sequationisatestfortheabsenceofcurvature.

LikeGauss’sformula,Riemann’sexpressionforcurvaturedependsonlyonthemanifold’smetric.Anantconfinedtothemanifoldcouldobservethemetricbymeasuringtinytrianglesandcomputingthecurvature.Curvatureisanintrinsicpropertyofamanifold,independentofanysurroundingspace.Indeed,themetricalreadydeterminesthegeometry,sonosurroundingspaceisrequired.Inparticular,wehumanantscanaskwhatshapeourvastandmysteriousuniverseis,andhopetoansweritbymakingobservationsthatdonotrequireustostepoutsidetheuniverse.Whichisjustaswell,becausewecan’t.

Riemannfoundhisformulabyusingforcestodefinegeometry.Fiftyyearslater,EinsteinturnedRiemann’sideaonitshead,usinggeometrytodefinetheforceofgravityinhisgeneraltheoryofrelativity,andinspiringnewideasabouttheshapeoftheuniverse:seeChapter13.It’sanastonishingprogressionofevents.Pythagoras’sequationfirstcameintobeingaround3500yearsagotomeasureafarmer’sland.Itsextensiontotriangleswithoutrightangles,andtrianglesonasphere,allowedustomapourcontinentsandmeasureourplanet.Andaremarkablegeneralisationletsusmeasuretheshapeoftheuniverse.Bigideashavesmallbeginnings.

2Shorteningtheproceedings

Logarithms

Whatdoesittellus?

Howtomultiplynumbersbyaddingrelatednumbersinstead.

Whyisthatimportant?

Additionismuchsimplerthanmultiplication.

Whatdiditleadto?

Efficientmethodsforcalculatingastronomicalphenomenasuchaseclipsesandplanetaryorbits.Quickwaystoperformscientificcalculations.Theengineers’faithfulcompanion,thesliderule.Radioactivedecayandthepsychophysicsofhumanperception.

Numbersoriginatedinpracticalproblems:recordingproperty,suchasanimalsorland,andfinancialtransactions,suchastaxationandkeepingaccounts.Theearliestknownnumbernotation,asidefromsimpletallyingmarkslike||||,isfoundontheoutsideofclayenvelopes.In8000BCMesopotamianaccountantskeptrecordsusingsmallclaytokensofvariousshapes.ThearchaeologistDeniseSchmandt-Besseratrealisedthateachshaperepresentedabasiccommodity–asphereforgrain,aneggforajarofoil,andsoon.Forsecurity,thetokensweresealedinclaywrappings.Butitwasanuisancetobreakaclayenvelopeopentofindouthowmanytokenswereinside,sotheancientaccountantsscratchedsymbolsontheoutsidetoshowwhatwasinside.Eventuallytheyrealisedthatonceyouhadthesesymbols,youcouldscrapthetokens.Theresultwasaseriesofwrittensymbolsfornumbers–theoriginofalllaternumbersymbols,andperhapsofwritingtoo.

Alongwithnumberscamearithmetic:methodsforadding,subtracting,multiplying,anddividingnumbers.Devicesliketheabacuswereusedtodothesums;thentheresultscouldberecordedinsymbols.Afteratime,wayswerefoundtousethesymbolstoperformthecalculationswithoutmechanicalassistance,althoughtheabacusisstillwidelyusedinmanypartsoftheworld,whileelectroniccalculatorshavesupplantedpenandpapercalculationsinmostothercountries.

Arithmeticprovedessentialinotherways,too,especiallyinastronomyandsurveying.Asthebasicoutlinesofthephysicalsciencesbegantoemerge,thefledgelingscientistsneededtoperformevermoreelaboratecalculations,byhand.Oftenthistookupmuchoftheirtime,sometimesmonthsoryears,gettinginthewayofmorecreativeactivities.Eventuallyitbecameessentialtospeeduptheprocess.Innumerablemechanicaldeviceswereinvented,butthemostimportantbreakthroughwasaconceptualone:thinkfirst,calculatelater.Usingclevermathematics,youcouldmakedifficultcalculationsmucheasier.

Thenewmathematicsquicklydevelopedalifeofitsown,turningouttohavedeeptheoreticalimplicationsaswellaspracticalones.Today,thoseearlyideashavebecomeanindispensabletoolthroughoutscience,reachingevenintopsychologyandthehumanities.Theywerewidelyuseduntilthe1980s,whencomputersrenderedthemobsoleteforpracticalpurposes,but,despitethat,theirimportanceinmathematicsandsciencehascontinuedtogrow.

Thecentralideaisamathematicaltechniquecalledalogarithm.Itsinventor

wasaScottishlaird,butittookageometryprofessorwithstronginterestsinnavigationandastronomytoreplacethelaird’sbrilliantbutflawedideabyamuchbetterone.

InMarch1615HenryBriggswrotealettertoJamesUssher,recordingacrucialeventinthehistoryofscience:

Napper,lordofMarkinston,hathsetmyheadandhandsaworkwithhisnewandadmirablelogarithms.Ihopetoseehimthissummer,ifitpleaseGod,forIneversawabookwhichpleasedmebetterormadememorewonder.

BriggswasthefirstprofessorofgeometryatGreshamCollegeinLondon,and‘Napper,lordofMarkinston’wasJohnNapier,eighthlairdofMerchiston,nowpartofthecityofEdinburghinScotland.Napierseemstohavebeenabitofamystic;hehadstrongtheologicalinterests,buttheymostlycentredonthebookofRevelation.Inhisview,hismostimportantworkwasAPlaineDiscoveryoftheWholeRevelationofStJohn,whichledhimtopredictthattheworldwouldendineither1688or1700.Heisthoughttohaveengagedinbothalchemyandnecromancy,andhisinterestsintheoccultlenthimareputationasamagician.Accordingtorumour,hecarriedablackspiderinasmallboxeverywherehewent,andpossesseda‘familiar’,ormagicalcompanion:ablackcockerel.Accordingtooneofhisdescendants,MarkNapier,Johnemployedhisfamiliartocatchservantswhowerestealing.Helockedthesuspectinaroomwiththecockerelandinstructedthemtostrokeit,tellingthemthathismagicalbirdwouldunerringlydetecttheguilty.ButNapier’smysticismhadarationalcore,whichinthisparticularinstanceinvolvedcoatingthecockerelwithathinlayerofsoot.Aninnocentservantwouldbeconfidentenoughtostrokethebirdasinstructed,andwouldgetsootontheirhands.Aguiltyone,fearingdetection,wouldavoidstrokingthebird.So,ironically,cleanhandsprovedyouwereguilty.

Napierdevotedmuchofhistimetomathematics,especiallymethodsforspeedingupcomplicatedarithmeticalcalculations.Oneinvention,Napier’sbones,wasasetoftenrods,markedwithnumbers,whichsimplifiedtheprocessoflongmultiplication.Evenbetterwastheinventionthatmadehisreputationandcreatedascientificrevolution:nothisbookonRevelation,ashehadhoped,buthisMirificiLogarithmorumCanonisDescriptio(‘DescriptionoftheWonderfulCanonofLogarithms’)of1614.TheprefaceshowsthatNapierknewexactlywhathehadproduced,andwhatitwasgoodfor.1

Sincenothingismoretedious,fellowmathematicians,inthepracticeofthemathematicalarts,thanthegreatdelayssufferedinthetediumoflengthymultiplicationsanddivisions,thefindingofratios,andintheextractionofsquareandcuberoots–and…themanyslipperyerrorsthatcanarise:Ihadthereforebeenturningoverinmymind,bywhatsureandexpeditiousart,Imightbeabletoimproveuponthesesaiddifficulties.Intheendaftermuchthought,finallyIhavefoundanamazingwayofshorteningtheproceedings…itisapleasanttasktosetoutthemethodforthepublicuseofmathematicians.

ThemomentBriggsheardoflogarithms,hewasenchanted.Likemanymathematiciansofhisera,hespentalotofhistimeperformingastronomicalcalculations.WeknowthisbecauseanotherletterfromBriggstoUssher,dated1610,mentionscalculatingeclipses,andbecauseBriggshadearlierpublishedtwobooksofnumericaltables,onerelatedtotheNorthPoleandtheothertonavigation.Alloftheseworkshadrequiredvastquantitiesofcomplicatedarithmeticandtrigonometry.Napier’sinventionwouldsaveagreatdealoftediouslabour.ButthemoreBriggsstudiedthebook,themoreconvincedhebecamethatalthoughNapier’sstrategywaswonderful,he’dgothistacticswrong.Briggscameupwithasimplebuteffectiveimprovement,andmadethelongjourneytoScotland.Whentheymet,‘almostonequarterofanhourwasspent,eachbeholdingtheotherwithadmiration,beforeonewordwasspoken’.2

Whatwasitthatexcitedsomuchadmiration?Thevitalobservation,obvioustoanyonelearningarithmetic,wasthataddingnumbersisrelativelyeasy,butmultiplyingthemisnot.Multiplicationrequiresmanymorearithmeticaloperationsthanaddition.Forexample,addingtwoten-digitnumbersinvolvesabouttensimplesteps,butmultiplicationrequires200.Withmoderncomputers,thisissueisstillimportant,butnowitistuckedawaybehindthescenesinthealgorithmsusedformultiplication.ButinNapier’sdayitallhadtobedonebyhand.Wouldn’titbegreatifthereweresomemathematicaltrickthatwouldconvertthosenastymultiplicationsintonice,quickadditionsums?Itsoundstoogoodtobetrue,butNapierrealisedthatitwaspossible.Thetrickwastoworkwithpowersofafixednumber.

Inalgebra,powersofanunknownxareindicatedbyasmallraisednumber.Thatis,xx=x2,xxx=x3,xxxx=x4,andsoon,whereasusualinalgebraplacingtwolettersnexttoeachothermeansyoushouldmultiplythemtogether.So,for

instance,104=10×10×10×10=10,000.Youdon’tneedtoplayaroundwithsuchexpressionsforlongbeforeyoudiscoveraneasywaytoworkout,say,104×103.Justwritedown

Thenumberof0sintheansweris7,whichequals4+3.Thefirststepinthecalculationshowswhyitis4+3:westickfour10sandthree10snexttoeachother.Inshort,

104×103=104+3=107

Inthesameway,whateverthevalueofxmightbe,ifwemultiplyitsathpowerbyitsbthpower,whereaandbarewholenumbers,thenwegetthe(a+b)thpower:

xaxb=xa+b

Thismayseemaninnocuousformula,butontheleftitmultipliestwoquantitiestogether,whileontherightthemainstepistoaddaandb,whichissimpler.

Supposeyouwantedtomultiply,say,2.67by3.51.Bylongmultiplicationyouget9.3717,whichtotwodecimalplacesis9.37.Whatifyoutrytousethepreviousformula?Thetrickliesinthechoiceofx.Ifwetakextobe1.001,thenabitofarithmeticrevealsthat

(1.001)983=2.67(1.001)1256=3.51

correcttotwodecimalplaces.Theformulathentellsusthat2.87×3.41is

(1.001)983+1256=(1.001)2239

which,totwodecimalplaces,is9.37.Thecoreofthecalculationisaneasyaddition:983+1256=2239.However,

ifyoutrytocheckmyarithmeticyouwillquicklyrealisethatifanythingI’vemadetheproblemharder,noteasier.Toworkout(1.001)983youhavetomultiply1.001byitself983times.Andtodiscoverthat983istherightpowerto

use,youhavetodoevenmorework.Soatfirstsightthisseemslikeaprettyuselessidea.

Napier’sgreatinsightwasthatthisobjectioniswrong.Buttoovercomeit,somehardysoulhastocalculatelotsofpowersof1.001,startingwith(1.001)2andgoinguptosomethinglike(1.001)10,000.Thentheycanpublishatableofallthesepowers.Afterthat,mostoftheworkhasbeendone.Youjusthavetorunyourfingersdownthesuccessivepowersuntilyousee2.67nextto983;yousimilarlylocate3.51nextto1256.Thenyouaddthosetwonumberstoget2239.Thecorrespondingrowofthetabletellsyouthatthispowerof1.001is9.37.Jobdone.

Reallyaccurateresultsrequirepowersofsomethingalotcloserto1,suchas1.000001.Thismakesthetablefarbigger,withamillionorsopowers.Doingthecalculationsforthattableisahugeenterprise.Butithastobedoneonlyonce.Ifsomeself-sacrificingbenefactormakestheeffortupfront,succeedinggenerationswillbesavedagiganticamountofarithmetic.

Inthecontextofthisexample,wecansaythatthepowers983and1256arethelogarithmsofthenumbers2.67and3.51thatwewishtomultiply.Similarly2239isthelogarithmoftheirproduct9.38.Writinglogasanabbreviation,whatwehavedoneamountstotheequation

logab=loga+logb

whichisvalidforanynumbersaandb.Theratherarbitrarychoiceof1.001iscalledthebase.Ifweuseadifferentbase,thelogarithmsthatwecalculatearealsodifferent,butforanyfixedbaseeverythingworksthesameway.

ThisiswhatNapiershouldhavedone.Butforreasonsthatwecanonlyguessat,hedidsomethingslightlydifferent.Briggs,approachingthetechniquefromafreshperspective,spottedtwowaystoimproveonNapier’sidea.

WhenNapierstartedthinkingaboutpowersofnumbers,inthelatesixteenthcentury,theideaofreducingmultiplicationtoadditionwasalreadycirculatingamongmathematicians.Arathercomplicatedmethodknownas‘prosthapheiresis’,basedonaformulainvolvingtrigonometricfunctions,wasinuseinDenmark.3Napier,intrigued,wassmartenoughtorealisethatpowersofafixednumbercoulddothesamejobmoresimply.Thenecessarytablesdidn’texist–butthatwaseasilyremedied.Somepublic-spiritedsoulmustcarryoutthework.Napiervolunteeredhimselfforthetask,buthemadeastrategicerror.

Insteadofusingabasethatwasslightlybiggerthan1,heusedabaseslightlysmallerthan1.Inconsequence,thesequenceofpowersstartedoutwithbignumbers,whichgotsuccessivelysmaller.Thismadethecalculationsslightlymoreclumsy.

Briggsspottedthisproblem,andsawhowtodealwithit:useabaseslightlylargerthan1.Healsospottedasubtlerproblem,anddealtwiththataswell.IfNapier’smethodweremodifiedtoworkwithpowersofsomethinglike1.0000000001,therewouldbenostraightforwardrelationbetweenthelogarithmsof,say,12.3456and1.23456.Soitwasn’tentirelyclearwhenthetablecouldstop.Thesourceoftheproblemwasthevalueoflog10,because

log10x=log10+logx

Unfortunatelylog10wasmessy:withthebase1.0000000001thelogarithmof10was23,025,850,929.Briggsthoughtitwouldbemuchnicerifthebasecouldbechosensothatlog10=1.Thenlog10x=1+logx,sothatwhateverlog1.23456mightbe,youjusthadtoadd1toittogetlog12.3456.Nowtablesoflogarithmsneedonlyrunfrom1to10.Ifbiggernumbersturnedup,youjustaddedtheappropriatewholenumber.

Tomakelog10=1,youdowhatNapierdid,usingabaseof1.0000000001,butthenyoudivideeverylogarithmbythatcuriousnumber23,025,850,929.Theresultingtableconsistsoflogarithmstobase10,whichI’llwriteaslog10x.Theysatisfy

log10xy=log10x+log10y

asbefore,butalso

log1010x=log10x+1

WithintwoyearsNapierwasdead,soBriggsstartedworkonatableofbase-10logarithms.In1617hepublishedLogarithmorumChiliasPrima(‘LogarithmsoftheFirstChiliad’),thelogarithmsoftheintegersfrom1to1000accurateto14decimalplaces.In1624hefolloweditupwithArithmeticLogarithmica(‘ArithmeticofLogarithms’),atableofbase-10logarithmsofnumbersfrom1to20,000andfrom90,000to100,000,tothesameaccuracy.OthersrapidlyfollowedBriggs’slead,fillinginthelargegapanddevelopingauxiliarytablessuchaslogarithmsoftrigonometricfunctionslikelogsinx.

Thesameideasthatinspiredlogarithmsallowustodefinepowersxaofapositivevariablexforvaluesofathatarenotpositivewholenumbers.Allwehavetodoisinsistthatourdefinitionsmustbeconsistentwiththeequationxaxb=xa+b,andfollowournoses.Toavoidnastycomplications,itisbesttoassumexispositive,andtodefinexasothatthisisalsopositive.(Fornegativex,it’sbesttointroducecomplexnumbers,asinChapter5.)

Forexample,whatisx0?Bearinginmindthatx1=x,theformulasaysthatx0mustsatisfyx0x=x0+1=x.Dividingbyxwefindthatx0=1.Nowwhataboutx-1?Well,theformulasaysthatx−1x=x−1+1=x0=1.Dividingbyx,wegetx−1=1/x.Similarlyx−2=1/x2,x−3=1/x3,andsoon.

Itstartstogetmoreinteresting,andpotentiallyveryuseful,whenwethinkaboutx1/2.Thishastosatisfyx1/2x1/2=x1/2+1/2=x1=x.Sox1/2,multipliedbyitself,isx.Theonlynumberwiththispropertyisthesquarerootofx.Sox1/2=

.Similarly,x1/3= ,thecuberoot.Continuinginthismannerwecandefinexp/qforanyfractionp/q.Then,usingfractionstoapproximaterealnumbers,wecandefinexaforanyreala.Andtheequationxaxb=xa+bstillholds.

Italsofollowsthatlog logxandlog logx,sowecancalculatesquarerootsandcuberootseasilyusingatableoflogarithms.Forexample,tofindthesquarerootofanumberweformitslogarithm,divideby2,andthenworkoutwhichnumberhastheresultasitslogarithm.Forcuberoots,dothesamebutdivideby3.Traditionalmethodsfortheseproblemsweretediousandcomplicated.YoucanseewhyNapiershowcasedsquareandcuberootsintheprefacetohisbook.

Assoonascompletetablesoflogarithmswereavailable,theybecameanindispensabletoolforscientists,engineers,surveyors,andnavigators.Theysavedtime,theysavedeffort,andtheyincreasedthelikelihoodthattheanswerwascorrect.Earlyon,astronomywasamajorbeneficiary,becauseastronomersroutinelyneededtoperformlonganddifficultcalculations.TheFrenchmathematicianandastronomerPierreSimondeLaplacesaidthattheinventionoflogarithms‘reducestoafewdaysthelabourofmanymonths,doublesthelifeoftheastronomer,andspareshimtheerrorsanddisgust’.Astheuseofmachineryinmanufacturinggrew,engineersstartedtomakemoreandmoreuseofmathematics–todesigncomplexgears,analysethestabilityofbridgesandbuildings,andconstructcars,lorries,ships,andaeroplanes.Logarithmswereafirmpartoftheschoolmathematicscurriculumafewdecadesago.And

engineerscarriedwhatwasineffectananaloguecalculatorforlogarithmsintheirpockets,aphysicalrepresentationofthebasicequationforlogarithmsforon-the-spotuse.Theycalleditasliderule,andtheyuseditroutinelyinapplicationsrangingfromarchitecturetoaircraftdesign.

ThefirstsliderulewasconstructedbyanEnglishmathematician,WilliamOughtred,in1630,usingcircularscales.Hemodifiedthedesignin1632,bymakingthetworulersstraight.Thiswasthefirstsliderule.Theideaissimple:whenyouplacetworodsendtoend,theirlengthsadd.Iftherodsaremarkedusingalogarithmicscale,inwhichnumbersarespacedaccordingtotheirlogarithms,thenthecorrespondingnumbersmultiply.Forinstance,setthe1ononerodagainstthe2ontheother.Thenagainstanynumberxonthefirstrod,wefind2xonthesecond.Soopposite3wefind6,andsoon,seeFigure11.Ifthenumbersaremorecomplicated,say2.67and3.51,weplace1opposite2.67andreadoffwhateverisopposite3.59,namely9.37.It’sjustaseasy.

Fig11Multiplying2by3onasliderule.

Engineersquicklydevelopedfancyslideruleswithtrigonometricfunctions,squareroots,log-logscales(logarithmsoflogarithms)tocalculatepowers,andsoon.Eventuallylogarithmstookabackseattodigitalcomputers,butevennowthelogarithmstillplaysahugeroleinscienceandtechnology,alongsideitsinseparablecompanion,theexponentialfunction.Forbase-10logarithms,thisisthefunction10x;fornaturallogarithms,thefunctionex,wheree=2.71828,approximately.Ineachpair,thetwofunctionsareinversetoeachother.Ifyoutakeanumber,formitslogarithm,andthenformtheexponentialofthat,yougetbackthenumberyoustartedwith.

Whydoweneedlogarithmsnowthatwehavecomputers?In2011amagnitude9.0earthquakejustofftheeastcoastofJapancauseda

gigantictsunami,whichdevastatedalargepopulatedareaandkilledaround25,000people.Onthecoastwasanuclearpowerplant,FukushimaDai-ichi(Fukushimanumber1powerplant,todistinguishitfromasecondnuclearpower

plantsituatednearby).Itcomprisedsixseparatenuclearreactors:threewereinoperationwhenthetsunamistruck;theotherthreehadtemporarilyceasedoperatingandtheirfuelhadbeentransferredtopoolsofwateroutsidethereactorsbutinsidethereactorbuildings.

Thetsunamioverwhelmedtheplant’sdefences,cuttingthesupplyofelectricalpower.Thethreeoperatingreactors(numbers1,2,and3)wereshutdownasasafetymeasure,buttheircoolingsystemswerestillneededtostopthefuelfrommelting.However,thetsunamialsowreckedtheemergencygenerators,whichwereintendedtopowerthecoolingsystemandothersafety-criticalsystems.Thenextlevelofbackup,batteries,quicklyranoutofpower.Thecoolingsystemstoppedandthenuclearfuelinseveralreactorsbegantooverheat.Improvising,theoperatorsusedfireenginestopumpseawaterintothethreeoperatingreactors,butthisreactedwiththezirconiumcladdingonthefuelrodstoproducehydrogen.Thebuild-upofhydrogencausedanexplosioninthebuildinghousingReactor1.Reactors2and3soonsufferedthesamefate.ThewaterinthepoolofReactor4drainedout,leavingitsfuelexposed.Bythetimetheoperatorsregainedsomesemblanceofcontrol,atleastonereactorcontainmentvesselhadcracked,andradiationwasleakingoutintothelocalenvironment.TheJapaneseauthoritiesevacuated200,000peoplefromthesurroundingareabecausetheradiationwaswellabovenormalsafetylimits.Sixmonthslater,thecompanyoperatingthereactors,TEPCO,statedthatthesituationremainedcriticalandmuchmoreworkwouldbeneededbeforethereactorscouldbeconsideredfullyundercontrol,butclaimedtheleakagehadbeenstopped.

Idon’twanttoanalysethemeritsorotherwiseofnuclearpowerhere,butIdowanttoshowhowthelogarithmanswersavitalquestion:ifyouknowhowmuchradioactivematerialhasbeenreleased,andofwhatkind,howlongwillitremainintheenvironment,whereitcouldbehazardous?

Radioactiveelementsdecay;thatis,theyturnintootherelementsthroughnuclearprocesses,emittingnuclearparticlesastheydoso.Itistheseparticlesthatconstitutetheradiation.Thelevelofradioactivityfallsawayovertimejustasthetemperatureofahotbodyfallswhenitcools:exponentially.So,inappropriateunits,whichIwon’tdiscusshere,thelevelofradioactivityN(t)attimetfollowstheequation

N(t)=N0e-kt

whereN0istheinitiallevelandkisaconstantdependingontheelementconcerned.Moreprecisely,itdependsonwhichform,orisotope,oftheelementweareconsidering.

Aconvenientmeasureofthetimeradioactivitypersistsisthehalf-life,aconceptfirstintroducedin1907.ThisisthetimeittakesforaninitiallevelN0todroptohalfthatsize.Tocalculatethehalf-life,wesolvetheequation

bytakinglogarithmsofbothsides.Theresultis

andwecanworkthisoutbecausekisknownfromexperiments.Thehalf-lifeisaconvenientwaytoassesshowlongtheradiationwillpersist.

Supposethatthehalf-lifeisoneweek,forinstance.Thentheoriginalrateatwhichthematerialemitsradiationhalvesafter1week,isdowntoonequarterafter2weeks,oneeighthafter3weeks,andsoon.Ittakes10weekstodroptoonethousandthofitsoriginallevel(actually1/1024),and20weekstodroptoonemillionth.

Inaccidentswithconventionalnuclearreactors,themostimportantradioactiveproductsareiodine-131(aradioactiveisotopeofiodine)andcaesium-137(aradioactiveisotopeofcaesium).Thefirstcancausethyroidcancer,becausethethyroidglandconcentratesiodine.Thehalf-lifeofiodine-131isonly8days,soitcauseslittledamageiftherightmedicationisavailable,anditsdangersdecreasefairlyrapidlyunlessitcontinuestoleak.Thestandardtreatmentistogivepeopleiodinetablets,whichreducetheamountoftheradioactiveformthatistakenupbythebody,butthemosteffectiveremedyistostopdrinkingcontaminatedmilk.

Caesium-137isverydifferent:ithasahalf-lifeof30years.Ittakesabout200yearsforthelevelofradioactivitytodroptoonehundredthofitsinitialvalue,soitremainsahazardforaverylongtime.Themainpracticalissueinareactoraccidentiscontaminationofsoilandbuildings.Decontaminationistosomeextentfeasible,butexpensive.Forexample,thesoilcanberemoved,cartedaway,andstoredsomewheresafe.Butthiscreateshugeamountsoflow-levelradioactivewaste.

RadioactivedecayisjustoneareaofmanyinwhichNapier’sandBriggs’s

logarithmscontinuetoservescienceandhumanity.Ifyouthumbthroughlaterchaptersyouwillfindthempoppingupinthermodynamicsandinformationtheory,forexample.Eventhoughfastcomputershavenowmadelogarithmsredundantfortheiroriginalpurpose,rapidcalculations,theyremaincentraltoscienceforconceptualratherthancomputationalreasons.

Anotherapplicationoflogarithmscomesfromstudiesofhumanperception:howwesensetheworldaroundus.Theearlypioneersofthepsychophysicsofperceptionmadeextensivestudiesofvision,hearing,andtouch,andtheyturnedupsomeintriguingmathematicalregularities.

Inthe1840saGermandoctor,ErnstWeber,carriedoutexperimentstodeterminehowsensitivehumanperceptionis.Hissubjectsweregivenweightstoholdintheirhands,andaskedwhentheycouldtellthatoneweightfeltheavierthananother.Webercouldthenworkoutwhatthesmallestdetectabledifferenceinweightwas.Perhapssurprisingly,thisdifference(foragivenexperimentalsubject)wasnotafixedamount.Itdependedonhowheavytheweightsbeingcomparedwere.Peopledidn’tsenseanabsoluteminimumdifference–50grams,say.Theysensedarelativeminimumdifference–1%oftheweightsundercomparison,say.Thatis,thesmallestdifferencethatthehumansensescandetectisproportionaltothestimulus,theactualphysicalquantity.

Inthe1850sGustavFechnerrediscoveredthesamelaw,andrecastitmathematically.Thisledhimtoanequation,whichhecalledWeber’slaw,butnowadaysitisusuallycalledFechner’slaw(ortheWeber–Fechnerlawifyou’reapurist).Itstatesthattheperceivedsensationisproportionaltothelogarithmofthestimulus.Experimentssuggestedthatthislawappliesnotonlytooursenseofweightbuttovisionandhearingaswell.Ifwelookatalight,thebrightnessthatweperceivevariesasthelogarithmoftheactualenergyoutput.Ifonesourceistentimesasbrightasanother,thenthedifferenceweperceiveisconstant,howeverbrightthetwosourcesreallyare.Thesamegoesfortheloudnessofsounds:abangwithtentimesasmuchenergysoundsafixedamountlouder.

TheWeber–Fechnerlawisnottotallyaccurate,butit’sagoodapproximation.Evolutionprettymuchhadtocomeupwithsomethinglikealogarithmicscale,becausetheexternalworldpresentsoursenseswithstimulioverahugerangeofsizes.Anoisemaybelittlemorethanamousescuttlinginthehedgerow,oritmaybeaclapofthunder;weneedtobeabletohearboth.Buttherangeofsoundlevelsissovastthatnobiologicalsensorydevicecanrespondinproportiontotheenergygeneratedbythesound.Ifanearthatcouldhearthemousedidthat,thenathunderclapwoulddestroyit.Ifittunedthesoundlevels

downsothatthethunderclapproducedacomfortablesignal,itwouldn’tbeabletohearthemouse.Thesolutionistocompresstheenergylevelsintoacomfortablerange,andthelogarithmdoesexactlythat.Beingsensitivetoproportionsratherthanabsolutesmakesexcellentsense,andmakesforexcellentsenses.

Ourstandardunitfornoise,thedecibel,encapsulatestheWeber–Fechnerlawinadefinition.Itmeasuresnotabsolutenoise,butrelativenoise.Amouseinthegrassproducesabout10decibels.Normalconversationbetweenpeopleametreaparttakesplaceat40–60decibels.Anelectricmixerdirectsabout60decibelsatthepersonusingit.Thenoiseinacar,causedbyengineandtyres,is60–80decibels.Ajetairlinerahundredmetresawayproduces110–140decibels,risingto150atthirtymetres.Avuvuzela(theannoyingplastictrumpet-likeinstrumentwidelyheardduringthefootballWorldCupin2010andbroughthomeassouvenirsbymisguidedfans)generates120decibelsatonemetre;amilitarystungrenadeproducesupto180decibels.

Scaleslikethesearewidelyencounteredbecausetheyhaveasafetyaspect.Thelevelatwhichsoundcanpotentiallycausehearingdamageisabout120decibels.Pleasethrowawayyourvuvuzela.

3Ghostsofdepartedquantities

Calculus

Whatdoesitsay?

Tofindtheinstantaneousrateofchangeofaquantitythatvarieswith(say)time,calculatehowitsvaluechangesoverashorttimeintervalanddividebythetimeconcerned.Thenletthatintervalbecomearbitrarilysmall.

Whyisthatimportant?

Itprovidesarigorousbasisforcalculus,themainwayscientistsmodelthenaturalworld.

Whatdiditleadto?

Calculationoftangentsandareas.Formulasforvolumesofsolidsandlengthsofcurves.Newton’slawsofmotion,differentialequations.Thelawsofconservationofenergyandmomentum.Mostofmathematicalphysics.

In1665CharlesIIwaskingofEnglandandhiscapitalcity,London,wasasprawlingmetropolisofhalfamillionpeople.Theartsflourished,andsciencewasintheearlystagesofanever-acceleratingascendancy.TheRoyalSociety,perhapstheoldestscientificsocietynowinexistence,hadbeenfoundedfiveyearsearlier,andCharleshadgranteditaroyalcharter.Therichlivedinimpressivehouses,andcommercewasthriving,butthepoorwerecrammedintonarrowstreetsovershadowedbyramshacklebuildingsthatjuttedouteverfurtherastheyrose,storeybystorey.Sanitationwasinadequate;ratsandotherverminwereeverywhere.Bytheendof1666,onefifthofLondon’spopulationhadbeenkilledbybubonicplague,spreadfirstbyratsandthenbypeople.Itwastheworstdisasterinthecapital’shistory,andthesametragedyplayedoutalloverEuropeandNorthAfrica.ThekingdepartedinhasteforthemoresanitarycountrysideofOxfordshire,returningearlyin1666.Nooneknewwhatcausedplague,andthecityauthoritiestriedeverything–burningfirescontinuallytocleansetheair,burninganythingthatgaveoffastrongsmell,buryingthedeadquicklyinpits.Theykilledmanydogsandcats,whichironicallyremovedtwocontrolsontheratpopulation.

Duringthosetwoyears,anobscureandunassumingundergraduateatTrinityCollege,Cambridge,completedhisstudies.Hopingtoavoidtheplague,hereturnedtothehouseofhisbirth,fromwhichhismothermanagedafarm.Hisfatherhaddiedshortlybeforehewasborn,andhehadbeenbroughtupbyhismaternalgrandmother.Perhapsinspiredbyruralpeaceandquiet,orlackinganythingbettertodowithhistime,theyoungmanthoughtaboutscienceandmathematics.Laterhewrote:‘InthosedaysIwasintheprimeofmylifeforinvention,andmindedmathematicsand[natural]philosophymorethanatanyothertimesince.’Hisresearchesledhimtounderstandtheimportanceoftheinversesquarelawofgravity,anideathathadbeenhangingaroundineffectuallyforatleast50years.Heworkedoutapracticalmethodforsolvingproblemsincalculus,anotherconceptthatwasintheairbuthadnotbeenformulatedinanygenerality.Andhediscoveredthatwhitesunlightiscomposedofmanydifferentcolours–allthecoloursoftherainbow.

Whentheplaguedieddown,hetoldnooneaboutthediscoverieshehadmade.HereturnedtoCambridge,tookamaster’sdegree,andbecameafellowatTrinity.ElectedtotheLucasianChairofMathematics,hefinallybegantopublishhisideasandtodevelopnewones.

TheyoungmanwasIsaacNewton.Hisdiscoveriescreatedarevolutioninscience,bringingaboutaworldthatCharlesIIwouldneverhavebelievedcouldexist:buildingswithmorethanahundredfloors,horselesscarriagesdoing80mphalongtheM6motorwaywhilethedriverlistenstomusicusingamagicdiscmadefromastrangeglasslikematerial,heavier-than-airflyingmachinesthatcrosstheAtlanticinsixhours,colourpicturesthatmove,andboxesyoucarryinyourpocketthattalktotheothersideoftheworld…

Previously,GalileoGalilei,JohannesKepler,andothershadturnedupthecornerofnature’srugandseenafewofthewondersconcealedbeneathit.NowNewtoncasttherugaside.Notonlydidherevealthattheuniversehassecretpatterns,lawsofnature;healsoprovidedmathematicaltoolstoexpressthoselawspreciselyandtodeducetheirconsequences.Thesystemoftheworldwasmathematical;theheartofGod’screationwasasoullessclockworkuniverse.

Theworldviewofhumanitydidnotsuddenlyswitchfromreligioustosecular.Itstillhasnotdonesocompletely,andprobablyneverwill.ButafterNewtonpublishedhisPhilosophiæNaturalisPrincipiaMathematica(‘MathematicalPrinciplesofNaturalPhilosophy’)the‘SystemoftheWorld’–thebook’ssubtitle–wasnolongersolelytheprovinceoforganisedreligion.Evenso,Newtonwasnotthefirstmodernscientist;hehadamysticalsidetoo,devotingyearsofhislifetoalchemyandreligiousspeculation.Innotesforalecture1theeconomistJohnMaynardKeynes,alsoaNewtonianscholar,wrote:

Newtonwasnotthefirstoftheageofreason.Hewasthelastofthemagicians,thelastoftheBabyloniansandSumerians,thelastgreatmindwhichlookedoutonthevisibleandintellectualworldwiththesameeyesasthosewhobegantobuildourintellectualinheritanceratherlessthan10,000yearsago.IsaacNewton,aposthumouschildbornwithnofatheronChristmasDay,1642,wasthelastwonderchildtowhomtheMagicoulddosincereandappropriatehomage.

TodaywemostlyignoreNewton’smysticaspect,andrememberhimforhisscientificandmathematicalachievements.Paramountamongthemarehisrealisationthatnatureobeysmathematicallawsandhisinventionofcalculus,themainwaywenowexpressthoselawsandderivetheirconsequences.TheGermanmathematicianandphilosopherGottfriedWilhelmLeibnizalsodevelopedcalculus,moreorlessindependently,atmuchthesametime,buthedidlittlewithit.Newtonusedcalculustounderstandtheuniverse,thoughhe

keptitunderwrapsinhispublishedwork,recastingitinclassicalgeometriclanguage.Hewasatransitionalfigurewhomovedhumanityawayfromamystical,medievaloutlookandusheredinthemodernrationalworldview.AfterNewton,scientistsconsciouslyrecognisedthattheuniversehasdeepmathematicalpatterns,andwereequippedwithpowerfultechniquestoexploitthatinsight.

Thecalculusdidnotarise‘outoftheblue’.Itcamefromquestionsinbothpureandappliedmathematics,anditsantecedentscanbetracedbacktoArchimedes.Newtonhimselffamouslyremarked,‘IfIhaveseenalittlefurtheritisbystandingontheshouldersofgiants.’2ParamountamongthosegiantswereJohnWallis,PierredeFermat,Galileo,andKepler.Wallisdevelopedaprecursortocalculusinhis1656ArithmeticaInfinitorum(‘ArithmeticoftheInfinite’).Fermat’s1679DeTangentibusLinearumCurvarum(‘OnTangentstoCurvedLines’)presentedamethodforfindingtangentstocurves,aproblemintimatelyrelatedtocalculus.Keplerformulatedthreebasiclawsofplanetarymotion,whichledNewtontohislawofgravity,thesubjectofthenextchapter.Galileomadebigadvancesinastronomy,buthealsoinvestigatedmathematicalaspectsofnaturedownontheground,publishinghisdiscoveriesinDeMotu(‘OnMotion’)in1590.Heinvestigatedhowafallingbodymoves,findinganelegantmathematicalpattern.Newtondevelopedthishintintothreegenerallawsofmotion.

TounderstandGalileo’spatternweneedtwoeverydayconceptsfrommechanics:velocityandacceleration.Velocityishowfastsomethingismoving,andinwhichdirection.Ifweignorethedirection,wegetthebody’sspeed.Accelerationisachangeinvelocity,whichusuallyinvolvesachangeinspeed(anexceptionariseswhenthespeedremainsthesamebutthedirectionchanges).Ineverydaylifeweuseaccelerationtomeanspeedingupanddecelerationforslowingdown,butinmechanicsbothchangesareaccelerations:thefirstpositive,thesecondnegative.Whenwedrivealongaroadthespeedofthecarisdisplayedonthespeedometer–itmight,forinstance,be50mph.Thedirectioniswhicheverwaythecarispointing.Whenweputourfootdown,thecaracceleratesandthespeedincreases;whenwestamponthebrakes,thecardecelerates–negativeacceleration.

Ifthecarismovingatafixedspeed,it’seasytoworkoutwhatthatspeedis.Theabbreviationmphgivesitaway:milesperhour.Ifthecartravels50milesin1hour,wedividethedistancebythetime,andthat’sthespeed.Wedon’tneedtodriveforanhour:ifthecargoes5milesin6minutes,bothdistanceandtime

aredividedby10,andtheirratioisstill50mph.Inshort,

speed=distancetravelleddividedbytimetaken.

Inthesameway,afixedrateofaccelerationisgivenby

acceleration=changeinspeeddividedbytimetaken.

Thisallseemsstraightforward,butconceptualdifficultiesarisewhenthespeedoraccelerationisnotfixed.Andtheycan’tbothbeconstant,becauseconstant(andnonzero)accelerationimpliesachangingspeed.Supposeyoudrivealongacountrylane,speedinguponthestraights,slowingforthecorners.Yourspeedkeepschanging,andsodoesyouracceleration.Howcanweworkthemoutatanygiveninstantoftime?Thepragmaticansweristotakeashortintervaloftime,sayasecond.Thenyourinstantaneousspeedat(say)11.30amisthedistanceyoutravelbetweenthatmomentandonesecondlater,dividedbyonesecond.Thesamegoesforinstantaneousacceleration.

Except…that’snotquiteyourinstantaneousspeed.It’sreallyanaveragespeed,overaone-secondintervaloftime.Therearecircumstancesinwhichonesecondisahugelengthoftime–aguitarstringplayingmiddleCvibrates440timeseverysecond;averageitsmotionoveranentiresecondandyou’llthinkit’sstandingstill.Theansweristoconsiderashorterintervaloftime–onetenthousandthofasecond,perhaps.Butthisstilldoesn’tcaptureinstantaneousspeed.Visiblelightvibratesonequadrillion(1015)timeseverysecond,sotheappropriatetimeintervalislessthanonequadrillionthofasecond.Andeventhen…well,tobepedantic,that’sstillnotaninstant.Pursuingthislineofthought,itseemstobenecessarytouseanintervaloftimethatisshorterthananyotherinterval.Buttheonlynumberlikethatis0,andthat’suseless,becausenowthedistancetravelledisalso0,and0/0ismeaningless.

Earlypioneersignoredtheseissuesandtookapragmaticview.Oncetheprobableerrorinyourmeasurementsexceedstheincreasedprecisionyouwouldtheoreticallygetbyusingsmallerintervalsoftime,there’snopointindoingso.TheclocksinGalileo’sdaywereveryinaccurate,sohemeasuredtimebyhummingtunestohimself–atrainedmusiciancansubdivideanoteintoveryshortintervals.Eventhen,timingafallingbodyistricky,soGalileohitonthetrickofslowingthemotiondownbyrollingballsdownaninclinedslope.Thenheobservedthepositionoftheballatsuccessiveintervalsoftime.Whathefound(I’msimplifyingthenumberstomakethepatternclear,butit’sthesame

pattern)isthatfortimes0,1,2,3,4,5,6,…thesepositionswere

0149162536

Thedistancewas(proportionalto)thesquareofthetime.Whataboutthespeeds?Averagedoversuccessiveintervals,thesewerethedifferences

1357911

betweenthesuccessivesquares.Ineachinterval,otherthanthefirst,theaveragespeedincreasedby2units.It’sastrikingpattern–allthemoresotoGalileowhenhedugsomethingverysimilaroutofdozensofmeasurementswithballsofmanydifferentmassesonslopeswithmanydifferentinclinations.

Fromtheseexperimentsandtheobservedpattern,Galileodeducedsomethingwonderful.Thepathofafallingbody,oronethrownintotheair,suchasacannonball,isaparabola.ThisisaU-shapedcurve,knowntotheancientGreeks.(TheUisupsidedowninthiscase.I’mignoringairresistance,whichchangestheshape:itdidn’thavemucheffectonGalileo’srollingballs.)Keplerencounteredarelatedcurve,theellipse,inhisanalysisofplanetaryorbits:thismusthaveseemedsignificanttoNewtontoo,butthatstorymustwaituntilthenextchapter.

Withonlythisparticularseriesofexperimentstogoon,it’snotclearwhatgeneralprinciplesunderlieGalileo’spattern.Newtonrealisedthatthesourceofthepatternisratesofchange.Velocityistherateatwhichpositionchangeswithrespecttotime;accelerationistherateatwhichvelocitychangeswithrespecttotime.InGalileo’sobservations,positionvariedaccordingtothesquareoftime,velocityvariedlinearly,andaccelerationdidn’tvaryatall.NewtonrealisedthatinordertogainadeeperunderstandingofGalileo’spatterns,andwhattheymeantforourviewofnature,hehadtocometogripswithinstantaneousratesofchange.Whenhedid,outpoppedcalculus.

Youmightexpectanideaasimportantascalculustobeannouncedwithafanfareoftrumpetsandparadesthroughthestreets.However,ittakestimeforthesignificanceofnovelideastosinkinandtobeappreciated,andsoitwaswithcalculus.Newton’sworkonthetopicdatesfrom1671orearlier,whenhewroteTheMethodofFluxionsandInfiniteSeries.Weareunsureofthedatebecausethebookwasnotpublisheduntil1736,nearlyadecadeafterhisdeath.SeveralothermanuscriptsbyNewtonalsorefertoideasthatwenowrecogniseasdifferentialandintegralcalculus,thetwomainbranchesofthesubject.

Leibniz’snotebooksshowthatheobtainedhisfirstsignificantresultsincalculusin1675,buthepublishednothingonthetopicuntil1684.

AfterNewtonhadrisentoscientificprominence,longafterbothmenhadworkedoutthebasicsofcalculus,someofNewton’sfriendssparkedalargelypointlessbutheatedcontroversyaboutpriority,accusingLeibnizofplagiarisingNewton’sunpublishedmanuscripts.AfewmathematiciansfromcontinentalEuroperespondedwithcounter-claimsofplagiarismbyNewton.Englishandcontinentalmathematicianswerescarcelyonspeakingtermsforacentury,whichcausedhugedamagetoEnglishmathematicians,butnonewhatsoevertothecontinentalones.TheydevelopedcalculusintoacentraltoolofmathematicalphysicswhiletheirEnglishcounterpartswereseethingaboutinsultstoNewtoninsteadofexploitinginsightsfromNewton.Thestoryistangledandstillsubjecttoscholarlydisputationbyhistoriansofscience,butbroadlyspeakingitseemsthatNewtonandLeibnizdiscoveredthebasicideasofcalculusindependently–atleast,asindependentlyastheircommonmathematicalandscientificculturepermitted.

Leibniz’snotationdiffersfromNewton’s,buttheunderlyingideasaremoreorlessidentical.Theintuitionbehindthem,however,isdifferent.Leibniz’sapproachwasaformalone,manipulatingalgebraicsymbols.Newtonhadaphysicalmodelatthebackofhismind,inwhichthefunctionunderconsiderationwasaphysicalquantitythatvarieswithtime.Thisiswherehiscuriousterm‘fluxion’comesfrom–somethingthatflowsastimepasses.

Newton’smethodcanbeillustratedusinganexample:aquantityythatisthesquarex2ofanotherquantityx.(ThisisthepatternthatGalileofoundforarollingball:itspositionisproportionaltothesquareofthetimethathaselapsed.Sothereywouldbepositionandxtime.Theusualsymbolfortimeist,butthestandardcoordinatesystemintheplaneusesxandy.)Startbyintroducinganewquantityo,denotingasmallchangeinx.Thecorrespondingchangeinyisthedifference

(x+o)2−x2

whichsimplifiesto2xo+o2.Therateofchange(averagedoverasmallintervaloflengtho,asxincreasestox+o)istherefore

Thisdependsono,whichisonlytobeexpectedsinceweareaveragingtherateofchangeoveranonzerointerval.However,ifobecomessmallerandsmaller,‘flowingtowards’zero,therateofchange2x+ogetscloserandcloserto2x.Thisdoesnotdependono,anditgivestheinstantaneousrateofchangeatx.

Leibnizperformedessentiallythesamecalculation,replacingobydx(‘smalldifferenceinx’),anddefiningdytobethecorrespondingsmallchangeiny.Whenavariableydependsonanothervariablex,therateofchangeofywithrespecttoxiscalledthederivativeofy.Newtonwrotethederivativeofyby

placingadotaboveit:ẏLeibnizwrote .Forhigherderivatives,Newtonused

moredots,whileLeibnizwrotethingslike Todaywesaythatyisafunctionofxandwritey=f(x),butthisconceptexistedonlyinrudimentaryformatthetime.WeeitheruseLeibniz’snotation,oravariantofNewton’sinwhichthedotisreplacedbyadash,whichiseasiertoprint:y′,y″.Wealsowritef′(x)andf″(x)toemphasisethatthederivativesarethemselvesfunctions.Calculatingthederivativeiscalleddifferentiation.

Integralcalculus–findingareas–turnsouttobetheinverseofdifferentialcalculus–findingslopes.Toseewhy,imagineaddingathinsliceontheendoftheshadedareaofFigure12.Thissliceisveryclosetoalongthinrectangle,ofwidthoandheighty.Itsareaisthereforeveryclosetooy.Therateatwhichtheareachanges,withrespecttox,istheratiooy/o,whichequalsy.Sothederivativeoftheareaistheoriginalfunction.BothNewtonandLeibnizunderstoodthatthewaytocalculatethearea,aprocesscalledintegration,isthereverseofdifferentiationinthissense.Leibnizfirstwrotetheintegralusingthesymbolomn.,shortforomnia,or‘sum’,inLatin.Laterhechangedthisto∫,anold-fashionedlongs,alsostandingfor‘sum’.Newtonhadnosystematicnotationfortheintegral.

Fig12Addingathinslicetotheareabeneaththecurvey=f(x).

Newtondidmakeonecrucialadvance,however.Wallishadcalculatedthederivativeofanypowerxa:itisaxa-1.Sothederivativesofx3,x4,x5are3x2,4x3,5x4,forexample.Hehadextendedthisresulttoanypolynomial–afinitecombinationofpowers,suchas3x7−25x4+x2−3.Thetrickistoconsidereachpowerseparately,findthecorrespondingderivatives,andcombinetheminthesamemanner.Newtonnoticedthatthesamemethodworkedforinfiniteseries,expressionsinvolvinginfinitelymanypowersofthevariable.Thislethimperformtheoperationsofcalculusonmanyotherexpressions,morecomplicatedthanpolynomials.

Giventheclosecorrespondencebetweenthetwoversionsofcalculus,differingmainlyinunimportantfeaturesofthenotation,itiseasytoseehowaprioritydisputemighthavearisen.However,thebasicideaisafairlydirectformulationoftheunderlyingquestion,soitisalsoeasytoseehowNewtonandLeibnizcouldhavearrivedattheirversionsindependently,despitethesimilarities.Inanycase,FermatandWallishadbeatenthembothtomanyoftheirresults.Thedisputewaspointless.

Amorefruitfulcontroversyconcernedthelogicalstructureofcalculus,ormoreprecisely,theillogicalstructureofcalculus.AleadingcriticwastheAnglo-IrishphilosopherGeorgeBerkeley,BishopofCloyne.Berkeleyhadareligiousagenda;hefeltthatthematerialistviewoftheworldthatwasdevelopingfromNewton’sworkrepresentedGodasadetachedcreatorwhostoodbackfromhiscreationassoonasitgotgoingandthereafterleftittoitsowndevices,quiteunlikethepersonal,immanentGodofChristianbelief.Soheattackedlogical

inconsistenciesinthefoundationsofcalculus,presumablyhopingtodiscredittheresultingscience.Hisattackhadnodiscernibleeffectontheprogressofmathematicalphysics,forastraightforwardreason:theresultsobtainedusingcalculusshedsomuchinsightintonature,andagreedsowellwithexperiment,thatthelogicalfoundationsseemedunimportant.Eventoday,physicistsstilltakethisview:ifitworks,whocaresaboutlogicalhair-splitting?

Berkeleyarguedthatitmakesnologicalsensetomaintainthatasmallquantity(Newton’so,Leibniz’sdx)isnonzeroformostofacalculation,andthentosetittozero,ifyouhavepreviouslydividedboththenumeratorandthedenominatorofafractionbythatveryquantity.Divisionbyzeroisnotanacceptableoperationinarithmetic,becauseithasnounambiguousmeaning.Forexample,0×1=0×2,sincebothare0,butifwedividebothsidesofthisequationby0weget1=2,whichisfalse.3Berkeleypublishedhiscriticismsin1734inapamphletTheAnalyst,aDiscourseAddressedtoanInfidelMathematician.

Newtonhad,infact,attemptedtosortoutthelogic,byappealingtoaphysicalanalogy.Hesawonotasafixedquantity,butassomethingthatflowed–variedwithtime–gettingcloserandclosertozerowithouteveractuallygettingthere.Thederivativewasalsodefinedbyaquantitythatflowed:theratioofthechangeinytothatofx.Thisratioalsoflowedtowardssomething,butnevergotthere;thatsomethingwastheinstantaneousrateofchange–thederivativeofywithrespecttox.Berkeleydismissedthisideaasthe‘ghostofadepartedquantity’.

Leibniztoohadapersistentcritic,thegeometerBernardNieuwentijt,whoputhiscriticismsintoprintin1694and1695.Leibnizhadnothelpedhiscasebytryingtojustifyhismethodintermsof‘infinitesimals’,atermopentomisinterpretation.However,hedidexplainthatwhathemeantbythistermwasnotafixednonzeroquantitythatcanbearbitrarilysmall(whichmakesnologicalsense)butavariablenonzeroquantitythatcanbecomearbitrarilysmall.Newton’sandLeibniz’sdefenceswereessentiallyidentical.Totheiropponents,bothmusthavesoundedlikeverbaltrickery.

Fortunately,thephysicistsandmathematiciansofthedaydidnotwaitforthelogicalfoundationsofcalculustobesortedoutbeforetheyappliedittothefrontiersofscience.Theyhadanalternativewaytomakesuretheyweredoingsomethingsensible:comparisonwithobservationsandexperiments.Newtonhimselfinventedcalculusforpreciselythispurpose.Hederivedlawsforhowbodiesmovewhenaforceisappliedtothem,andcombinedthesewithalawfortheforceexertedbygravitytoexplainmanyriddlesabouttheplanetsandother

bodiesoftheSolarSystem.Hislawofgravityissuchapivotalequationinphysicsandastronomythatitdeserves,andgets,achapterofitsown(thenextone).Hislawofmotion–strictly,asystemofthreelaws,oneofwhichcontainedmostofthemathematicalcontent–ledfairlydirectlytocalculus.

Ironically,whenNewtonpublishedtheselawsandtheirscientificapplicationsinhisPrincipia,heeliminatedalltracesofcalculusandreplaceditbyclassicalgeometricarguments.Heprobablythoughtthatgeometrywouldbemoreacceptabletohisintendedaudience,andifhedid,hewasalmostcertainlyright.However,manyofhisgeometricproofsareeithermotivatedbycalculus,ordependontheuseofcalculustechniquestodeterminethecorrectanswers,uponwhichthestrategyofthegeometricproofrelies.Thisisespeciallyclear,tomoderneyes,inhistreatmentofwhathecalled‘generatedquantities’inBookIIofPrincipia.Thesearequantitiesthatincreaseordecreaseby‘continualmotionorflux’,thefluxionsofhisunpublishedbook.Todaywewouldcallthemcontinuous(indeeddifferentiable)functions.Inplaceofexplicitoperationsofthecalculus,Newtonsubstitutedageometricmethodof‘primeandultimateratios’.Hisopeninglemma(thenamegiventoanauxiliarymathematicalresultthatisusedrepeatedlybuthasnointrinsicinterestinitsownright)givesthegameaway,becauseitdefinesequalityoftheseflowingquantitieslikethis:

Quantities,andtheratiosofquantities,whichinanyfinitetimeconvergecontinuallytoequality,andbeforetheendofthattimeapproachnearertoeachotherthanbyanygivendifference,becomeultimatelyequal.

InNeveratRest,Newton’sbiographerRichardWestfallexplainshowradicalandnovelthislemmawas:‘Whateverthelanguage,theconcept…wasthoroughlymodern;classicalgeometryhadcontainednothinglikeit.’4Newton’scontemporariesmusthavestruggledtofigureoutwhatNewtonwasgettingat.Berkeleypresumablyneverdid,because–aswewillshortlysee–itcontainsthebasicideaneededtodisposeofhisobjection.

Calculus,then,wasplayinganinfluentialrolebehindthescenesofthePrincipia,butitmadenoappearanceonstage.Assoonascalculuspeepedoutfrombehindthecurtains,however,Newton’sintellectualsuccessorsquicklyreverse-engineeredhisthoughtprocesses.Theyrephrasedhismainideasinthelanguageofcalculus,becausethisprovidedamorenaturalandmorepowerfulframework,andsetouttoconquerthescientificworld.

ThecluewasalreadyvisibleinNewton’slawsofmotion.ThequestionthatledNewtontotheselawswasaphilosophicalone:whatcausesabodytomove,ortochangeitsstateofmotion?TheclassicalanswerwasAristotle’s:abodymovesbecauseaforceisappliedtoit,andthisaffectsitsvelocity.Aristotlealsostatedthatinordertokeepabodymoving,theforcemustcontinuetobeapplied.YoucantestAristotle’sstatementsbyplacingabookorsimilarobjectonatable.Ifyoupushthebook,itstartstomove,andifyoukeeppushingwithmuchthesameforceitcontinuestoslideoverthetableataroughlyconstantvelocity.Ifyoustoppushing,thebookstopsmoving.SoAristotle’sviewsseemtoagreewithexperiment.However,theagreementissuperficial,becausethepushisnottheonlyforcethatactsonthebook.Thereisalsofrictionwiththesurfaceofthetable.Moreover,thefasterthebookmoves,thegreaterthefrictionbecomes–atleast,whilethebook’svelocityremainsreasonablysmall.Whenthebookismovingsteadilyacrossthetable,propelledbyasteadyforce,thefrictionalresistancecancelsouttheappliedforce,andthetotalforceactingonthebodyisactuallyzero.

Newton,followingearlierideasofGalileoandDescartes,realisedthis.TheresultingtheoryofmotionisverydifferentfromAristotle’s.Newton’sthreelawsare:

Firstlaw.Everybodycontinuesinitsstateofrest,orofuniformmotioninaright[straight]line,unlessitiscompelledtochangethatstatebyforcesimpresseduponit.Secondlaw.Thechangeofmotionisproportionaltothemotivepowerimpressed,andismadeinthedirectionoftherightlineinwhichthatforceisimpressed.(Theconstantofproportionalityisthereciprocalofthebody’smass;thatis,1dividedbythatmass.)Thirdlaw.Toeveryactionthereisalwaysopposedanequalreaction.

ThefirstlawexplicitlycontradictsAristotle.Thethirdlawsaysthatifyoupushsomething,itpushesback.Thesecondlawiswherecalculuscomesin.By‘changeofmotion’Newtonmeanttherateatwhichthebody’svelocitychanges:itsacceleration.Thisisthederivativeofvelocitywithrespecttotime,andthesecondderivativeofposition.SoNewton’ssecondlawofmotionspecifiestherelationbetweenabody’sposition,andtheforcesthatactonit,intheformofadifferentialequation:

secondderivativeofposition=force/mass

Tofindthepositionitself,wehavetosolvethisequation,deducingthepositionfromitssecondderivative.

ThislineofthoughtleadstoasimpleexplanationofGalileo’sobservationsofarollingball.Thecrucialpointisthattheaccelerationoftheballisconstant.Istatedthispreviously,usingarough-and-readycalculationappliedatdiscreteintervalsoftime;nowwecandoitproperly,allowingtimetovarycontinuously.Theconstantisrelatedtotheforceofgravityandtheangleoftheslope,butherewedon’tneedthatmuchdetail.Supposethattheconstantaccelerationisa.Integratingthecorrespondingfunction,thevelocitydowntheslopeattimetisat+b,wherebisthevelocityattimezero.Integratingagain,thepositiondowntheslopeis ,wherecisthepositionattimezero.Inthespecialcasea=2,b=0,c=0thesuccessivepositionsfitmysimplifiedexample:thepositionattimetist2.AsimilaranalysisrecoversGalileo’smajorresult:thepathofaprojectileisaparabola.

Newton’slawsofmotiondidnotjustprovideawaytocalculatehowbodiesmove.Theyledtodeepandgeneralphysicalprinciples.Paramountamongtheseare‘conservationlaws’,tellingusthatwhenasystemofbodies,nomatterhowcomplicated,moves,certainfeaturesofthatsystemdonotchange.Amidthetumultofthemotion,afewthingsremainserenelyunaffected.Threeoftheseconservedquantitiesareenergy,momentum,andangularmomentum.

Energycanbedefinedasthecapacitytodowork.Whenabodyisraisedtoacertainheight,againstthe(constant)forceofgravity,theworkdonetoputitthereisproportionaltothebody’smass,theforceofgravity,andtheheighttowhichitisraised.Conversely,ifwethenletthebodygo,itcanperformthesameamountofworkwhenitfallsbacktoitsoriginalheight.Thistypeofenergyiscalledpotentialenergy.

Onitsown,potentialenergywouldnotbeterriblyinteresting,butthereisabeautifulmathematicalconsequenceofNewton’ssecondlawofmotionleadingtoasecondkindofenergy:kineticenergy.Asabodymoves,bothitspotentialenergyanditskineticenergychange.Butthechangeinoneexactlycompensatesforthechangeintheother.Asthebodydescendsundergravity,itspeedsup.Newton’slawallowsustocalculatehowitsvelocitychangeswithheight.Itturnsoutthatthedecreaseinpotentialenergyisexactlyequaltohalfthemasstimesthesquareofthevelocity.Ifwegivethatquantityaname–kineticenergy–thenthetotalenergy,potentialpluskinetic,isconserved.ThismathematicalconsequenceofNewton’slawsprovesthatperpetualmotionmachinesareimpossible:nomechanicaldevicecankeepgoingindefinitelyanddowork

withoutsomeexternalinputofenergy.Physically,potentialandkineticenergyseemtobetwodifferentthings;

mathematically,wecantradeonefortheother.Itisasifmotionsomehowconvertspotentialenergyintokinetic.‘Energy’,asatermapplicabletoboth,isaconvenientabstraction,carefullydefinedsothatitisconserved.Asananalogy,travellerscanconvertpoundsintodollars.Currencyexchangeshavetablesofexchangerates,assertingthat,say,1poundisofequalvalueto1.4693dollars.Theyalsodeductasumofmoneyforthemselves.Subjecttotechnicalitiesofbankchargesandsoon,thetotalmonetaryvalueinvolvedinthetransactionissupposedtobalanceout:thetravellergetsexactlytheamountindollarsthatcorrespondstotheiroriginalsuminpounds,minusvariousdeductions.However,thereisn’taphysicalthingbuiltintobanknotesthatsomehowgetsswappedoutofapoundnoteintoadollarnoteandsomecoins.Whatgetsswappedisthehumanconventionthattheseparticularitemshavemonetaryvalue.

Energyisanewkindof‘physical’quantity.FromaNewtonianviewpoint,quantitiessuchasposition,time,velocity,acceleration,andmasshavedirectphysicalinterpretations.Youcanmeasurepositionwitharuler,timewithaclock,velocityandaccelerationusingbothpiecesofapparatus,andmasswithabalance.Butyoudon’tmeasureenergyusinganenergymeter.Agreed,youcanmeasurecertainspecifictypesofenergy.Potentialenergyisproportionaltoheight,soarulerwillsufficeifyouknowtheforceofgravity.Kineticenergyishalfthemasstimesthesquareofthevelocity:useabalanceandaspeedometer.Butenergy,asaconcept,isnotsomuchaphysicalthingasaconvenientfictionthathelpstobalancethemechanicalbooks.

Momentum,thesecondconservedquantity,isasimpleconcept:masstimesvelocity.Itcomesintoplaywhenthereareseveralbodies.Animportantexampleisarocket;hereonebodyistherocketandtheotherisitsfuel.Asfuelisexpelledbytheengine,conservationofmomentumimpliesthattherocketmustmoveintheoppositedirection.Thisishowarocketworksinavacuum.

Angularmomentumissimilar,butitrelatestospinratherthanvelocity.Itisalsocentraltorocketry,indeedthewholeofmechanics,terrestrialorcelestial.OneofthebiggestpuzzlesabouttheMoonisitslargeangularmomentum.ThecurrenttheoryisthattheMoonwassplashedoffwhenaMars-sizedplanethittheEarthabout4.5billionyearsago.Thisexplainstheangularmomentum,anduntilrecentlywasgenerallyaccepted,butitnowseemsthattheMoonhastoomuchwaterinitsrocks.Suchanimpactshouldhaveboiledalotofthewater

away.5Whatevertheeventualoutcome,angularmomentumisofcentralimportancehere.Calculusworks.Itsolvesproblemsinphysicsandgeometry,gettingtherightanswers.Itevenleadstonewandfundamentalphysicalconceptslikeenergyandmomentum.Butthatdoesn’tanswerBishopBerkeley’sobjection.Calculushastoworkasmathematics,notjustagreewithphysics.BothNewtonandLeibnizunderstoodthatoordxcannotbebothzeroandnonzero.Newtontiredtoescapefromthelogicaltrapbyemployingthephysicalimageofafluxion.Leibniztalkedofinfinitesimals.Bothreferredtoquantitiesthatapproachzerowithoutevergettingthere–butwhatarethesethings?Ironically,Berkeley’sgibeabout‘ghostsofdepartedquantities’comesclosetoresolvingtheissue,butwhathefailedtotakeaccountof–andwhatbothNewtonandLeibnizemphasised–washowthequantitiesdeparted.Makethemdepartintherightwayandyoucanleaveaperfectlywell-formedghost.IfeitherNewtonorLeibnizhadframedtheirintuitioninrigorousmathematicallanguage,Berkeleymighthaveunderstoodwhattheyweregettingat.

ThecentralquestionisonethatNewtonfailedtoanswerexplicitlybecauseitseemedobvious.Recallthatintheexamplewherey=x2,Newtonobtainedthederivativeas2x+o,andthenassertedthatasoflowstowardszero,2x+oflowstowards2x.Thismayseemobvious,butwecan’tseto=0toproveit.Itistruethatwegettherightresultbydoingthat,butthisisaredherring.6InPrincipiaNewtonslidroundthisissuealtogether,replacing2x+obyhis‘primeratio’and2xbyhis‘ultimateratio’.Buttherealkeytoprogressistotackletheissueheadon.Howdoweknowthatthecloseroapproacheszero,thecloser2x+oapproaches2x?Itmayseemaratherpedanticpoint,butifI’dusedmorecomplicatedexamplesthecorrectanswermightnotseemsoplausible.

Whenmathematiciansreturnedtothelogicofcalculus,theyrealisedthatthisapparentlysimplequestionwastheheartofthematter.Whenwesaythatoapproacheszero,wemeanthatgivenanynonzeropositivenumber,ocanbechosentobesmallerthanthatnumber.(Thisisobvious:letobehalfthatnumber,forinstance.)Similarly,whenwesaythat2x+oapproaches2x,wemeanthatthedifferenceapproacheszero,intheprevioussense.Sincethedifferencehappenstobeoitselfinthiscase,that’sevenmoreobvious:whatever‘approacheszero’means,clearlyoapproacheszerowhenoapproacheszero.Amorecomplicatedfunctionthanthesquarewouldrequireamorecomplicatedanalysis.

Theanswertothiskeyquestionistostatetheprocessinformalmathematicalterms,avoidingideasof‘flow’altogether.Thisbreakthroughcameabout

throughtheworkoftheBohemianmathematicianandtheologianBernardBolzanoandtheGermanmathematicianKarlWeierstrass.Bolzano’sworkdatesfrom1816,butitwasnotappreciateduntilabout1870whenWeierstrassextendedtheformulationtocomplexfunctions.TheiranswertoBerkeleywastheconceptofalimit.I’llstatethedefinitioninwordsandleavethesymbolicversiontotheNotes.7Saythatafunctionf(h)ofavariablehtendstoalimitLashtendstozeroif,givenanypositivenonzeronumber,thedifferencebetweenf(h)andLcanbemadesmallerthanthatnumberbychoosingsufficientlysmallnonzerovaluesofh.Insymbols,

Theideaattheheartofcalculusistoapproximatetherateofchangeofafunctionoverasmallintervalh,andthentakethelimitashtendstozero.Forageneralfunctiony=f(x)thisprocedureleadstotheequationthatdecoratestheopeningofthischapter,butusingageneralvariablexinsteadoftime:

Inthenumeratorweseethechangeinf;thedenominatoristhechangeinx.Thisequationdefinesthederivativef′(x)uniquely,providedthelimitexists.Thathastobeprovedforanyfunctionunderconsideration:thelimitdoesexistformostofthestandardfunctions–squares,cubes,higherpowers,logarithms,exponentials,trigonometricfunctions.

Nowhereinthecalculationdoweeverdividebyzero,becauseweneverseth=0.Moreover,nothinghereactuallyflows.Whatmattersistherangeofvaluesthathcanassume,nothowitmovesthroughthatrange.SoBerkeley’ssarcasticcharacterisationisactuallyspoton.ThelimitListheghostofthedepartedquantity–myh,Newton’so.Butthemannerofthequantity’sdeparture–approachingzero,notreachingit–leadstoaperfectlysensibleandlogicallywell-definedghost.

Calculusnowhadasoundlogicalbasis.Itdeserved,andacquired,anewnametoreflectitsnewstatus:analysis.

Itisnomorepossibletolistallthewaysthatcalculuscanbeappliedthanitistolisteverythingintheworldthatdependsonusingascrewdriver.Onasimplecomputationallevel,applicationsofcalculusincludefindinglengthsofcurves,areasofsurfacesandcomplicatedshapes,volumesofsolids,maximumand

minimumvalues,andcentresofmass.Inconjunctionwiththelawsofmechanics,calculustellsushowtoworkoutthetrajectoryofaspacerocket,thestressesinrockatasubductionzonethatmightproduceanearthquake,thewayabuildingwillvibrateifanearthquakehits,thewayacarbouncesupanddownonitssuspension,thetimeittakesabacterialinfectiontospread,thewayasurgicalwoundheals,andtheforcesthatactonasuspensionbridgeinahighwind.

ManyoftheseapplicationsstemfromthedeepstructureofNewton’slaws:theyaremodelsofnaturestatedasdifferentialequations.Theseareequationsinvolvingderivativesofanunknownfunction,andtechniquesfromcalculusareneededtosolvethem.Iwillsaynomorehere,becauseeverychapterfromChapter8onwardsinvolvescalculusexplicitly,mainlyintheguiseofdifferentialequations.ThesoleexceptionisChapter15oninformationtheory,andeventhereotherdevelopmentsthatIdon’tmentionalsoinvolvecalculus.Likethescrewdriver,calculusissimplyanindispensabletoolintheengineer’sandscientist’stoolkits.Morethananyothermathematicaltechnique,ithascreatedthemodernworld.

4Thesystemoftheworld

Newton’sLawofGravity

Whatdoesitsay?

Itdeterminestheforceofgravitationalattractionbetweentwobodiesintermsoftheirmassesandthedistancebetweenthem.

Whyisthatimportant?

Itcanbeappliedtoanysystemofbodiesinteractingthroughtheforceofgravity,suchastheSolarSystem.Ittellsusthattheirmotionisdeterminedbyasimplemathematicallaw.

Whatdiditleadto?

Accuratepredictionofeclipses,planetaryorbits,thereturnofcomets,therotationofgalaxies.Artificialsatellites,surveysoftheEarth,theHubbletelescope,observationsofsolarflares.Interplanetaryprobes,Marsrovers,satellitecommunicationsandtelevision,theGlobalPositioningSystem.

Newton’slawsofmotioncapturetherelationshipbetweentheforcesthatactonabodyandhowitmovesinresponsetothoseforces.Calculusprovidesmathematicaltechniquesforsolvingtheresultingequations.Onefurtheringredientisneededtoapplythelaws:specifyingtheforces.ThemostambitiousaspectofNewton’sPrincipiawastodopreciselythatforthebodiesoftheSolarSystem–theSun,planets,moons,asteroids,andcomets.Newton’slawofgravitationsynthesised,inonesimplemathematicalformula,millenniaofastronomicalobservationsandtheories.Itexplainedmanypuzzlingfeaturesofplanetarymotion,andmadeitpossibletopredictthefuturemovementsoftheSolarSystemwithgreataccuracy.Einstein’stheoryofgeneralrelativityeventuallysupersededtheNewtoniantheoryofgravity,asfarasfundamentalphysicsisconcerned,butforalmostallpracticalpurposesthesimplerNewtonianapproachstillreignssupreme.Todaytheworld’sspaceagencies,suchasNASAandESA,stilluseNewton’slawsofmotionandgravitationtoworkoutthemosteffectivetrajectoriesforspacecraft.

ItwasNewton’slawofgravitation,aboveallelse,thatjustifiedhissubtitle:TheSystemoftheWorld.Thislawdemonstratedtheenormouspowerofmathematicstofindhiddenpatternsinnatureandtorevealhiddensimplicitiesbehindtheworld’scomplexities.Andintime,asmathematiciansandastronomersaskedharderquestions,torevealthehiddencomplexitiesimplicitinNewton’ssimplelaw.ToappreciatewhatNewtonachieved,wemustfirstgobackintime,toseehowpreviousculturesviewedthestarsandplanets.

Humanshavebeenwatchingthenightskysincethedawnofhistory.Theirinitialimpressionwouldhavebeenarandomscatteringofbrightpointsoflight,buttheywouldsoonhavenoticedthatacrossthisbackgroundtheglowingorboftheMoontracedaregularpath,changingshapeasitdidso.Theywouldalsohaveseenthatmostofthosetinybrightspecksoflightremaininthesamerelativepatterns,whichwenowcallconstellations.Starsmoveacrossthenightsky,buttheymoveasasinglerigidunit,asiftheconstellationsarepaintedontheinsideofagigantic,rotatingbowl.1However,asmallnumberofstarsbehavequitedifferently:theyseemtowanderaroundthesky.Theirpathsarequitecomplicated,andsomeappeartoloopbackonthemselvesfromtimetotime.Thesearetheplanets,awordthatcomesfromtheGreekfor‘wanderer’.Theancientsrecognisedfiveofthem,nowcalledMercury,Venus,Mars,Jupiter,andSaturn.Theymoverelativetothefixedstarsatdifferentspeeds,withSaturn

beingtheslowest.Othercelestialphenomenawereevenmorepuzzling.Fromtimetotimea

cometwouldappear,asiffromnowhere,trailingalong,curvedtail.‘Shootingstars’wouldseemtofallfromtheheavens,asiftheyhadbecomedetachedfromtheirsupportingbowl.Itisnowonderthatearlyhumansattributedtheirregularitiesoftheheavenstothecapricesofsupernaturalbeings.

Theregularitiescouldbesummedupintermssoobviousthatfewwouldeverdreamofdisputingthem.TheSun,stars,andplanetsrevolvearoundastationaryEarth.That’swhatitlookslike,that’swhatitfeelslike,sothat’showitmustbe.Totheancients,thecosmoswasgeocentric–Earth-centred.Onelonevoicedisputedtheobvious:AristarchusofSamos.Usinggeometricalprinciplesandobservations,AristarchuscalculatedthesizesoftheEarth,theSun,andtheMoon.Around270BCheputforwardthefirstheliocentrictheory:theEarthandplanetsrevolveroundtheSun.Histheoryquicklyfelloutoffavourandwasnotrevivedfornearly2000years.

BythetimeofPtolemy,aRomanwholivedinEgyptaround120AD,theplanetshadbeentamed.Theirmovementswerenotcapricious,butpredictable.Ptolemy’sAlmagest(‘GreatTreatise’)proposedthatweliveinageocentricuniverseinwhicheverythingliterallyrevolvesaroundhumanityincomplexcombinationsofcirclescalledepicycles,supportedbygiantcrystalspheres.Histheorywaswrong,butthemotionsthatitpredictedweresufficientlyaccuratefortheerrorstoremainundetectedforcenturies.Ptolemy’ssystemhadanadditionalphilosophicalattraction:itrepresentedthecosmosintermsofperfectgeometricfigures–spheresandcircles.ItcontinuedthePythagoreantradition.InEurope,thePtolemaictheoryremainedunchallengedfor1400years.

WhileEuropedawdled,newscientificadvanceswerebeingmadeelsewhere,especiallyinArabia,China,andIndia.In499theIndianastronomerAryabhataputforwardamathematicalmodeloftheSolarSysteminwhichtheEarthspunonitsaxisandtheperiodsofplanetaryorbitswerestatedrelativetothepositionoftheSun.IntheIslamicworld,AlhazenwroteastingingcriticismofthePtolemaictheory,thoughthiswasprobablynotfocusedonitsgeocentricnature.Around1000AbuRayhanBirunigaveseriousconsiderationtothepossibilityofaheliocentricSolarSystem,withtheEarthspinningonitsaxis,buteventuallyplumpedfortheorthodoxyofthetime,astationaryEarth.Around1300,Najmal-Dinal-Qazwinial-Katibiproposedaheliocentrictheory,butsoonchangedhismind.

ThebigbreakthroughcamewiththeworkofNicolausCopernicus,published

in1543asDeRevolutionibusOrbiumCoelestium(‘OntheRevolutionsoftheCelestialSpheres’).Thereisevidence,notablytheoccurrenceofalmostidenticaldiagramslabelledwiththesameletters,tosuggestthatCopernicuswas,tosaytheleast,influencedbyal-Katibi,buthewentmuchfurther.Hesetoutanexplicitlyheliocentricsystem,arguedthatitfittedtheobservationsbetterandmoreeconomicallythanPtolemy’sgeocentrictheorydid,andlaidoutsomeofthephilosophicalimplications.Paramountamongthemwasthenovelthoughtthathumanswerenotatthecentreofthings.TheChristianChurchviewedthissuggestionascontrarytodoctrineanddiditsbesttodiscourageit.Explicitheliocentrismwasheresy.

Itprevailednevertheless,becausetheevidencewassostrong.Newandbetterheliocentrictheoriesappeared.Thenthesphereswerethrownawayaltogether,infavourofadifferentshapefromclassicalgeometry:theellipse.Ellipsesareovalshapes,andindirectevidencesuggeststheywerefirststudiedinGreekgeometrybyMenaechmusaround350BC,alongwithhyperbolasandparabolas,assectionsofacone,Figure13.Euclidissaidtohavewrittenfourbooksonconicsections,thoughnothinghassurvivedifhedid,andArchimedesinvestigatedsomeoftheirproperties.Greekresearchonthetopicreacheditsclimaxinabout240BCwiththeeight-volumeConicSectionsbyApolloniusofPerga,whofoundawaytodefinethesecurvespurelywithinaplane,avoidingthethirddimension.However,thePythagoreanviewthatcirclesandspheresattainedahigherdegreeofperfectionthanellipsesandothermorecomplexcurvespersisted.

Fig13Conicsections.

Ellipsescementedtheirroleinastronomyaround1600,withtheworkofKepler.Hisastronomicalinterestsbeganinchildhood;attheageofsixhewitnessedthegreatcometof1577,2andthreeyearslaterhesawaneclipseoftheMoon.AttheUniversityofTübingen,Keplershowedgreattalentfor

mathematicsandputittoprofitableusecastinghoroscopes.Inthosedaysmathematics,astronomy,andastrologyoftenwenttogether.Hecombinedaheadylevelofmysticismwithalevel-headedattentiontomathematicaldetail.AtypicalexampleishisMysteriumCosmographicum(‘TheCosmographicMystery’),aspiriteddefenceoftheheliocentricsystempublishedin1596.ItcombinesacleargraspofCopernicus’stheorywithwhattomoderneyesisaverystrangespeculationrelatingthedistancesoftheknownplanetsfromtheSuntotheregularsolids.ForalongtimeKeplerregardedthisdiscoveryasoneofhisgreatest,revealingtheCreator’splanfortheuniverse.Hesawhislaterresearches,whichwenowconsidertobefarmoresignificant,asmereelaborationsofthisbasicplan.Atthetime,oneadvantageofthetheorywasthatitexplainedwhytherewerepreciselysixplanets(MercurythroughSaturn).Betweenthesesixorbitsliefivegaps,oneforeachregularsolid.WiththediscoveryofUranusandlaterNeptuneandPluto(untilitsrecentdemotionfromplanetarystatus)thisfeaturequicklybecameafatalflaw.

Kepler’slastingcontributionhasitsrootsinhisemploymentbyTychoBrahe.Thetwofirstmetin1600.Afteratwo-monthstayandaheatedargumentKeplernegotiatedanacceptablesalary.FollowingaspateofproblemsinhishomecityofGrazhemovedtoPrague,assistingTychointheanalysisofhisplanetaryobservations,especiallyofMars.WhenTychounexpectedlydiedin1601Keplertookoverhisemployer’spositionasimperialmathematiciantoRudolphII.Hisprimaryrolewascastingimperialhoroscopes,buthealsohadtimetocontinuehisanalysisoftheorbitofMars.Followingtraditionalepicyclicprinciplesherefinedhismodeltothepointatwhichitserrors,comparedwithobservation,wereusuallyameretwominutesofarc,thetypicalerrorintheobservationsthemselves.However,hedidn’tstoptherebecausesometimestheerrorswerebigger,uptoeightminutesofarc.

Hissearcheventuallyledhimtotwolawsofplanetarymotion,publishedinAstronomiaNova(‘ANewAstronomy’).FormanyyearshehadtriedtofittheorbitofMarstoanovoid–anegg-shapedcurve,sharperatoneendthantheother–withoutsuccess.PerhapsheexpectedtheorbittobemorecurvedclosertotheSun.In1605itoccurredtoKeplertotryanellipse,equallyroundedatbothends,andtohissurprisethisdidamuchbetterjob.Heconcludedthatallplanetaryorbitsareellipses,hisfirstlaw.Hissecondlawdescribedhowtheplanetmovesalongitsorbit,statingthatplanetssweepoutequalareasinequaltimes.Thebookappearedin1609.Keplerthendevotedmuchofhisefforttopreparingvariousastronomicaltables,buthereturnedtotheregularitiesofplanetaryorbitsin1619inhisHarmonicesMundi(‘TheHarmonyofthe

World’).Thisbookhadsomeideaswenowfindstrange,forexamplethattheplanetsemitmusicalsoundsastheyrollroundtheSun.Butitalsoincludeshisthirdlaw:thesquaresoftheorbitalperiodsareproportionaltothecubesofthedistancesfromtheSun.

Kepler’sthreelawswereallbutburiedamidamassofmysticism,religioussymbolism,andphilosophicalspeculation.Buttheyrepresentedagiantleapforward,leadingNewtontooneofthegreatestscientificdiscoveriesofalltime.

NewtonderivedhislawofgravityfromKepler’sthreelawsofplanetarymotion.Itstatesthateveryparticleintheuniverseattractseveryotherparticlewithaforcethatisproportionaltotheproductoftheirmassesandinverselyproportionaltothesquareofthedistancebetweenthem.Insymbols,

HereFistheattractiveforce,disthedistance,themsarethetwomasses,andGisaspecificnumber,thegravitationalconstant.3

WhodiscoveredNewton’slawofgravity?Itsoundslikeoneofthoseself-answeringquestions,like‘whosestatuestandsontopofNelson’scolumn?’.ButareasonableansweristhecuratorofexperimentsattheRoyalSociety,RobertHooke.WhenNewtonpublishedthelawin1687,inhisPrincipia,Hookeaccusedhimofplagiarism.However,Newtonprovidedthefirstmathematicalderivationofellipticalorbitsfromthelaw,whichwasvitalinestablishingitscorrectness,andHookeacknowledgedthis.Moreover,NewtonhadcitedHooke,alongwithseveralothers,inthebook.PresumablyHookefelthedeservedmorecredit;hehadsufferedsimilarproblemsseveraltimesbeforeanditwasasorepoint.

Theideathatbodiesattracteachotherhadbeenfloatingaroundforawhile,andsohaditslikelymathematicalexpression.In1645theFrenchastronomerIsmaëlBoulliau(Bullialdus)wrotehisAstronomiaPhilolaica(‘PhilolaicAstronomy’–PhilolauswasaGreekphilosopherwhothoughtthatacentralfire,nottheEarth,wasthecentreoftheuniverse).Inithewrote:

AsforthepowerbywhichtheSunseizesorholdstheplanets,andwhich,beingcorporeal,functionsinthemannerofhands,itisemittedinstraightlinesthroughoutthewholeextentoftheworld,andlikethespeciesoftheSun,itturnswiththebodyoftheSun;now,seeingthatitiscorporeal,itbecomesweakerandattenuatedat

agreaterdistanceorinterval,andtheratioofitsdecreaseinstrengthisthesameasinthecaseoflight,namely,theduplicateproportion,butinversely,ofthedistances.

Thisisthefamous‘inversesquare’dependencyoftheforceondistance.Therearesimple,thoughnaive,reasonstoexpectsuchaformula,becausethesurfaceareaofaspherevariesasthesquareofitsradius.Ifthesameamountofgravitational‘stuff’spreadsoutoverever-increasingspheresasitdepartsfromtheSun,thentheamountofitreceivedatanypointmustvaryintheinverseproportiontothesurfacearea.Exactlythishappenswithlight,andBoulliauassumed,withoutmuchevidence,thatgravitymustbeanalogous.Healsothoughtthattheplanetsmovealongtheirorbitsundertheirownpower,sotospeak:‘Nokindofmotionpressesupontheremainingplanets,[which]aredrivenroundbyindividualformswithwhichtheywereprovided.’

Hooke’scontributiondatesto1666,whenhepresentedapapertotheRoyalSocietywiththetitle‘Ongravity’.HerehesortedoutwhatBoulliauhadgotwrong,arguingthatanattractiveforcefromtheSuncouldinterferewithaplanet’snaturaltendencytomoveinastraightline(asspecifiedbyNewton’sthirdlawofmotion)andcauseittofollowacurve.Healsostatedthat‘theseattractivepowersaresomuchthemorepowerfulinoperating,byhowmuchthenearerthebodywroughtuponistotheirownCenters’,showingthathethoughttheforcefelloffwithdistance.Buthedidn’ttellanyoneelsethemathematicalformforthisdecreaseuntil1679,whenhewrotetoNewton:‘TheAttractionalwaysisinaduplicateproportiontotheDistancefromtheCenterReciprocall.’InthesameletterhesaidthatthisimpliesthatthevelocityofaplanetvariesasthereciprocalofitsdistancefromtheSun.Whichiswrong.

WhenHookecomplainedthatNewtonhadstolenhislaw,Newtonwashavingnoneofit,pointingoutthathehaddiscussedtheideawithChristopherWrenbeforeHookehadsenthisletter.Todemonstratepriorart,hecitedBoulliau,andalsoGiovanniBorelli,anItalianphysiologistandmathematicalphysicist.Borellihadsuggestedthatthreeforcescombinetocreateplanetarymotion:aninwardforcecausedbytheplanet’sdesiretoapproachtheSun,asidewaysforcecausedbysunlight,andanoutwardforcecausedbytheSun’srotation.Scoreoneoutofthree,andthat’sgenerous.

Newton’smainpoint,generallyconsidereddecisive,isthatwhateverelseHookehaddone,hehadnotdeducedtheexactformoforbitsfrominversesquarelawattraction.Newtonhad.Infact,hehaddeducedallthreeofKepler’slawsofplanetarymotion:ellipticalorbits,sweepingoutequalareasinequal

intervalsoftime,withthesquareoftheperiodbeingproportionaltothecubeofthedistance.‘WithoutmyDemonstrations,’Newtoninsisted,theinversesquarelaw‘cannotbebelievedbyajudiciousphilosophertobeanywhereaccurate.’Buthedidalsoacceptthat‘MrHookisyetastranger’tothisproof.AkeyfeatureofNewton’sargumentisthatitappliesnotjusttoapointparticle,buttoasphere.Thisextension,whichiscrucialtoplanetarymotion,hadcausedNewtonconsiderableeffort.Hisgeometricproofisadisguisedapplicationofintegralcalculus,andhewasjustifiablyproudofit.ThereisalsodocumentaryevidencethatNewtonhadbeenthinkingaboutsuchquestionsforquiteawhile.

Atanyrate,wenamethelawafterNewton,andthisdoesjusticetotheimportanceofhiscontribution.

ThemostimportantaspectofNewton’slawofgravitationisnottheinversesquarelawassuch.Itistheassertionthatgravitationactsuniversally.Anytwobodies,anywhereintheuniverse,attracteachother.Ofcourseyouneedanaccurateforcelaw(inversesquare)togetaccurateresults,butwithoutuniversality,youdon’tknowhowtowritedowntheequationsforanysystemwithmorethantwobodies.Almostalloftheinterestingsystems,suchastheSolarSystemitself,orthefinestructureofthemotionoftheMoonundertheinfluenceof(atleast)theSunandtheEarth,involvemorethantwobodies,soNewton’slawwouldhavebeenalmostuselessifithadappliedonlytothecontextinwhichhefirstdeducedit.

Whatmotivatedthisvisionofuniversality?Inhis1752MemoirsofSirIsaacNewton’sLife,WilliamStukeleyreportedataleNewtonhadtoldhimin1726:

Thenotionofgravitation…wasoccasionedbythefallofanapple,ashesatincontemplativemood.Whyshouldthatapplealwaysdescendperpendicularlytotheground,thoughthetohimself.Whyshoulditnotgosidewaysorupwards,butconstantlytotheEarth’scentre?Assuredlythereasonis,thattheEarthdrawsit.Theremustbeadrawingpowerinmatter.AndthesumofthedrawingpowerinthematteroftheEarthmustbeintheEarth’scentre,notinanysideoftheEarth.Thereforedoesthisapplefallperpendicularlyortowardsthecentre?Ifmatterthusdrawsmatter;itmustbeinproportionofitsquantity.ThereforetheappledrawstheEarth,aswellastheEarthdrawstheapple.

Whetherthestoryistheliteraltruth,oraconvenientfictionthatNewton

inventedtohelphimexplainhisideaslateron,isnotentirelyclear,butitseemsreasonabletotakethetaleatfacevaluebecausetheideadoesnotendwithapples.TheapplewasimportanttoNewtonbecauseitmadehimrealisethatthesamelawofforcescanexplainboththemotionoftheappleandthatoftheMoon.TheonlydifferenceisthattheMoonalsomovessideways;thisiswhyitstaysup.Actually,itisalwaysfallingtowardstheEarth,butthesidewaysmotioncausestheEarth’ssurfacetofallawayaswell.Newton,beingNewton,didn’tstopwiththisqualitativeargument.Hedidthesums,comparedthemwithobservations,andwassatisfiedthathisideamustbecorrect.

Ifgravityactsontheapple,theMoon,andtheEarth,asaninherentfeatureofmatter,thenpresumablyitactsoneverything.

Itisnotpossibletoverifytheuniversalityofgravitationalforcesdirectly;youwouldhavetostudyallpairsofbodiesintheentireuniverse,andfindawaytoremovetheinfluenceofalltheotherbodies.Butthat’snothowscienceworks.Instead,itemploysamixtureofinferenceandobservations.Universalityisahypothesis,capableofbeingfalsifiedeverytimeitisapplied.Everytimeitsurvivesfalsification–afancywaytosayitgivesgoodresults–thejustificationforusingitbecomesalittlestronger.If(asinthiscase)itsurvivesthousandsofsuchtests,thejustificationbecomesverystrongindeed.However,thehypothesiscanneverbeprovedtrue:forallweknow,thenextexperimentmightproduceincompatibleresults.Perhapssomewhereinagalaxyfar,farawaythereisonespeckofmatter,oneatom,thatisnotattractedtoeverythingelse.Ifso,wewillneverfindit;equally,itwon’tupsetourcalculations.Theinversesquarelawitselfisexceedinglydifficulttoverifydirectly,thatis,byactuallymeasuringtheattractiveforce.Instead,weapplythelawtosystemsthatwecanmeasurebyusingittopredictorbits,andthencheckwhetherthepredictionsagreewithobservations.

Evengrantinguniversality,itisnotenoughtowritedownanaccuratelawofattraction.Thatjustproducesanequationdescribingthemotion.Inordertofindthemotionitself,youhavetosolvetheequation.Evenfortwobodies,thisisnotstraightforward,andevenbearinginmindthatheknewinadvancewhatanswertoexpect,Newton’sdeductionofellipticalorbitsisatourdeforce.ItexplainswhyKepler’sthreelawsprovideaveryaccuratedescriptionofeachplanet’sorbit.Italsoexplainswhythatdescriptionisnotexact:otherbodiesinthesolarsystem,otherthantheSunandtheplanetitself,affectthemotion.Inordertoaccountforthesedisturbances,youhavetosolvetheequationsofmotionforthreeormorebodies.Inparticular,ifyouwanttopredictthemotionoftheMoonwithhighprecision,youhavetoincludetheSunandtheEarthinyour

equations.Theeffectsoftheotherplanets,especiallyJupiter,arenotentirelynegligibleeither,buttheyshowuponlyinthelongterm.So,freshfromNewton’ssuccesswiththemotionoftwobodiesundergravity,mathematiciansandphysicistsmovedontothenextcase:threebodies.Theirinitialoptimismdissipatedrapidly:thethree-bodycaseturnedouttobeverydifferentfromthetwo-bodycase.Infact,itdefiedsolution.

Itwasoftenpossibletocalculategoodapproximationstothemotion(whichoftensolvedtheproblemforpracticalpurposes),buttherenolongerseemedtobeanexactformula.Thisproblembedevilledevensimplifiedversions,suchastherestrictedthree-bodyproblem.Supposethataplanetorbitsastarinaperfectcircle:howwillaspeckofdust,ofnegligiblemass,move?

Calculatingapproximateorbitsforthreeormorebodies,byhand,usingpencilandpaper,wasjustaboutfeasible,butverylaborious.Mathematiciansdevisedinnumerabletricksandshortcuts,leadingtoareasonableunderstandingofseveralastronomicalphenomena.Onlyinthelatenineteenthcenturydidthetruecomplexityofthethree-bodyproblembecomeapparent,whenHenriPoincarérealisedthatthegeometryinvolvedwasnecessarilyextraordinarilyintricate.Andonlyinthelatetwentiethcenturydidtheadventofpowerfulcomputersreducethelabourofhandcalculations,permittingaccuratelong-termpredictionsofthemotionoftheSolarSystem.

Poincaré’sbreakthrough–ifitcanbecalledthat,sinceatthetimeitseemedtobetellingeveryonethattheproblemwashopelessanditwaspointlesstoseekasolution–cameaboutbecausehecompetedforamathematicalprize.OscarII,kingofSwedenandNorway,announcedacompetitiontocelebratehis60thbirthdayin1889.TakingadvicefromthemathematicianGöstaMittag-Leffler,thekingchosethegeneralproblemofarbitrarilymanybodiesmovingunderNewtoniangravitation.Sinceitwaswellunderstoodthatanexplicitformulaakintothetwo-bodyellipsewasanunrealisticaim,therequirementwasrelaxed:theprizewouldbeawardedforanapproximationmethodofaveryspecifickind.Namely,themotionmustbedeterminedasaninfiniteseries,givingresultsasaccurateaswepleaseifenoughtermsareincluded.

Poincarédidnotanswerthisquestion.Instead,hismemoironthetopic,publishedin1890,providedevidencethatitmightnotpossessthatkindofanswer,evenforjustthreebodies–star,planet,anddustparticle.Bythinkingaboutthegeometryofhypotheticalsolutions,Poincarédiscoveredthatinsomecasestheorbitofthedustparticlemustbeexceedinglycomplexandtangled.Hethen,ineffect,threwuphishandsinhorrorandmadethepessimisticstatement

that‘Whenonetriestodepictthefigureformedbythesetwocurvesandtheirinfinityofintersections,eachofwhichcorrespondstoadoublyasymptoticsolution,theseintersectionsformakindofnet,weborinfinitelytightmesh…OneisstruckbythecomplexityofthisfigurethatIamnotevenattemptingtodraw.’

WenowseePoincaré’sworkasabreakthrough,anddiscounthispessimism,becausethecomplicatedgeometrythatledhimtodespairofeversolvingtheproblemactuallyprovidespowerfulinsightsifitisproperlydevelopedandunderstood.Thecomplexgeometryoftheassociateddynamicsturnedouttobeoneoftheearliestexamplesofchaos:theoccurrence,innon-randomequations,ofsolutionssocomplicatedthatinsomerespectstheyappeartoberandom,seeChapter16.

Thereareseveralironiesinthestory.MathematicalhistorianJuneBarrow-GreendiscoveredthatthepublishedversionofPoincaré’sprizewinningmemoirwasnottheonethatwontheprize.4Thisearlierversioncontainedamajorerror,overlookingthechaoticsolutions.TheworkwasatproofstagewhenanembarrassedPoincarérealisedhisblunder,andhepaidforanewprintingofacorrectedversion.Almostallcopiesoftheoriginalweredestroyed,butoneremainedtuckedawayinthearchivesoftheMittag-LefflerInstituteinSweden,whereBarrow-Greenfoundit.

Italsoturnedoutthatthepresenceofchaosdoesnot,infact,ruleoutseriessolutions,butthesearevalidalmostalwaysratherthanalways.KarlFrithiofSundman,aFinnishmathematician,discoveredthisin1912forthethree-bodyproblem,usingseriesformedfrompowersofthecuberootoftime.(Powersoftimewon’thackit.)Theseriesconverge–haveasensiblesum–unlesstheinitialstatehaszeroangularmomentum,butsuchstatesareinfinitelyrare,inthesensethatarandomchoiceofangularmomentumisalmostalwaysnonzero.In1991theChinesemathematicianQiudongWangextendedtheseresultstoanynumberofbodies,butdidnotclassifytherareexceptionswhentheseriesfailtoconverge.Suchaclassificationislikelytobeverycomplicated:itmustincludesolutionswherebodiesescapetoinfinityinfinitetime,oroscillateeverfaster,bothofwhichcanhappenforfiveormorebodies.

Newton’slawofgravityisroutinelyappliedtodesignorbitsforspacemissions.Hereeventwo-bodydynamicsisusefulinitsownright.Initsearlydays,theexplorationoftheSolarSystemmainlyusedtwo-bodyorbits,segmentsofellipses.Byburningitsrocketsthespacecraftcouldbeswitchedfromoneellipsetoadifferentone.Butastheaimsofspaceprogrammesgotmoreambitious,

moreefficientmethodswereneeded.Theycamefrommany-bodydynamics,usuallythreebodiesbutoccasionallyasgreatasfive.Thenewmethodsofchaosandtopologicaldynamicsbecamethebasisofpracticalsolutionstoengineeringproblems.

Fig14Hohmanntransferellipsefromlow-Earthorbittolunarorbit.

Itallstartedwithasimplequestion:WhatisthemostefficientroutefromtheEarthtotheMoonortheplanets?Theclassicanswer,knownasaHohmanntransferellipse(Figure14),startsfromacircularorbitroundtheEarth,andthenfollowspartofalong,thinellipsetojoinupwithasecondcircularorbitroundthedestination.ThismethodwasemployedfortheApollomissionsofthe1960sand1970s,butformanytypesofmissionithasonedisadvantage.ThespacecraftmustbeboostedoutofEarthorbitandslowedagaintoenterlunarorbit;thiswastesfuel.TherearealternativesinvolvingmanyloopsroundtheEarth,atransitionthroughthepointbetweenEarthandMoonwheretheirgravitationalfieldscancel,andmanyloopsroundtheMoon.ButtrajectorieslikethattakelongerthanHohmannellipses,sotheywerenotusedforthemannedApollomissionswherefoodandoxygen,hencetime,wereoftheessence.Forunmannedmissions,however,timeisrelativelycheap,whereasanythingthataddstotheoverallweightofthespacecraft,includingfuel,costsmoney.

BytakingafreshlookatNewton’slawofgravityandhissecondlawofmotion,mathematiciansandspaceengineershaverecentlydiscoveredanew,andremarkable,approachtofuel-efficientinterplanetarytravel.

Gobytube.It’sanideastraightoutofsciencefiction.Inhis2004Pandora’sStar,Peter

Hamiltonportraysafuturewherepeopletraveltoplanetsencirclingdistantstarsbytrain,runningtherailwaylinesthroughawormhole,ashortcutthroughspace-time.InhisLensmanseriesfrom1934to1948,EdwardElmer‘Doc’Smithcameupwiththehyperspatialtube,whichmalevolentaliensusedtoinvadehumanworldsfromthefourthdimension.

Althoughwedon’tyethavewormholesoraliensfromthefourthdimension,ithasbeendiscoveredthattheplanetsandmoonsoftheSolarSystemaretiedtogetherbyanetworkoftubes,whosemathematicaldefinitionrequiresmanymoredimensionsthanfour.Thetubesprovideenergy-efficientroutesfromoneworldtoanother.Theycanbeseenonlythroughmathematicaleyes,becausetheyarenotmadeofmatter:theirwallsareenergylevels.Ifwecouldvisualisetheever-changinglandscapeofgravitationalfieldsthatcontrolshowtheplanetsmove,wewouldbeabletoseethetubes,swirlingalongwiththeplanetsastheyorbittheSun.

Tubesexplainsomepuzzlingorbitaldynamics.Consider,forexample,thecometcalledOterma.Acenturyago,Oterma’sorbitwaswelloutsidethatofJupiter.Butafteracloseencounterwiththegiantplanet,thecomet’sorbitshiftedinsidethatofJupiter.Afteranothercloseencounter,itswitchedbackoutsideagain.WecanconfidentlypredictthatOtermawillcontinuetoswitchorbitsinthiswayeveryfewdecades:notbecauseitbreaksNewton’slaw,butbecauseitobeysit.

Thisisafarcryfromtidyellipses.TheorbitspredictedbyNewtoniangravityareellipticalonlywhennootherbodiesexertasignificantgravitationalpull.ButtheSolarSystemisfullofotherbodies,andtheycanmakeahuge–andsurprising–difference.Itisherethatthetubesenterthestory.Oterma’sorbitliesinsidetwotubes,whichmeetnearJupiter.OnetubeliesinsideJupiter’sorbit,theotheroutside.Theyenclosespecialorbitsin3:2and2:3resonancewithJupiter,meaningthatabodyinsuchanorbitwillgoroundtheSunthreetimesforeverytworevolutionsofJupiter,ortwotimesforeverythree.AtthetubejunctionnearJupiter,thecometcanswitchtubes,ornot,dependingonrathersubtleeffectsofJovianandsolargravity.Butonceinsideatube,Otermaisstuckthereuntilthetubereturnstothejunction.Likeatrainthathastostayontherails,butcanchangeitsroutetoanothersetofrailsifsomeoneswitchesthepoints,Otermahassomefreedomtochangeitsitinerary,butnotalot(Figure15).

Fig15Left:Twoperiodicorbits,in2:3and3:2resonancewithJupiter,connectedviaLagrangepoints.Right:ActualorbitofcometOterma,1910–1980.

Thetubesandtheirjunctionsmayseembizarre,buttheyarenaturalandimportantfeaturesofthegravitationalgeographyoftheSolarSystem.Victorianrailway-buildersunderstoodtheneedtoexploitnaturalfeaturesofthelandscape,runningrailwaysthroughvalleysandalongcontourlines,anddiggingtunnelsthroughhillsratherthantakingthetrainoverthetop.Onereasonwasthattrainstendtosliponsteepgradients,butthemainonewasenergy.Climbingahill,againsttheforceofgravity,costsenergy,whichshowsupasincreasedfuelconsumption,whichcostsmoney.

It’smuchthesamewithinterplanetarytravel.Imagineaspacecraftmovingthroughspace.Whereitgoesnextdoesnotdependsolelyonwhereitisnow:italsodependsonhowfastitismovingandinwhichdirection.Ittakesthreenumberstospecifythespacecraft’sposition–forexampleitsdirectionfromtheEarth,whichrequirestwonumbers(astronomersuserightascensionanddeclination,whichareanalogoustolongitudeandlatitudeonthecelestialsphere,theapparentsphereformedbythenightsky),anditsdistancefromtheEarth.Ittakesafurtherthreenumberstospecifyitsvelocityinthosethreedirections.Sothespacecrafttravelsthroughamathematicallandscapethathassixdimensionsratherthantwo.

Anaturallandscapeisnotflat:ithashillsandvalleys.Ittakesenergytoclimbahill,butatraincangainenergybyrollingdownintoavalley.Infact,twotypesofenergycomeintoplay.Theheightabovesea-leveldeterminesthetrain’spotentialenergy,whichrepresentsworkdoneagainsttheforceofgravity.

Thehigheryougo,themorepotentialenergyyoumustcreate.Thesecondkindiskineticenergy,whichcorrespondstospeed.Thefasteryougo,thegreateryourkineticenergybecomes.Whenthetrainrollsdownhillandaccelerates,ittradespotentialenergyforkinetic.Whenitclimbsahillandslowsdown,thetradeisinthereversedirection.Thetotalenergyisconstant,sothetrain’strajectoryisanalogoustoacontourlineintheenergylandscape.However,trainshaveathirdsourceofenergy:coal,diesel,orelectricity.Byexpendingfuel,atraincanclimbagradientorspeedup,freeingitselffromitsnaturalfree-runningtrajectory.Thetotalenergystillcannotchange,butallelseisnegotiable.

Itismuchthesamewithspacecraft.ThecombinedgravitationalfieldsoftheSun,planets,andotherbodiesoftheSolarSystemprovidepotentialenergy.Thespeedofthespacecraftcorrespondstokineticenergy.Anditsmotivepower–beitrocketfuel,ions,orlight-pressure–addsafurtherenergysource,whichcanbeswitchedonoroffasrequired.Thepathfollowedbythespacecraftisakindofcontourlineinthecorrespondingenergylandscape,andalongthatpaththetotalenergyremainsconstant.Andsometypesofcontourlinearesurroundedbytubes,correspondingtonearbyenergylevels.

ThoseVictorianrailwayengineerswerealsoawarethattheterrestriallandscapehasspecialfeatures–peaks,valleys,mountainpasses–whichhaveabigeffectonefficientroutesforrailwaylines,becausetheyconstituteakindofskeletonfortheoverallgeometryofthecontours.Forinstance,nearapeakoravalleybottomthecontoursformclosedcurves.Atpeaks,potentialenergyislocallyatamaximum;inavalley,itisatalocalminimum.Passescombinefeaturesofboth,beingatamaximuminonedirection,butaminimuminanother.Similarly,theenergylandscapeoftheSolarSystemhasspecialfeatures.Themostobviousaretheplanetsandmoonsthemselves,whichsitatthebottomofgravitywells,likevalleys.Equallyimportant,butlessvisible,arethepeaksandpassesoftheenergylandscape.Allthesefeaturesorganisetheoverallgeometry,andwithit,thetubes.

Theenergylandscapehasotherattractivefeaturesforthetourist,notablyLagrangepoints.ImagineasystemconsistingonlyoftheEarthandtheMoon.In1772Joseph-LouisLagrangediscoveredthatatanyinstanttherearepreciselyfiveplaceswherethegravitationalfieldsofthetwobodies,togetherwithcentrifugalforce,canceloutexactly.ThreeareinlinewithbothEarthandMoon–L1liesbetweenthem,L2isonthefarsideoftheMoon,andL3isonthefarsideoftheEarth.TheSwissmathematicianLeonhardEulerhadalreadydiscoveredthesearound1750.ButtherearealsoL4andL5,knownasTrojanpoints,whichlieinthesameorbitastheMoonbut60degreesaheadofitor

behindit.AstheMoonrotatesroundtheEarth,theLagrangepointsrotatewithit.OtherpairsofbodiesalsohaveLagrangepoints–Earth/Sun,Jupiter/Sun,Titan/Saturn.

Theold-fashionedHohmanntransferorbitisbuiltfrompiecesofcirclesandellipses,whicharethenaturaltrajectoriesfortwo-bodysystems.Thenewtube-basedpathsarebuiltfrompiecesofthenaturaltrajectoriesofthree-bodysystems,suchasSun/Earth/spacecraft.Lagrangepointsplayaspecialrole,justaspeaksandpassesdidforrailways:theyarethejunctionswheretubesmeet.L1isagreatplacetomakesmallcoursechanges,becausethenaturaldynamicsofaspacecraftnearL1ischaotic,Figure16.Chaoshasausefulfeature(seeChapter16):verysmallchangesinpositionorspeedcancreatelargechangestothetrajectory.Soitiseasytoredirectthespacecraftinafuel-efficient,thoughpossiblyslow,manner.

ThefirstpersontotakethisideaseriouslywastheGerman-bornmathematicianEdwardBelbruno,anorbitalanalystattheJetPropulsionLaboratoryfrom1985to1990.Herealisedthatchaoticdynamicsinmany-bodysystemsprovidedanopportunityfornovellow-energytransferorbits,namingthetechniquefuzzyboundarytheory.In1991heputhisideasintopractice.Hiten,aJapaneseprobe,hadbeensurveyingtheMoon,andhadcompleteditsintendedmission,returningtoorbittheEarth.BelbrunodesignedaneworbitthatwouldtakeitbacktotheMoondespitehavingprettymuchrunoutoffuel.AfterapproachingtheMoonasintended,HitenvisiteditsL4andL5pointstosearchforcosmicdustthatmighthavebeentrappedthere.

Asimilartrickwasusedin1985toredirectthealmost-deadInternationalSun–EarthExplorerISEE-3torendezvouswithcometGiacobini–Zinner,anditwasusedagainforNASA’sGenesismissiontobringbacksamplesofthesolarwind.Mathematiciansandengineerswantedtorepeatthetrick,andtofindothersofthesamekind,whichmeantfindingoutwhatreallymadeitwork.Itturnedouttobetubes.

Fig16ChaosnearJupiter.Thediagramshowsacross-sectionoforbits.Thenestedloopsarequasiperiodicorbitsandtheremainingstippledregionisachaoticorbit.Thetwothinloopscrossingeachotherattherightarecross-sectionsoftubes.

Theunderlyingideaissimplebutclever.Thosespecialplacesintheenergylandscapethatresemblemountainpassescreatebottlenecksthatwould-betravellerscannoteasilyavoid.Ancienthumansdiscovered,thehardway,thateventhoughittakesenergytoclimbapass,ittakesmoreenergytofollowanyotherroute–unlessyoucangoroundthemountaininatotallydifferentdirection.Thepassmakesthebestofabadchoice.

Intheenergylandscape,theanaloguesofpassesincludeLagrangepoints.Associatedwiththemareveryspecificinboundpaths,whicharelikethemostefficientwaytoclimbupthepass.Therearealsoequallyspecificoutboundpaths,analogoustothenaturalroutesdownfromthepass.Tofollowtheseinboundandoutboundpathsexactly,youhavetotravelatjusttherightspeed,butifyourspeedisslightlydifferentyoucanstillstaynearthosepaths.Inthelate1960sAmericanmathematiciansCharlesConleyandRichardMcGeheefollowedupBelbruno’spioneeringwork,pointingoutthateachsuchpathissurroundedbyanestedsetoftubes,oneinsidetheother.Eachtubecorrespondstoaparticularchoiceofspeed;thefurtherawayitisfromtheoptimalspeed,the

widerthetubeis.Onthesurfaceofanygiventube,thetotalenergyisconstant,buttheconstantsdifferfromonetubetoanother.Muchasacontourlineisataconstantheight,butthatheightisdifferentforeachcontour.

Thewaytoplananefficientmissionprofile,then,istoworkoutwhichtubesarerelevanttoyourchoiceofdestination.Thenyourouteyourspacecraftalongtheinsideofthefirstinboundtube,andwhenitgetstotheassociatedLagrangepointyoufireaquickburstonthemotorstoredirectitalongthemostsuitableoutboundtube,Figure17.Thattubenaturallyflowsintothecorrespondinginboundtubeofthenextswitchingpoint…andsoitgoes.

Fig17Left:TubesmeetingnearJupiter.Right:Close-upofregionwherethetubesjoin.

Plansforfuturetubularmissionsarealreadybeingdrawnup.In2000WangSangKoon,MartinLo,JerroldMarsden,andShaneRossusedthetubetechniquetofinda‘PetitGrandTour’ofthemoonsofJupiter,endingwithacaptureorbitroundEuropa,whichwasverytrickywithpreviousmethods.ThepathinvolvesagravitationalboostnearGanymedefollowedbyatubetriptoEuropa.Amorecomplexroute,requiringevenlessenergy,includesCallistoaswell.Itmakesuseofanotherfeatureoftheenergylandscape–resonances.Theseoccurwhen,say,twomoonsrepeatedlyreturntothesamerelativepositions,butonerevolvestwiceroundJupiterwhiletheotherrevolvesthreetimes.Anysmallnumberscanreplace2and3here.Thisrouteusesfive-bodydynamics:Jupiter,thethreemoons,andthespacecraft.

In2005,MichaelDellnitz,OliverJunge,MarcusPost,andBiancaThiereusedtubestoplananenergy-efficientmissionfromtheEarthtoVenus.The

maintubeherelinkstheSun/EarthL1pointtotheSun/VenusL2point.Asacomparison,thisrouteusesonlyonethirdofthefuelrequiredbytheEuropeanSpaceAgency’sVenusExpressmission,becauseitcanuselow-thrustengines;thepricepaidisalengtheningofthetransittimefrom150daystoabout650days.

Theinfluenceoftubesmaygofurther.Inunpublishedwork,DellnitzhasdiscoveredevidenceofanaturalsystemoftubesconnectingJupitertoeachoftheinnerplanets.Thisremarkablestructure,nowcalledtheInterplanetarySuperhighway,hintsthatJupiter,longknowntobethedominantplanetoftheSolarSystem,alsoplaystheroleofacelestialGrandCentralStation.ItstubesmaywellhaveorganisedtheformationoftheentireSolarSystem,determiningthespacingsoftheinnerplanets.

Whywerethetubesnotspottedsooner?Untilveryrecently,twovitalthingsweremissing.Onewaspowerfulcomputers,capableofcarryingoutthenecessarymany-bodycalculations.Theyarefartoocumbersomebyhand.Buttheother,evenmoreimportant,wasadeepmathematicalunderstandingofthegeographyoftheenergylandscape.Withoutthisimaginativetriumphofmodernmathematicalmethods,therewouldbenothingforthecomputerstocalculate.AndwithoutNewton’slawofgravity,themathematicalmethodswouldneverhavebeendevised.

5Portentoftheidealworld

TheSquareRootofMinusOne

Whatdoesitsay?

Eventhoughitoughttobeimpossible,thesquareofthenumberiisminusone.

Whyisthatimportant?

Itledtothecreationofcomplexnumbers,whichinturnledtocomplexanalysis,oneofthemostpowerfulareasofmathematics.

Whatdiditleadto?

Improvedmethodstocalculatetrigonometrictables.Generalisationsofalmostallmathematicstothecomplexrealm.Morepowerfulmethodstounderstandwaves,heat,electricity,andmagnetism.Themathematicalbasisofquantummechanics.

RenaissanceItalywasahotbedofpoliticsandviolence.Thenorthofthecountrywascontrolledbyadozenwarringcity-states,amongthemMilan,Florence,Pisa,Genoa,andVenice.Inthesouth,GuelphsandGibellineswereinconflictasPopesandHolyRomanEmperorsbattledforsupremacy.Bandsofmercenariesroamedtheland,villageswerelaidwaste,coastalcitieswagednavalwarfareagainsteachother.In1454Milan,Naples,andFlorencesignedtheTreatyofLodi,andpeacereignedforthenextfourdecades,butthepapacyremainedembroiledincorruptpolitics.ThiswasthetimeoftheBorgias,notoriousforpoisoninganyonewhogotinthewayoftheirquestforpoliticalandreligiouspower,butitwasalsothetimeofLeonardodaVinci,Brunelleschi,PierodellaFrancesca,Titian,andTintoretto.Againstabackdropofintrigueandmurder,long-heldassumptionswerecomingintoquestion.Greatartandgreatscienceflourishedinsymbiosis,eachfeedingofftheother.

Greatmathematicsflourishedaswell.In1545thegamblingscholarGirolamoCardanowaswritinganalgebratext,andheencounteredanewkindofnumber,onesobafflingthathedeclaredit‘assubtleasitisuseless’anddismissedthenotion.RafaelBombellihadasolidgraspofCardano’salgebrabook,buthefoundtheexpositionconfusing,anddecidedhecoulddobetter.By1572hehadnoticedsomethingintriguing:althoughthesebafflingnewnumbersmadenosense,theycouldbeusedinalgebraiccalculationsandledtoresultsthatweredemonstrablycorrect.

Forcenturiesmathematiciansengagedinalove–haterelationshipwiththese‘imaginarynumbers’,astheyarestillcalledtoday.Thenamebetraysanambivalentattitude:they’renotrealnumbers,theusualnumbersencounteredinarithmetic,butinmostrespectstheybehavelikethem.Themaindifferenceisthatwhenyousquareanimaginarynumber,theresultisnegative.Butthatoughtnottobepossible,becausesquaresarealwayspositive.

Onlyintheeighteenthcenturydidmathematiciansfigureoutwhatimaginarynumberswere.Onlyinthenineteenthdidtheystarttofeelcomfortablewiththem.Butbythetimethelogicalstatusofimaginarynumberswasseentobeentirelycomparabletothatofthemoretraditionalrealnumbers,imaginarieshadbecomeindispensablethroughoutmathematicsandscience,andthequestionoftheirmeaninghardlyseemedinterestinganymore.Inthelatenineteenthandearlytwentiethcenturies,revivedinterestinthefoundationsofmathematicsledtoarethinkoftheconceptofnumber,andtraditional‘real’numberswereseen

tobenomorerealthanimaginaryones.Logically,thetwokindsofnumberwereasalikeasTweedledumandTweedledee.Bothwereconstructsofthehumanmind,bothrepresented–butwerenotsynonymouswith–aspectsofnature.Buttheyrepresentedrealityindifferentwaysandindifferentcontexts.

Bythesecondhalfofthetwentiethcentury,imaginarynumbersweresimplypartandparcelofeverymathematician’sandeveryscientist’smentaltoolkit.TheywerebuiltintoquantummechanicsinsuchafundamentalwaythatyoucouldnomoredophysicswithoutthemthanyoucouldscalethenorthfaceoftheEigerwithoutropes.Evenso,imaginarynumbersareseldomtaughtinschools.Thesumsareeasyenough,butthementalsophisticationneededtoappreciatewhyimaginariesareworthstudyingisstilltoogreatforthevastmajorityofstudents.Veryfewadults,eveneducatedones,areawareofhowdeeplytheirsocietydependsonnumbersthatdonotrepresentquantities,lengths,areas,oramountsofmoney.Yetmostmoderntechnology,fromelectriclightingtodigitalcameras,couldnothavebeeninventedwithoutthem.

Letmebacktracktoacrucialquestion.Whyaresquaresalwayspositive?InRenaissancetimes,whereequationsweregenerallyrearrangedtomake

everynumberinthempositive,theywouldn’thavephrasedthequestionquitethisway.Theywouldhavesaidthatifyouaddanumbertoasquarethenyouhavetogetabiggernumber–youcan’tgetzero.Butevenifyouallownegativenumbers,aswenowdo,squaresstillhavetobepositive.Here’swhy.

Realnumberscanbepositiveornegative.However,thesquareofanyrealnumber,whateveritssign,isalwayspositive,becausetheproductoftwonegativenumbersispositive.Soboth3×3and−3×−3yieldthesameresult:9.Therefore9hastwosquareroots,3and−3.

Whatabout—9?Whatareitssquareroots?Itdoesn’thaveany.Itallseemsterriblyunfair:thepositivenumbershogtwosquarerootseach,

whilethenegativenumbersgowithout.Itistemptingtochangetheruleformultiplyingtwonegativenumbers,sothat,say,−3×−3=−9.Thenpositiveandnegativenumberseachgetonesquareroot;moreover,thishasthesamesignasitssquare,whichseemsneatandtidy.Butthisseductivelineofreasoninghasanunintendeddownside:itwreckstheusualrulesofarithmetic.Theproblemisthat—9alreadyoccursas3×−3itselfaconsequenceoftheusualrulesofarithmetic,andafactthatalmosteveryoneishappytoaccept.Ifweinsistthat−3×−3isalso9,then−3×−3=3×−3.Thereareseveralwaystoseethatthis

causesproblems;thesimplestistodividebothsidesby−3,toget3=−3.Ofcourseyoucanchangetherulesofarithmetic.Butnowitallgets

complicatedandmessy.Amorecreativesolutionistoretaintherulesofarithmetic,andtoextendthesystemofrealnumbersbypermittingimaginaries.Remarkably–andnoonecouldhaveanticipatedthis,youjusthavetofollowthelogicthrough–thisboldstepleadstoabeautiful,consistentsystemofnumbers,withamyriaduses.Nowallnumbersexcept0havetwosquareroots,onebeingminustheother.Thisistrueevenforthenewkindsofnumber;oneenlargementofthesystemsuffices.Ittookawhileforthistobecomeclear,butinretrospectithasanairofinevitability.Imaginarynumbers,impossiblethoughtheywere,refusedtogoaway.Theyseemedtomakenosense,buttheykeptcroppingupincalculations.Sometimestheuseofimaginarynumbersmadethecalculationssimpler,andtheresultwasmorecomprehensiveandmoresatisfactory.Wheneverananswerthathadbeenobtainedusingimaginarynumbers,butdidnotexplicitlyinvolvethem,couldbeverifiedindependently,itturnedouttoberight.Butwhentheanswerdidinvolveexplicitimaginarynumbersitseemedtobemeaningless,andoftenlogicallycontradictory.Theenigmasimmeredfortwohundredyears,andwhenitfinallyboiledover,theresultswereexplosive.

Cardanoisknownasthegamblingscholarbecausebothactivitiesplayedaprominentroleinhislife.Hewasbothgeniusandrogue.Hislifeconsistsofabewilderingseriesofveryhighhighsandverylowlows.Hismothertriedtoaborthim,hissonwasbeheadedforkillinghis(theson’s)wife,andhe(Cardano)gambledawaythefamilyfortune.HewasaccusedofheresyforcastingthehoroscopeofJesus.YetinbetweenhealsobecameRectoroftheUniversityofPadua,waselectedtotheCollegeofPhysiciansinMilan,gained2000goldcrownsforcuringtheArchbishopofStAndrews’asthma,andreceivedapensionfromPopeGregoryXIII.Heinventedthecombinationlockandgimbalstoholdagyroscope,andhewroteanumberofbooks,includinganextraordinaryautobiographyDeVitaPropria(‘TheBookofMyLife’).ThebookthatisrelevanttoourtaleistheArsMagnaof1545.Thetitlemeans‘greatart’,andreferstoalgebra.Init,Cardanoassembledthemostadvancedalgebraicideasofhisday,includingnewanddramaticmethodsforsolvingequations,someinventedbyastudentofhis,someobtainedfromothersincontroversialcircumstances.

Algebra,initsfamiliarsensefromschoolmathematics,isasystemforrepresentingnumberssymbolically.ItsrootsgobacktotheGreekDiophantusaround250AD,whoseArithmeticaemployedsymbolstodescribewaystosolve

equations.Mostoftheworkwasverbal–‘findtwonumberswhosesumis10andwhoseproductis24’.ButDiophantussummarisedthemethodsheusedtofindthesolutions(here4and6)symbolically.Thesymbols(seeTable1)wereverydifferentfromthoseweusetoday,andmostwereabbreviations,butitwasastart.Cardanomainlyusedwords,withafewsymbolsforroots,andagainthesymbolsscarcelyresemblethoseincurrentuse.Laterauthorshomedin,ratherhaphazardly,ontoday’snotation,mostofwhichwasstandardisedbyEulerinhisnumeroustextbooks.However,Gaussstillusedxxinsteadofx2aslateas1800.

Table1Thedevelopmentofalgebraicnotation.

ThemostimportanttopicsintheArsMagnawerenewmethodsforsolvingcubicandquarticequations.Thesearelikequadraticequations,whichmostofusmeetinschoolalgebra,butmorecomplicated.Aquadraticequationstatesarelationshipinvolvinganunknownquantity,normallysymbolisedbytheletterx,anditssquarex2.‘Quadratic’comesfromtheLatinfor‘square’.Atypicalexampleis

x2−5x+6=0

Verbally,thissays:‘Squaretheunknown,subtract5timestheunknown,andadd6:theresultiszero.’Givenanequationinvolvinganunknown,ourtaskistosolvetheequation–tofindthevalueorvaluesoftheunknownthatmaketheequationcorrect.

Forarandomlychosenvalueofx,thisequationwillusuallybefalse.Forexample,ifwetryx=1,thenx2−5x+6=1−5+6=2,whichisn’tzero.Butforrarechoicesofx,theequationistrue.Forexample,whenx=2wehavex2–

5x+6=4−10+6=0.Butthisisnottheonlysolution!Whenx=3wehavex2−5x+6=9−15+6=0aswell.Therearetwosolutions,x=2andx=3,anditcanbeshownthattherearenoothers.Aquadraticequationcanhavetwosolutions,one,ornone(inrealnumbers).Forexample,x2−2x+1=0hasonlythesolutionx=1,andx2+1=0hasnosolutionsinrealnumbers.

Cardano’smasterworkprovidesmethodsforsolvingcubicequations,whichalongwithxandx2alsoinvolvethecubex3oftheunknown,andquarticequations,wherex4turnsupaswell.Thealgebragetsverycomplicated;evenwithmodernsymbolismittakesapageortwotoderivetheanswers.Cardanodidnotgoontoquinticequations,involvingx5,becausehedidnotknowhowtosolvethem.Muchlateritwasprovedthatnosolutions(ofthetypeCardanowouldhavewanted)exist:althoughhighlyaccuratenumericalsolutionscanbecalculatedinanyparticularcase,thereisnogeneralformulaforthem,unlessyouinventnewsymbolsspecificallyforthetask.

I’mgoingtowritedownafewalgebraicformulas,becauseIthinkthetopicmakesmoresenseifwedon’ttrytoavoidthem.Youdon’tneedtofollowthedetails,butI’dliketoshowyouwhateverythinglookslike.Usingmodernsymbols,wecanwriteoutCardano’ssolutionofthecubicequationinaspecialcase,whenx3+ax+b=0forspecificnumbersaandb.(Ifx2ispresent,acunningtrickgetsridofit,sothiscaseactuallydealswitheverything.)Theansweris:

Thismayappearabitofamouthful,butit’salotsimplerthanmanyalgebraicformulas.Ittellsushowtocalculatetheunknownxbyworkingoutthesquareofbandthecubeofa,addingafewfractions,andtakingacoupleofsquareroots(the symbol)andacoupleofcuberoots(the symbol).Thecuberootofanumberiswhateveryouhavetocubetogetthatnumber.

Thediscoveryofthesolutionforcubicequationsinvolvesatleastthreeothermathematicians,oneofwhomcomplainedbitterlythatCardanohadpromisednottorevealhissecret.Thestory,thoughfascinating,istoocomplicatedtorelatehere.1ThequarticwassolvedbyCardano’sstudentLodovicoFerrari.I’llspareyoutheevenmorecomplicatedformulaforquarticequations.

TheresultsreportedintheArsMagnawereamathematicaltriumph,the

culminationofastorythatspannedmillennia.TheBabyloniansknewhowtosolvequadraticequationsaround1500BC,perhapsearlier.TheancientGreeksandOmarKhayyamknewgeometricmethodsforsolvingcubics,butalgebraicsolutionsofcubicequations,letalonequartics,wereunprecedented.Atastroke,mathematicsoutstrippeditsclassicalorigins.

Therewasonetinysnag,however.Cardanonoticedit,andseveralpeopletriedtoexplainit;theyallfailed.Sometimesthemethodworksbrilliantly;atothertimes,theformulaisasenigmaticastheDelphicoracle.SupposeweapplyCardano’sformulatotheequationx3−15x−4=0.Theresultis

However,−121isnegative,soithasnosquareroot.Tocompoundthemystery,thereisaperfectlygoodsolution,x=4.Theformuladoesn’tgiveit.

Lightofakindwasshedin1572whenBombellipublishedL’Algebra.HismainaimwastoclarifyCardano’sbook,butwhenhecametothisparticularthornyissuehespottedsomethingCardanohadmissed.Ifyouignorewhatthesymbolsmean,andjustperformroutinecalculations,thestandardrulesofalgebrashowthat

Thereforeyouareentitledtowrite

Similarly,

NowtheformulathatbaffledCardanocanberewrittenas

whichisequalto4becausethetroublesomesquarerootscancelout.SoBombelli’snonsensicalformalcalculationsgottherightanswer.Andthatwasaperfectlynormalrealnumber.

Somehow,pretendingthatsquarerootsofnegativenumbersmadesense,even

thoughtheyobviouslydidnot,couldleadtosensibleanswers.Why?

Toanswerthisquestion,mathematicianshadtodevelopgoodwaystothinkaboutsquarerootsofnegativequantities,anddocalculationswiththem.Earlywriters,amongthemDescartesandNewton,interpretedthese‘imaginary’numbersasasignthataproblemhasnosolutions.Ifyouwantedtofindanumberwhosesquarewasminusone,theformalsolution‘squarerootofminusone’wasimaginary,sonosolutionexisted.ButBombelli’scalculationimpliedthattherewasmoretoimaginariesthanthat.Theycouldbeusedtofindsolutions;theycouldariseaspartofthecalculationofsolutionsthatdidexist.

Leibnizhadnodoubtabouttheimportanceofimaginarynumbers.In1702hewrote:‘TheDivineSpiritfoundasublimeoutletinthatwonderofanalysis,thatportentoftheidealworld,thatamphibianbetweenbeingandnon-being,whichwecalltheimaginaryrootofnegativeunity.’Buttheeloquenceofhisstatementfailstoobscureafundamentalproblem:hedidn’thaveacluewhatimaginarynumbersactuallywere.

OneofthefirstpeopletocomeupwithasensiblerepresentationofcomplexnumberswasWallis.Theimageofrealnumberslyingalongaline,likemarkedpointsonaruler,wasalreadycommonplace.In1673Wallissuggestedthatacomplexnumberx+iyshouldbethoughtofasapointinaplane.Drawalineintheplane,andidentifypointsonthislinewithrealnumbersintheusualway.Thenthinkofx+iyasapointlyingtoonesideoftheline,distanceyawayfromthepointx.

Wallis’sideawaslargelyignored,orworse,criticised.FrançoisDavietdeFoncenex,writingaboutimaginariesin1758,saidthatthinkingofimaginariesasformingalineatrightanglestothereallinewaspointless.Buteventuallytheideawasrevivedinaslightlymoreexplicitform.Infact,threepeoplecameupwithexactlythesamemethodforrepresentingcomplexnumbers,atintervalsofafewyears,Figure18.OnewasaNorwegiansurveyor,oneaFrenchmathematician,andoneaGermanmathematician.Respectively,theywereCasparWessel,whopublishedin1797,Jean-RobertArgandin1806,andGaussin1811.TheybasicallysaidthesameasWallis,buttheyaddedasecondlinetothepicture,animaginaryaxisatrightanglestotherealone.Alongthissecondaxislivedtheimaginarynumbersi,2i,3i,andsoon.Ageneralcomplexnumber,suchas3+2i,livedoutintheplane,threeunitsalongtherealaxisandtwoalongtheimaginaryone.

Fig18Thecomplexplane.Left:accordingtoWallis.Right:accordingtoWessel,Argand,andGauss.

Thisgeometricrepresentationwasallverywell,butitdidn’texplainwhycomplexnumbersformalogicallyconsistentsystem.Itdidn’ttellusinwhatsensetheyarenumbers.Itjustprovidedawaytovisualisethem.Thisnomoredefinedwhatacomplexnumberisthanadrawingofastraightlinedefinesarealnumber.Itdidprovidesomesortofpsychologicalprop,aslightlyartificiallinkbetweenthosecrazyimaginariesandtherealworld,butnothingmore.

Whatconvincedmathematiciansthattheyshouldtakeimaginarynumbersseriouslywasn’talogicaldescriptionofwhattheywere.Itwasoverwhelmingevidencethatwhatevertheywere,mathematicscouldmakegooduseofthem.Youdon’taskdifficultquestionsaboutthephilosophicalbasisofanideawhenyouareusingiteverydaytosolveproblemsandyoucanseethatitgivestherightanswers.Foundationalquestionsstillhavesomeinterest,ofcourse,buttheytakeabackseattothepragmaticissuesofusingthenewideatosolveoldandnewproblems.

Imaginarynumbers,andthesystemofcomplexnumbersthattheyspawned,cementedtheirplaceinmathematicswhenafewpioneersturnedtheirattentiontocomplexanalysis:calculus(Chapter3)butwithcomplexnumbersinsteadofrealones.Thefirststepwastoextendalltheusualfunctions–powers,logarithms,exponentials,trigonometricfunctions–tothecomplexrealm.Whatissinzwhenz=x+iyiscomplex?Whatisezorlogz?

Logically,thesethingscanbewhateverwewish.Weareoperatinginanewdomainwheretheoldideasdon’tapply.Itdoesn’tmakemuchsense,forinstance,tothinkofaright-angledtrianglewhosesideshavecomplexlengths,sothegeometricdefinitionofthesinefunctionisirrelevant.Wecouldtakeadeep

breath,insistthatsinzhasitsusualvaluewhenzisreal,butequals42wheneverzisn’treal:jobdone.Butthatwouldbeaprettysillydefinition:notbecauseit’simprecise,butbecauseitbearsnosensiblerelationshiptotheoriginaloneforrealnumbers.Onerequirementforanextendeddefinitionmustbethatitagreeswiththeoldonewhenappliedtorealnumbers,butthat’snotenough.It’strueformysillyextensionofthesine.Anotherrequirementisthatthenewconceptshouldretainasmanyfeaturesoftheoldoneaswecanmanage;itshouldsomehowbe‘natural’.

Whatpropertiesofsineandcosinedowewanttopreserve?Presumablywe’dlikealltheprettyformulasoftrigonometrytoremainvalid,suchassin2z=2sinzcosz.Thisimposesaconstraintbutdoesn’thelp.Amoreinterestingproperty,derivedusinganalysis(therigorousformulationofcalculus),istheexistenceofaninfiniteseries:

(Thesumofsuchaseriesisdefinedtobethelimitofthesumoffinitelymanytermsasthenumberoftermsincreasesindefinitely.)Thereisasimilarseriesforthecosine:

andthetwoareobviouslyrelatedinsomewaytotheseriesfortheexponential:

Theseseriesmayseemcomplicated,buttheyhaveanattractivefeature:weknowhowtomakesenseofthemforcomplexnumbers.Alltheyinvolveisintegerpowers(whichweobtainbyrepeatedmultiplication)andatechnicalissueofconvergence(makingsenseoftheinfinitesum).Bothoftheseextendnaturallyintothecomplexrealmandhavealloftheexpectedproperties.Sowecandefinesinesandcosinesofcomplexnumbersusingthesameseriesthatworkintherealcase.

Sincealloftheusualformulasintrigonometryareconsequencesoftheseseries,thoseformulasautomaticallycarryoveraswell.Sodothebasicfactsofcalculus,suchas‘thederivativeofsineiscosine’.Sodoesez+w=ezew.Thisis

allsopleasantthatmathematicianswerehappytosettleontheseriesdefinitions.Andoncethey’ddonethat,agreatdealelsenecessarilyhadtofitinwithit.Ifyoufollowedyournose,youcoulddiscoverwhereitled.

Forexample,thosethreeserieslookverysimilar.Indeed,ifyoureplacezbyizintheseriesfortheexponential,youcansplittheresultingseriesintotwoparts,andwhatyougetarepreciselytheseriesforsineandcosine.Sotheseriesdefinitionsimplythat

eiz=cosz+isinz:

Youcanalsoexpressbothsineandcosineusingexponentials:

Thishiddenrelationshipisextraordinarilybeautiful.Butyou’dneversuspectanythinglikeitcouldexistifyouremainedstuckintherealmofthereals.Curioussimilaritiesbetweentrigonometricformulasandexponentialones(forexample,theirinfiniteseries)wouldremainjustthat.Viewedthroughcomplexspectacles,everythingsuddenlyslotsintoplace.

Oneofthemostbeautiful,yetenigmatic,equationsinthewholeofmathematicsemergesalmostbyaccident.Inthetrigonometricseries,thenumberz(whenreal)hastobemeasuredinradians,forwhichafullcircleof360°becomes2πradians.Inparticular,theangle180°isπradians.Moreover,sinπ=0andcosπ=1.Therefore

eiπ=cosπ+isinπ=−1

Theimaginarynumberiunitesthetwomostremarkablenumbersinmathematics,eandπ,inasingleelegantequation.Ifyou’veneverseenthisbefore,andhaveanymathematicalsensitivity,thehairsonyourneckraiseandpricklesrundownyourspine.Thisequation,attributedtoEuler,regularlycomestopofthelistinpollsforthemostbeautifulequationinmathematics.Thatdoesn’tmeanthatitisthemostbeautifulequation,butitdoesshowhowmuchmathematiciansappreciateit.Armedwithcomplexfunctionsandknowingtheirproperties,themathematiciansofthenineteenthcenturydiscoveredsomethingremarkable:theycouldusethesethingstosolvedifferentialequationsinmathematicalphysics.Theycouldapplythemethodtostaticelectricity,magnetism,andfluidflow.Notonlythat:itwaseasy.

InChapter3wetalkedoffunctions–mathematicalrulesthatassign,toanygivennumber,acorrespondingnumber,suchasitssquareorsine.Complexfunctionsaredefinedinthesameway,butnowweallowthenumbersinvolvedtobecomplex.Themethodforsolvingdifferentialequationswasdelightfullysimple.Allyouhadtodowastakesomecomplexfunction,callitf(z),andsplititintoitsrealandimaginaryparts:

f(z)=u(z)+iv(z)

Nowyouhavetworeal-valuedfunctionsuandv,definedforanyzinthecomplexplane.Moreover,whateverfunctionyoustartwith,thesetwocomponentfunctionssatisfydifferentialequationsfoundinphysics.Inafluid-flowinterpretation,forexample,uandvdeterminetheflow-lines.Inanelectrostaticinterpretation,thetwocomponentsdeterminetheelectricfieldandhowasmallchargedparticlewouldmove;inamagneticinterpretation,theydeterminethemagneticfieldandthelinesofforce.

I’llgivejustoneexample:abarmagnet.Mostofusrememberseeingafamousexperimentinwhichamagnetisplacedbeneathasheetofpaper,andironfilingsarescatteredoverthepaper.Theyautomaticallylineuptoshowthelinesofmagneticforceassociatedwiththemagnet–thepathsthatatinytestmagnetwouldfollowifplacedinthemagneticfield.ThecurveslooklikeFigure19(left).

Fig19Left:Magneticfieldofbarmagnet.Right:Fieldderivedusingcomplexanalysis.

Toobtainthispictureusingcomplexfunctions,wejustletf(z)=1/z.Thelinesofforceturnouttobecircles,tangenttotherealaxis,asinFigure19(right).Thisiswhatthemagneticfieldslinesofaverytinybarmagnetwouldlooklike.Amorecomplicatedchoiceoffunctioncorrespondstoamagnetoffinitesize:I

chosethisfunctiontokeepeverythingassimpleaspossible.Thiswaswonderful.Therewereendlessfunctionstoworkwith.Youdecided

whichfunctiontolookat,founditsrealandimaginaryparts,workedouttheirgeometry…and,loandbehold,youhadsolvedaprobleminmagnetism,orelectricity,orfluidflow.Experiencesoontoldyouwhichfunctiontouseforwhichproblem.Thelogarithmwasapointsource,minusthelogarithmwasasinkthroughwhichfluiddisappearedliketheplugholeinakitchensink,itimesthelogarithmwasapointvortexwherethefluidspunroundandround…Itwasmagic!Herewasamethodthatcouldchurnoutsolutionaftersolutiontoproblemsthatwouldotherwisebeopaque.Yetitcamewithaguaranteeofsuccess,andifyouwereworriedaboutallthatcomplexanalysisstuff,youcouldcheckdirectlythattheresultsyouobtainedreallydidrepresentsolutions.

Thiswasjustthebeginning.Aswellasspecialsolutions,youcouldprovegeneralprinciples,hiddenpatternsinthephysicallaws.Youcouldanalysewavesandsolvedifferentialequations.Youcouldtransformshapesintoothershapes,usingcomplexequations,andthesameequationstransformedtheflow-linesroundthem.Themethodwaslimitedtosystemsintheplane,becausethatwaswhereacomplexnumbernaturallylived,butthemethodwasagodsendwhenpreviouslyevenproblemsintheplanewereoutofreach.Today,everyengineeristaughthowtousecomplexanalysistosolvepracticalproblems,earlyintheiruniversitycourse.TheJoukowskitransformationz+1/zturnsacircleintoanaerofoilshape,thecross-sectionofarudimentaryaeroplanewing,seeFigure20.Itthereforeturnstheflowpastacircle,easytofindifyouknewthetricksofthetrade,intotheflowpastanaerofoil.Thiscalculation,andmorerealisticimprovements,wereimportantintheearlydaysofaerodynamicsandaircraftdesign.

Thiswealthofpracticalexperiencemadethefoundationalissuesmoot.Whylookagifthorseinthemouth?Therehadtobeasensiblemeaningforcomplexnumbers–theywouldn’tworkotherwise.Mostscientistsandmathematiciansweremuchmoreinterestedindiggingoutthegoldthantheywereinestablishingexactlywhereithadcomefromandwhatdistinguisheditfromfools’gold.Butafewpersisted.Eventually,theIrishmathematicianWilliamRowanHamiltonknockedthewholethingonthehead.HetookthegeometricrepresentationproposedbyWessel,Argand,andGauss,andexpresseditincoordinates.Acomplexnumberwasapairofrealnumbers(x,y).Therealnumberswerethoseoftheform(x,0).Theimaginaryiwas(0,1).Thereweresimpleformulasforaddingandmultiplyingthesepairs.Ifyouwereworriedaboutsomelawof

algebra,suchasthecommutativelawab=ba,youcouldroutinelyworkoutbothsidesaspairs,andmakesuretheywerethesame.(Theywere.)Ifyouidentified(x,0)withplainx,youembeddedtherealnumbersintothecomplexones.Betterstill,x+iythenworkedoutasthepair(x,y).

Fig20FlowpastawingderivedfromtheJoukowskitransformation.

Thiswasn’tjustarepresentation,butadefinition.Acomplexnumber,saidHamilton,isnothingmorenorlessthanapairofordinaryrealnumbers.Whatmadethemsousefulwasaninspiredchoiceoftherulesforaddingandmultiplyingthem.Whattheyactuallywerewastrite;itwashowyouusedthemthatproducedthemagic.Withthissimplestrokeofgenius,Hamiltoncutthroughcenturiesofheatedargumentandphilosophicaldebate.Butbythen,mathematicianshadbecomesousedtoworkingwithcomplexnumbersandfunctionsthatnoonecaredanymore.Allyouneededtorememberwasthati2=−1.

6Muchadoaboutknotting

Euler’sFormulaforPolyhedra

Whatdoesitsay?

Thenumbersoffaces,edges,andverticesofasolidarenotindependent,butarerelatedinasimplemanner.

Whyisthatimportant?

Itdistinguishesbetweensolidswithdifferenttopologiesusingtheearliestexampleofatopologicalinvariant.Thispavedthewaytomoregeneralandmorepowerfultechniques,creatinganewbranchofmathematics.

Whatdiditleadto?

Oneofthemostimportantandpowerfulareasofpuremathematics:topology,whichstudiesgeometricpropertiesthatareunchangedbycontinuousdeformations.Examplesincludesurfaces,knots,andlinks.Mostapplicationsareindirect,butitsinfluencebehindthescenesisvital.IthelpsusunderstandhowenzymesactonDNAinacell,andwhythemotionofcelestialbodiescanbechaotic.

Asthenineteenthcenturyapproacheditsend,mathematiciansbegantodevelopanewkindofgeometry,oneinwhichfamiliarconceptssuchaslengthsandanglesplayednorolewhatsoeverandnodistinctionwasmadebetweentriangles,squares,andcircles.Initiallyitwascalledanalysissitus,theanalysisofposition,butmathematiciansquicklysettledonanothername:topology.

TopologyhasitsrootsinacuriousnumericalpatternthatDescartesnoticedin1639whenthinkingaboutEuclid’sfiveregularsolids.DescarteswasaFrench-bornpolymathwhospentmostofhislifeintheDutchRepublic,present-dayNetherlands.Hisfamemainlyrestsonhisphilosophy,whichprovedsoinfluentialthatforalongtimeWesternphilosophyconsistedlargelyofresponsestoDescartes.Notalwaysinagreement,youappreciate,butmotivatedbyhisargumentsnonetheless.Hissoundbitecogitoergosum–‘Ithink,thereforeIam’–hasbecomecommonculturalcurrency.ButDescartes’sinterestsextendedbeyondphilosophyintoscienceandmathematics.

In1639Descartesturnedhisattentiontotheregularsolids,andthiswaswhenhenoticedhiscuriousnumericalpattern.Acubehas6faces,12edges,and8vertices;thesum6–12+8equals2.Adodecahedronhas12faces,30edges,and20vertices;thesum12–30+20=2.Anicosahedronhas20faces,30edges,and12vertices;thesum20–30+12=2.Thesamerelationshipholdsforthetetrahedronandoctahedron.Infact,itappliestoasolidofanyshape,regularornot.IfthesolidhasFfaces,Eedges,andVvertices,thenF–E+V=2.Descartesviewedthisformulaasaminorcuriosityanddidnotpublishit.Onlymuchlaterdidmathematiciansseethissimplelittleequationasoneofthefirsttentativestepstowardsthegreatsuccessstoryintwentieth-centurymathematics,theinexorableriseoftopology.Inthenineteenthcentury,thethreepillarsofpuremathematicswerealgebra,analysis,andgeometry.Bytheendofthetwentieth,theywerealgebra,analysis,andtopology.Topologyisoftencharacterisedas‘rubber-sheetgeometry’becauseitisthekindofgeometrythatwouldbeappropriateforfiguresdrawnonasheetofelastic,sothatlinescanbend,shrink,orstretch,andcirclescanbesquashedsothattheyturnintotrianglesorsquares.Allthatmattersiscontinuity:youarenotallowedtoripthesheetapart.Itmayseemremarkablethatanythingsoweirdcouldhaveanyimportance,butcontinuityisabasicaspectofthenaturalworldandafundamentalfeatureofmathematics.Todaywemostlyusetopologyindirectly,asonemathematicaltechniqueamongmany.Youdon’tfindanythingobviously

topologicalinyourkitchen.However,aJapanesecompanydidmarketachaoticdishwasher,whichaccordingtotheirmarketingpeoplecleaneddishesmoreefficiently,andourunderstandingofchaosrestsontopology.SodosomeimportantaspectsofquantumfieldtheoryandthaticonicmoleculeDNA.But,whenDescartescountedthemostobviousfeaturesoftheregularsolidsandnoticedthattheywerenotindependent,allthiswasfarinthefuture.

ItwaslefttotheindefatigableEuler,themostprolificmathematicianinhistory,toproveandpublishthisrelationship,whichhedidin1750and1751.I’llsketchamodernversion.TheexpressionF–E+Vmayseemfairlyarbitrary,butithasaveryinterestingstructure.Faces(F)arepolygons,ofdimension2,edges(E)arelines,sohavedimension1,andvertices(V)arepoints,ofdimension0.Thesignsintheexpressionalternate,+–+,with+beingassignedtofeaturesofevendimensionand–tothoseofodddimension.Thisimpliesthatyoucansimplifyasolidbymergingitsfacesorremovingedgesandvertices,andthesechangeswillnotalterthenumberF–E+Vprovidedthateverytimeyougetridofafaceyoualsoremoveanedge,oreverytimeyougetridofavertexyoualsoremoveanedge.Thealternatingsignsmeanthatchangesofthiskindcancelout.

NowI’llexplainhowthiscleverstructuremakestheproofwork.Figure21showsthekeystages.Takeyoursolid.Deformitintoaniceroundsphere,withitsedgesbeingcurvesonthatsphere.Iftwofacesmeetalongacommonedge,thenyoucanremovethatedgeandmergethefacesintoone.SincethismergerreducesbothFandEby1,itdoesn’tchangeF–E+V.Keepdoingthisuntilyougetdowntoasingleface,whichcoversalmostallofthesphere.Asidefromthisface,youareleftwithonlyedgesandvertices.Thesemustformatree,anetworkwithnoclosedloops,becauseanyclosedlooponasphereseparatesatleasttwofaces:oneinsideit,theotheroutsideit.Thebranchesofthistreearetheremainingedgesofthesolid,andtheyjointogetherattheremainingvertices.Atthisstageonlyonefaceremains:theentiresphere,minusthetree.Somebranchesofthistreeconnecttootherbranchesatbothends,butsome,attheextremes,terminateinavertex,towhichnootherbranchesattach.Ifyouremoveoneoftheseterminatingbranchestogetherwiththatvertex,thenthetreegetssmaller,butsincebothEandVdecreaseby1,F–E+Vagainremainsunchanged.

Thisprocesscontinuesuntilyouareleftwithasinglevertexsittingonanotherwisefeaturelesssphere.NowV=1,E=0,andF=1.SoF–E+V=1–0+1=2.ButsinceeachstepleavesF–E+Vunchanged,itsvalueatthebeginningmustalsohavebeen2,whichiswhatwewanttoprove.

Fig21Keystagesinsimplifyingasolid.Lefttoright:(1)Start.(2)Mergingadjacentfaces.(3)Treethatremainswhenallfaceshavebeenmerged.(4)Removinganedgeandavertexfromthetree.(5)End.

It’sacunningidea,anditcontainsthegermofafar-reachingprinciple.Theproofhastwoingredients.Oneisasimplificationprocess:removeeitherafaceandanadjacentedgeoravertexandanedgethatmeetsit.Theotherisaninvariant,amathematicalexpressionthatremainsunchangedwheneveryoucarryoutastepinthesimplificationprocess.Wheneverthesetwoingredientscoexist,youcancomputethevalueoftheinvariantforanyinitialobjectbysimplifyingitasfarasyoucan,andthencomputingthevalueoftheinvariantforthissimplifiedversion.Becauseitisaninvariant,thetwovaluesmustbeequal.Becausetheendresultissimple,theinvariantiseasytocalculate.

NowIhavetoadmitthatI’vebeenkeepingonetechnicalissueupmysleeve.Descartes’sformuladoesnot,infact,applytoanysolid.Themostfamiliarsolidforwhichitfailsisapictureframe.Thinkofapictureframemadefromfourlengthsofwood,eachrectangularincross-section,joinedatthefourcornersby45°mitresasinFigure22(left).Eachlengthofwoodcontributes4faces,soF=16.Eachlengthalsocontributes4edges,butthemitrejointcreates4moreateachcorner,soE=32.Eachcornercomprises4vertices,soV=16.ThereforeF–E+V=0.

Whatwentwrong?

Fig22Left:ApictureframewithF–E+V=0.Right:Finalconfigurationwhenthepictureframeissmoothedandthensimplified.

There’snoproblemwithF–E+Vbeinginvariant.Neitheristheremuchofaproblemwiththesimplificationprocess.Butifyouworkthroughitfortheframe,alwayscancellingonefaceagainstoneedge,oronevertexagainstoneedge,thenthefinalsimplifiedconfigurationisnotasinglevertexsittinginasingleface.Performingthecancellationinthemostobviousway,whatyougetisFigure22(right),withF=1,V=1,E=2.I’vesmoothedthefacesandedgesforreasonsthatwillquicklybecomeapparent.Atthisstageremovinganedgejustmergesthesoleremainingfacewithitself,sothechangestothenumbersnolongercancel.Thisiswhywestop,butwe’rehomeanddryanyway:forthisconfiguration,F–E+V=0.Sothemethodperformsperfectly.Itjustyieldsadifferentresultforthepictureframe.Theremustbesomefundamentaldifferencebetweenapictureframeandacube,andtheinvariantF–E+Vispickingitup.

Thedifferenceturnsouttobeatopologicalone.EarlyinmyversionofEuler’sproof,Itoldyoutotakethesolidand‘deformitintoaniceroundsphere’.Butthisisnotpossibleforthepictureframe.It’snotshapedlikeasphere,evenafterbeingsimplified.Itisatorus,whichlookslikeaninflatablerubberringwithaholethroughthemiddle.Theholeisalsoclearlyvisibleintheoriginalshape:it’swherethepicturewouldgo.Asphere,incontrast,hasnoholes.Theholeintheframeiswhythesimplificationprocessleadstoadifferentresult.However,wecanwrestvictoryfromthejawsofdefeat,becauseF–E+Visstillaninvariant.SotheprooftellsusthatanysolidthatisdeformableintoatoruswillsatisfytheslightlydifferentequationF–E+V=0.Inconsequence,wehavethebasisofarigorousproofthatatoruscannotbedeformedintoasphere:thatis,thetwosurfacesaretopologicallydifferent.

Ofcoursethisisintuitivelyobvious,butnowwecansupportintuitionwithlogic.JustasEuclidstartedfromobviouspropertiesofpointsandlines,andformalisedthemintoarigoroustheoryofgeometry,themathematiciansofthenineteenthandtwentiethcenturiescouldnowdeveloparigorousformaltheoryoftopology.

Fig23Left:2-holedtorus.Right:3-holedtorus.

Wheretostartwasano-brainer.Thereexistsolidslikeatorusbutwithtwoor

moreholes,asinFigure23,andthesameinvariantshouldtellussomethingusefulaboutthose.Itturnsoutthatanysoliddeformableintoa2-holedtorussatisfiesF–E+V=–2,anysoliddeformableintoa3-holedtorussatisfiesF–E+V=–4,andingeneralanysoliddeformableintoag-holedtorussatisfiesF–E+V=2–2g.Thesymbolgisshortfor‘genus’,thetechnicalnamefor‘numberofholes’.PursuingthelineofthoughtthatDescartesandEulerbeganleadstoaconnectionbetweenaquantitativepropertyofsolids,thenumberoffaces,vertices,andedges,andaqualitativeproperty,possessingholes.WecallF–E+VtheEulercharacteristicofthesolid,andobservethatitdependsonlyonwhichsolidweareconsideringandnotonhowwecutitintofaces,edges,andvertices.Thismakesitanintrinsicfeatureofthesoliditself.

Agreed,wecountthenumberofholes,aquantitativeoperation,but‘hole’itselfisqualitativeinthesensethatit’snotobviouslyafeatureofthesolidatall.Intuitively,it’saregioninspacewherethesolidisn’t.Butnotanysuchregion.Afterall,thatdescriptionappliestoallofthespacesurroundingthesolid,andnoonewouldconsideritalltobeahole.Anditalsoappliestoallofthespacesurroundingasphere…whichdoesn’thaveahole.Infact,themoreyoustarttothinkaboutwhataholeis,themoreyourealisethatit’squitetrickytodefineone.MyfavouriteexampletoshowjusthowconfusingitallgetsistheshapeinFigure24,knownasahole-through-a-hole-in-a-hole.Apparentlyyoucanthreadaholethroughanotherhole,whichisactuallyaholeinathirdhole.

Thiswayliesmadness.Itwouldn’tmuchmatterifsolidswithholesinthemneverturnedup

anywhereimportant.Butbytheendofthenineteenthcenturytheywereturningupallovermathematics–incomplexanalysis,algebraicgeometry,andRiemann’sdifferentialgeometry.Worse,higher-dimensionalanaloguesofsolidsweretakingcentrestage,inallareasofpureandappliedmathematics;asalreadynoted,thedynamicsoftheSolarSystemrequires6dimensionsperbody.Andtheyhadhigher-dimensionalanaloguesofholes.Somehowitwasnecessarytobringamodicumoforderintothearea.Andtheanswerturnedouttobe…invariants.

Fig24Hole-through-a-hole-in-a-hole.

TheideaofatopologicalinvariantgoesbacktoGauss’sworkonmagnetism.Hewasinterestedinhowmagneticandelectricalfieldlinescouldlinkwitheachother,andhedefinedthelinkingnumber,whichcountshowmanytimesonefieldlinewindsroundanother.Thisisatopologicalinvariant:itremainsthesameifthecurvesarecontinuouslydeformed.Hefoundaformulaforthisnumberusingintegralcalculus,andeverysooftenheexpressedawishforabetterunderstandingofthe‘basicgeometricproperties’ofdiagrams.ItisnocoincidencethatthefirstseriousinroadsintosuchanunderstandingcamethroughtheworkofoneofGauss’sstudents,JohannListing,andGauss’sassistantAugustMöbius.Listing’sVorstudienzurTopologie(‘StudiesinTopology’)of1847introducedtheword‘topology’,andMöbiusmadetheroleofcontinuoustransformationsexplicit.

Listinghadabrightidea:seekgeneralisationsofEuler’sformula.TheexpressionF–E+Visacombinatorialinvariant:afeatureofaspecificwayofdescribingasolid,basedoncuttingitintofaces,edges,andvertices.Thenumbergofholesisatopologicalinvariant:somethingthatdoesnotchangehoweverthesolidisdeformed,aslongasthedeformationiscontinuous.Atopologicalinvariantcapturesaqualitativeconceptualfeatureofashape;acombinatorialoneprovidesamethodforcalculatingit.Thetwotogetherareverypowerful,becausewecanusetheconceptualinvarianttothinkaboutshapes,andthecombinatorialversiontopindownwhatwearetalkingabout.

Infact,theformulaletsussidestepthetrickyissueofdefining‘hole’altogether.Instead,wedefine‘numberofholes’asapackage,withouteitherdefiningaholeorcountinghowmanythereare.How?Easy.Justrewritethe

generalisedversionofEuler’sformulaF–E+V=2–2gintheform

g=1F/2+E/2V/2

Nowwecancalculategbydrawingfacesandsoforthonoursolid,countingF,E,andV,andsubstitutingthosevaluesintotheformula.Sincetheexpressionisaninvariant,itdoesn’tmatterhowwecutthesolidup:wealwaysgetthesameanswer.Butnothingthatwedodependsonhavingadefinitionofahole.Instead,‘numberofholes’becomesaninterpretation,inintuitiveterms,derivedbylookingatsimpleexampleswherewefeelweknowwhatthephraseshouldmean.

Itmayseemlikeacheat,butitmakessignificantinroadsintoacentralquestionintopology:whencanoneshapebecontinuouslydeformedintoanother?Thatis,asfarastopologistsareconcerned,arethetwoshapesthesameornot?Iftheyarethesame,theirinvariantsmustalsobethesame;conversely,iftheinvariantsaredifferent,soaretheshapes.(However,sometimestwoshapesmighthavethesameinvariant,butbedifferent;itdependsontheinvariant.)SinceaspherehasEulercharacteristic2,butatorushasEulercharacteristic0,thereisnowaytodeformaspherecontinuouslyintoatorus.Thismayseemobvious,becauseofthehole…butwe’veseentheturbulentwatersintowhichthatwayofthinkingcanlead.Youdon’thavetointerprettheEulercharacteristicinordertouseittodistinguishshapes,andhereitisdecisive.

Lessobviously,theEulercharacteristicshowsthatthepuzzlinghole-through-a-hole-in-a-hole(Figure24)isactuallyjusta3-holedtorusindisguise.Mostoftheapparentcomplexitystemsnotfromtheintrinsictopologyofthesurface,butfromthewayIhavechosentoembeditinspace.

ThefirstreallysignificanttheoremintopologygrewoutoftheformulafortheEulercharacteristic.Itwasacompleteclassificationofsurfaces,curvedtwo-dimensionalshapeslikethesurfaceofasphereorthatofatorus.Acoupleoftechnicalconditionswerealsoimposed:thesurfaceshouldhavenoboundary,anditshouldbeoffiniteextent(thejargonis‘compact’).

Forthispurposeasurfaceisdescribedintrinsically;thatis,itisnotconceivedasexistinginsomesurroundingspace.Onewaytodothisistoviewthesurfaceasanumberofpolygonalregions(whichtopologicallyareequivalenttocirculardiscs)thataregluedtogetheralongtheiredgesaccordingtospecifiedrules,likethe‘gluetabAtotabB’instructionsyougetwhenassemblingacardboardcut-out.Asphere,forinstance,canbedescribedusingtwodiscs,gluedtogether

alongtheirboundaries.Onediscbecomesthenorthernhemisphere,theotherthesouthernhemisphere.Atorushasanespeciallyelegantdescriptionasasquarewithoppositeedgesgluedtoeachother.Thisconstructioncanbevisualisedinasurroundingspace(Figure25),whichexplainswhyitcreatesatorus,butthemathematicscanbecarriedoutusingjustthesquaretogetherwiththegluingrules,andthisoffersadvantagespreciselybecauseitisintrinsic.

Fig25Gluingtheedgesofasquaretomakeatorus.

Thepossibilityofgluingbitsofboundarytogetherleadstoaratherstrangephenomenon:surfaceswithonlyoneside.ThemostfamousexampleistheMöbiusband,introducedbyMöbiusandListingin1858,whichisarectangularstripwhoseendsaregluedtogetherwitha180°turn(usuallycalledahalf-twist,ontheconventionthat360°constitutesafulltwist).TheMöbiusband,seeFigure26(left),hasanedge,comprisingtheedgesoftherectanglethatdon’tgetgluedtoanything.Thisistheonlyedge,becausethetwoseparateedgesoftherectangleareconnectedtogetherintoaclosedloopbythehalf-twist,whichgluesthemendtoend.

ItispossibletomakeamodelofaMöbiusbandfrompaper,becauseitembedsnaturallyinthree-dimensionalspace.Thebandhasonlyoneside,inthesensethatifyoustartpaintingoneofitssurfaces,andkeepgoing,youwilleventuallycovertheentiresurface,frontandback.Thishappensbecausethehalf-twistconnectsthefronttotheback.That’snotanintrinsicdescription,becauseitreliesonembeddingthebandinspace,butthereisanequivalent,moretechnicalpropertyknownasorientability,whichisintrinsic.

Fig26Left:Möbiusband.Right:Kleinbottle.Theapparentself-intersectionoccursbecausethedrawingembedsitinthree-dimensionalspace.

Thereisarelatedsurfacewithonlyoneside,havingnoedgesatall,Figure26(right).ItarisesifwegluetwosidesofarectangletogetherlikeaMöbiusband,andgluetheothertwosidestogetherwithoutanytwisting.Anymodelinthree-dimensionalspacehastopassthroughitself,eventhoughfromanintrinsicpointofviewthegluingrulesdonotintroduceanyself-intersections.Ifthissurfaceispicturedwithsuchacrossing,itlookslikeabottlewhoseneckhasbeenpokedthroughthesidewallandjoinedtothebottom.ItwasinventedbyFelixKlein,andisknownasaKleinbottle–almostcertainlyajokebasedonaGermanpun,changingKleinscheFlaäche(Klein’ssurface)toKleinscheFlasche(Klein’sbottle).

TheKleinbottlehasnoboundaryandiscompact,soanyclassificationofsurfacesmustincludeit.Itisthebestknownofanentirefamilyofonesidedsurfaces,andsurprisinglyitisnotthesimplest.Thishonourgoestotheprojectiveplane,whicharisesifyougluebothpairsofoppositesidesofasquaretogether,withahalf-twistforeach.(Thisisdifficulttodowithpaperbecausepaperistoorigid;liketheKleinbottleitrequiresthesurfacetointersectitself.Itisbestdone‘conceptually’,thatis,bydrawingpicturesonthesquarebutrememberingthegluingruleswhenlinesgoofftheedgeand‘wrapround’.)Theclassificationtheoremforsurfaces,provedbyJohannListingaround1860,leadstotwofamiliesofsurfaces.Thosewithtwosidesarethesphere,torus,2-holedtorus,3-holedtorus,andsoon.Thosewithonlyonesideformasimilarinfinitefamily,startingwiththeprojectiveplaneandtheKleinbottle.Theycanbeobtainedbycuttingasmalldiscoutofthecorrespondingtwo-sidedsurfaceandgluinginaMöbiusbandinstead.

Surfacesturnupnaturallyinmanyareasofmathematics.Theyareimportantincomplexanalysis,wheresurfacesareassociatedwithsingularities,pointsatwhichfunctionsbehavestrangely–forinstance,thederivativefailstoexist.

Singularitiesarethekeytomanyproblemsincomplexanalysis;inasensetheycapturetheessenceofthefunction.Sincesingularitiesareassociatedwithsurfaces,thetopologyofsurfacesprovidesanimportanttechniqueforcomplexanalysis.Historically,thismotivatedtheclassification.

Mostmoderntopologyishighlyabstract,andalotofithappensinfourormoredimensions.Wecangetafeelforthesubjectinamorefamiliarsetting:knots.Intherealworld,aknotisatangletiedinalengthofstring.Topologistsneedawaytostoptheknotescapingofftheendsonceithasbeentied,sotheyjointheendsofthestringtogethertoformaclosedloop.Nowaknotisjustacircleembeddedinspace.Intrinsically,aknotistopologicallyidenticaltoacircle,butonthisoccasionwhatcountsishowthecirclesitsinsideitssurroundingspace.Thismightseemcontrarytothespiritoftopology,buttheessenceofaknotliesintherelationbetweentheloopofstringandthespacethatsurroundsit.Byconsideringnotjusttheloop,buthowitrelatestospace,topologycantackleimportantquestionsaboutknots.Amongtheseare:

Howdoweknowaknotisreallyknotted?Howcanwedistinguishtopologicallydifferentknots?Canweclassifyallpossibleknots?

Experiencetellsusthattherearemanydifferenttypesofknot.Figure27showsafewofthem:theoverhandortrefoilknot,reefknot,grannyknot,figure-8,stevedore’sknot,andsoon.Thereisalsotheunknot,anordinarycircularloop;asthenamereflects,thisloopisnotknotted.Manydifferentkindsofknothavebeenusedbygenerationsofmariners,mountaineers,andboyscouts.Anytopologicaltheoryshouldofcoursereflectthiswealthofexperience,buteverythinghastobeproved,rigorously,withintheformalsettingoftopology,justasEuclidhadtoprovePythagoras’stheoreminsteadofjustdrawingafewtrianglesandmeasuringthem.Remarkably,thefirsttopologicalproofthatknotsexist,inthesensethatthereisanembeddingofthecirclethatcannotbedeformedintotheunknot,firstappearedin1926intheGermanmathematicianKurtReidemeister’sKnotenundGruppen(‘KnotsandGroups’).Theword‘group’isatechnicalterminabstractalgebra,whichquicklybecamethemosteffectivesourceoftopologicalinvariants.In1927Reidemeister,andindependentlytheAmericanJamesWaddellAlexander,incollaborationwithhisstudentG.B.Briggs,foundasimplerproofoftheexistenceofknotsusingthe‘knotdiagram’.Thisisacartoonimageoftheknot,drawnwithtinybreaksinthelooptoshowhowtheseparatestrandsoverlap,asinFigure27.Thebreaks

arenotpresentintheknottedloopitself,buttheyrepresentitsthree-dimensionalstructureinatwo-dimensionaldiagram.Nowwecanusethebreakstosplittheknotdiagramintoanumberofdistinctpieces,itscomponents,andthenwecanmanipulatethediagramandseewhathappenstothecomponents.

Fig27Fiveknotsandtheunknot.

IfyoulookbackathowIusedtheinvarianceoftheEulercharacteristic,you’llseethatIsimplifiedthesolidusingaseriesofspecialmoves:mergetwofacesbyremovinganedge,mergetwoedgesbyremovingapoint.Thesametrickappliestoknotdiagrams,butnowyouneedthreetypesofmovetosimplifythem,calledReidemeistermoves,Figure28.Eachmovecanbecarriedoutineitherdirection:addorremoveatwist,overlaptwostrandsorpullthemapart,moveonestrandthroughtheplacewheretwootherscross.

Fig28Reidemeistermoves.

Withsomepreliminaryfiddlingtotidyuptheknotdiagram,suchasmodifyingplaceswherethreecurvesoverlapifthateverhappens,itcanbe

provedthatanydeformationofaknotcanberepresentedasafiniteseriesofReidemeistermovesappliedtoitsdiagram.NowwecanplaytheEulergame;allwehavetodoisfindaninvariant.Amongthemistheknotgroup,butthereisafarsimplerinvariantthatprovesthetrefoilreallyisaknot.Icanexplainitintermsofcolouringtheseparatecomponentsinaknotdiagram.I’mstartingwithaslightlymorecomplicateddiagramthanIhaveto,withanextraloop,inordertoillustratesomefeaturesoftheidea,Figure29.

Fig29Colouringatrefoilknotwithanextratwist.

Theextratwistcreatesfourseparatecomponents.SupposeIcolourthecomponentsusingthreecoloursforeach,sayred,yellow,andblue(showninthefigureasblack,lightgrey,anddarkgrey).Thenthiscolouringobeystwosimplerules:

Atleasttwodistinctcoloursareused.(Actuallyallthreeare,butthat’sextrainformationthatIdon’tneed.)Ateachcrossing,eitherthethreestrandsnearthecrossingallhavedifferentcoloursortheyareallthesamecolour.Nearthecrossingcausedbymyextraloop,allthreecomponentsareyellow.Twoofthesecomponents(inyellow)joinupelsewhere,butnearthecrossingtheyareseparate.

Thewonderfulobservationisthatifaknotdiagramcanbecolouredusingthreecolours,obeyingthesetworules,thenthesameistrueafteranyReidemeistermove.YoucanprovethisveryeasilybyworkingouthowtheReidemeistermovesaffectthecolours.Forexample,ifIuntwisttheextraloopinmypicturethenIcanleavethecoloursunchangedandeverythingstillworks.Whyisthiswonderful?Becauseitprovesthatthetrefoilreallyisknotted.Supposeforthesakeofargumentthatitcanbeunknotted;thensomeseriesofReidemeistermovesturnsitintoanunknottedloop.Sincethetrefoilcanbecolouredtoobeythetworules,thesamemustapplytotheunknottedloop.Butanunknottedloopconsistsofasinglestrandwithnooverlaps,sotheonlywaytocolouritistouse

thesamecoloureverywhere.Butthisviolatesthefirstrule.Bycontradiction,nosuchseriesofReidemeistermovescanexist;thatis,thetrefoilcan’tbeunknotted.

Thisprovesthatthetrefoilisknotted,butdoesn’tdistinguishitfromotherknotssuchasthereefknotorthestevedore’sknot.OneoftheearliesteffectivewaystodothiswasinventedbyAlexander.ItwasderivedfromReidemeister’sabstractalgebramethods,butitleadstoaninvariantthatisalgebraicinthemorefamiliarsenseofschoolalgebra.It’scalledtheAlexanderpolynomial,anditassociatestoanyknotaformulaformedfrompowersofavariablex.Strictlyspeaking,theterm‘polynomial’appliesonlywhenthepowersarepositiveintegers,butherewealsoallownegativepowers.Table2listsafewAlexanderpolynomials.IftwoknotsinthelisthavedifferentAlexanderpolynomials,andhereallbutthereefandgrannydo,thentheknotsmustbetopologicallydifferent.Theconverseisnottrue:thereefandgrannyhavethesameAlexanderpolynomials,butin1952RalphFoxprovedthattheyaretopologicallydifferent.Theproofrequiredsurprisinglysophisticatedtopology.Itwasfarmoredifficultthananyoneexpected.

Table2Alexanderpolynomialsofknots.

Afterabout1960knottheoryenteredthetopologicaldoldrums,becalmedinavastoceanofunsolvedquestions,awaitingabreathofcreativeinsight.Itcamein1984,whentheNewZealandmathematicianVaughanJoneshadanideasosimplethatitcouldhaveoccurredtoanyonefromReidemeisteronwards.Joneswasn’taknottheorist;hewasn’tevenatopologist.Hewasananalyst,workingonoperatoralgebras,anareawithstronglinkstomathematicalphysics.Itwasn’tatotalsurprisethattheideasappliedtoknots,becausemathematiciansandphysicistsalreadyknewofinterestingconnectionsbetweenoperatoralgebrasandbraids,whichareaspecialkindofmulti-strandedknot.Thenewknotinvariantheinvented,calledtheJonespolynomial,isalsodefinedusingtheknot

diagramandthreetypesofmove.However,themovesdonotpreservethetopologicaltypeoftheknot;theydonotpreservethenew‘Jonespolynomial’.Amazingly,however,theideacanstillbemadetowork,andtheJonespolynomialisaknotinvariant.

Fig30Jonesmoves.

Forthisinvariantwehavetochooseaspecificdirectionalongtheknot,shownbyanarrow.TheJonespolynomialV(x)isdefinedtobeonefortheunknot.GivenanyknotL0,movetwoseparatestrandsclosetogetherwithoutchanginganycrossingsinitsdiagram.Becarefultoalignthedirectionsasshown:that’swhythearrowisneeded,andtheprocessdoesn’tworkwithoutit.ReplacethatregionofL0bytwostrandsthatcrossinthetwopossibleways(Figure30).LettheresultingknotdiagramsbeL+andL–.Nowdefine

(x1/2−x−1/2)V(L0)=x−1V(L+)−xV(L−)

Bystartingwiththeunknotandapplyingsuchmovesintherightway,youcanworkouttheJonespolynomialforanyknot.Mysteriously,itturnsouttobeatopologicalinvariant.AnditoutperformsthetraditionalAlexanderpolynomial;forinstance,itcandistinguishreeffromgranny,becausetheyhavedifferentJonespolynomials.

Jones’sdiscoverywonhimtheFieldsmedal,themostprestigiousprizeinmathematics.Italsotriggeredanoutburstofnewknotinvariants.In1985fourdifferentgroupsofmathematicians,eightpeopleintotal,simultaneouslydiscoveredthesamegeneralisationoftheJonespolynomialandsubmittedtheirpapersindependentlytothesamejournal.Allfourproofsweredifferent,andtheeditorpersuadedtheeightauthorstojoinforcesandpublishonecombined

article.TheirinvariantisoftencalledtheHOMFLYpolynomial,basedontheirinitials.ButeventheJonesandHOMFLYpolynomialshavenotfullyansweredthethreeproblemsofknottheory.ItisnotknownwhetheraknotwithJonespolynomial1mustbeunknotted,thoughmanytopologiststhinkthisisprobablytrue.ThereexisttopologicallydistinctknotswiththesameJonespolynomial;thesimplestexamplesknownhavetencrossingsintheirknotdiagrams.Asystematicclassificationofallpossibleknotsremainsamathematician’spipedream.

It’spretty,butisituseful?Topologyhasmanyuses,buttheyareusuallyindirect.Topologicalprinciplesprovideinsightintoother,moredirectlyapplicable,areas.Forinstance,ourunderstandingofchaosisfoundedontopologicalpropertiesofdynamicalsystems,suchasthebizarrebehaviourthatPoincarénotedwhenherewrotehisprizewinningmemoir(Chapter4).TheInterplanetarySuperhighwayisatopologicalfeatureofthedynamicsoftheSolarSystem.

Moreesotericapplicationsoftopologyariseatthefrontiersoffundamentalphysics.Herethemainconsumersoftopologyarequantumfieldtheorists,becausethetheoryofsuperstrings,thehoped-forunificationofquantummechanicsandrelativity,isbasedontopology.HereanaloguesoftheJonespolynomialinknottheoryariseinthecontextofFeynmandiagrams,whichshowhowquantumparticlessuchaselectronsandphotonsmovethroughspace-time,colliding,merging,andbreakingapart.AFeynmandiagramisabitlikeaknotdiagram,andJones’sideascanbeextendedtothiscontext.

Tomeoneofthemostfascinatingapplicationsoftopologyisitsgrowinguseinbiology,helpingustounderstandtheworkingsofthemoleculeoflife,DNA.TopologyturnsupbecauseDNAisadoublehelix,liketwospiralstaircaseswindingaroundeachother.Thetwostrandsareintricatelyintertwined,andimportantbiologicalprocesses,inparticularthewayacellcopiesitsDNAwhenitdivides,havetotakeaccountofthiscomplextopology.WhenFrancisCrickandJamesWatsonpublishedtheirworkonthemolecularstructureofDNAin1953theyendedwithabriefallusiontoapossiblecopyingmechanism,presumablyinvolvedincelldivision,inwhichthetwostrandswerepulledapartandeachwasusedasthetemplateforanewcopy.Theywerereluctanttoclaimtoomuch,becausetheywereawarethattherearetopologicalobstaclestopullingapartintertwinedstrands.Beingtoospecificabouttheirproposalmighthavemuddiedthewatersatsuchanearlystage.

Asthingsturnedout,CrickandWatsonwereright.Thetopologicalobstacles

arereal,butevolutionhasprovidedmethodsforovercomingthem,suchasspecialenzymesthatcut-and-pastestrandsofDNA.Itisnocoincidencethatoneoftheseiscalledtopoisomerase.Inthe1990smathematiciansandmolecularbiologistsusedtopologytoanalysethetwistsandturnsofDNA,andtostudyhowitworksinthecell,wheretheusualmethodofX-raydiffractioncan’tbeusedbecauseitrequirestheDNAtobeincrystallineform.

Fig31LoopofDNAformingatrefoilknot.

Someenzymes,calledrecombinases,cutthetwoDNAstrandsandrejointheminadifferentway.Todeterminehowsuchanenzymeactswhenitisinacell,biologistsapplytheenzymetoaclosedloopofDNA.Thentheyobservetheshapeofthemodifiedloopusinganelectronmicroscope.Iftheenzymejoinsdistinctstrandstogether,theimageisaknot,Figure31.Iftheenzymekeepsthestrandsseparate,theimageshowstwolinkedloops.Methodsfromknottheory,suchastheJonespolynomialandanothertheoryknownas‘tangles’,makeitpossibletoworkoutwhichknotsandlinksoccur,andthisprovidesdetailedinformationaboutwhattheenzymedoes.Theyalsomakenewpredictionsthathavebeenverifiedexperimentally,givingsomeconfidencethatthemechanismindicatedbythetopologicalcalculationsiscorrect.1

Onethewhole,youwon’trunintotopologyineverydaylife,asidefromthatdishwasherImentionedatthestartofthischapter.Butbehindthescenes,topologyinformsthewholeofmainstreammathematics,enablingthedevelopmentofothertechniqueswithmoreobviouspracticaluses.Thisiswhymathematiciansconsidertopologytobeofvastimportance,whiletherestoftheworldhashardlyheardofit.

7Patternsofchance

NormalDistribution

Whatdoesitsay?

Theprobabilityofobservingaparticulardatavalueisgreatestnearthemeanvalue–theaverage–anddiesawayrapidlyasthedifferencefromthemeanincreases.Howrapidlydependsonaquantitycalledthestandarddeviation.

Whyisthatimportant?

Itdefinesaspecialfamilyofbell-shapedprobabilitydistributions,whichareoftengoodmodelsofcommonreal-worldobservations.

Whatdiditleadto?

Theconceptofthe‘averageman’,testsofthesignificanceofexperimentalresults,suchasmedicaltrials,andanunfortunatetendencytodefaulttothebellcurveasifnothingelseexisted.

Mathematicsisaboutpatterns.Therandomworkingsofchanceseemtobeaboutasfarfrompatternsasyoucanget.Infact,oneofthecurrentdefinitionsof‘random’boilsdownto‘lackinganydiscerniblepattern’.Mathematicianshadbeeninvestigatingpatternsingeometry,algebra,andanalysisforcenturiesbeforetheyrealisedthatevenrandomnesshasitsownpatterns.Butthepatternsofchancedonotconflictwiththeideathatrandomeventshavenopattern,becausetheregularitiesofrandomeventsarestatistical.Theyarefeaturesofawholeseriesofevents,suchastheaveragebehaviouroveralongrunoftrials.Theytellusnothingaboutwhicheventoccursatwhichinstant.Forexample,ifyouthrowadice1repeatedly,thenaboutonesixthofthetimeyouwillroll1,andthesameholdsfor2,3,4,5,and6–aclearstatisticalpattern.Butthistellsyounothingaboutwhichnumberwillturnuponthenextthrow.

Onlyinthenineteenthcenturydidmathematiciansandscientistsrealisetheimportanceofstatisticalpatternsinchanceevents.Evenhumanactions,suchassuicideanddivorce,aresubjecttoquantitativelaws,onaverageandinthelongrun.Ittooktimetogetusedtowhatseemedatfirsttocontradictfreewill.Buttodaythesestatisticalregularitiesformthebasisofmedicaltrials,socialpolicy,insurancepremiums,riskassessments,andprofessionalsports.

Andgambling,whichiswhereitallbegan.

Appropriately,itwasallstartedbythegamblingscholar,GirolamoCardano.Beingsomethingofawastrel,Cardanobroughtinmuchneededcashbytakingwagersongamesofchessandgamesofchance.Heappliedhispowerfulintellecttoboth.Chessdoesnotdependonchance:winningdependsonagoodmemoryforstandardpositionsandmoves,andanintuitivesenseoftheoverallflowofthegame.Inagameofchance,however,theplayerissubjecttothewhimsofLadyLuck.Cardanorealisedthathecouldapplyhismathematicaltalentstogoodeffecteveninthistempestuousrelationship.Hecouldimprovehisperformanceatgamesofchancebyhavingabettergraspoftheodds–thelikelihoodofwinningorlosing–thanhisopponentsdid.Heputtogetherabookonthetopic,LiberdeLudoAleae(‘BookonGamesofChance’).Itremainedunpublisheduntil1633.Itsscholarlycontentisthefirstsystematictreatmentofthemathematicsofprobability.Itslessreputablecontentisachapteronhowtocheatandgetawaywithit.

OneofCardano’sfundamentalprincipleswasthatinafairbet,thestakes

shouldbeproportionaltothenumberofwaysinwhicheachplayercanwin.Forexample,supposetheplayersrolladice,andthefirstplayerwinsifhethrowsa6,whilethesecondplayerwinsifhethrowsanythingelse.Thegamewouldbehighlyunfairifeachbetthesameamounttoplaythegame,becausethefirstplayerhasonlyonewaytowin,whereasthesecondhasfive.Ifthefirstplayerbets£1andthesecondbets£5,however,theoddsbecomeequitable.Cardanowasawarethatthismethodofcalculatingfairoddsdependsonthevariouswaysofwinningbeingequallylikely,butingamesofdice,cards,orcoin-tossingitwasclearhowtoensurethatthisconditionapplied.Tossingacoinhastwooutcomes,headsortails,andtheseareequallylikelyifthecoinisfair.Ifthecointendstothrowmoreheadsthantails,itisclearlybiased–unfair.Similarlythesixoutcomesforafairdiceareequallylikely,asarethe52outcomesforacarddrawnfromapack.

Thelogicbehindtheconceptoffairnesshereisslightlycircular,becauseweinferbiasfromafailuretomatchtheobviousnumericalconditions.Butthoseconditionsaresupportedbymorethanmerecounting.Theyarebasedonafeelingofsymmetry.Ifthecoinisaflatdiscofmetal,ofuniformdensity,thenthetwooutcomesarerelatedbyasymmetryofthecoin(flipitover).Fordice,thesixoutcomesarerelatedbysymmetriesofthecube.Andforcards,therelevantsymmetryisthatnocarddifferssignificantlyfromanyother,exceptforthevaluewrittenonitsface.Thefrequencies1/2,1/6,and1/52foranygivenoutcomerestonthesebasicsymmetries.Abiasedcoinorbiaseddicecanbecreatedbythecovertinsertionofweights;abiasedcardcanbecreatedusingsubtlemarksontheback,whichrevealitsvaluetothoseintheknow.

Thereareotherwaystocheat,involvingsleightofhand–say,toswapabiaseddiceintoandoutofthegamebeforeanyonenoticesthatitalwaysthrowsa6.Butthesafestwayto‘cheat’–towinbysubterfuge–istobeperfectlyhonest,buttoknowtheoddsbetterthanyouropponent.Inonesenseyouaretakingthemoralhighground,butyoucanimproveyourchancesoffindingasuitablynaiveopponentbyriggingnottheoddsbutyouropponent’sexpectationoftheodds.Therearemanyexampleswheretheactualoddsinagameofchancearesignificantlydifferentfromwhatmanypeoplewouldnaturallyassume.

Anexampleisthegameofcrownandanchor,widelyplayedbyBritishseamenintheeighteenthcentury.Itusesthreedice,eachbearingnotthenumbers1–6butsixsymbols:acrown,ananchor,andthefourcardsuitsofdiamond,spade,club,andheart.Thesesymbolsarealsomarkedonamat.Playersbetbyplacingmoneyonthematandthrowingthethreedice.Ifanyofthesymbolsthattheyhavebetonshowsup,thebankerpaysthemtheirstake,

multipliedbythenumberofdiceshowingthatsymbol.Forexample,iftheybet£1onthecrown,andtwocrownsturnup,theywin£2inadditiontotheirstake;ifthreecrownsturnup,theywin£3inadditiontotheirstake.Itallsoundsveryreasonable,butprobabilitytheorytellsusthatinthelongrunaplayercanexpecttolose8%ofhisstake.

ProbabilitytheorybegantotakeoffwhenitattractedtheattentionofBlaisePascal.PascalwasthesonofaRouentaxcollectorandachildprodigy.In1646hewasconvertedtoJansenism,asectofRomanCatholicismthatPopeInnocentXdeemedhereticalin1655.Theyearbefore,Pascalhadexperiencedwhathecalledhis‘secondconversion’,probablytriggeredbyanear-fatalaccidentwhenhishorsesfellofftheedgeofNeuillybridgeandhiscarriagenearlydidthesame.Mostofhisoutputfromthenonwasreligiousphilosophy.Butjustbeforetheaccident,heandFermatwerewritingtoeachotheraboutamathematicalproblemtodowithgambling.TheChevalierdeMeré,aFrenchwriterwhocalledhimselfaknighteventhoughhewasn’t,wasafriendofPascal’s,andheaskedhowthestakesinaseriesofgamesofchanceshouldbedividedifthecontesthadtobeabandonedpartwaythrough.Thisquestionwasnotnew:itgoesbacktotheMiddleAges.Whatwasnewwasitssolution.Inanexchangeofletters,PascalandFermatfoundthecorrectanswer.Alongthewaytheycreatedanewbranchofmathematics:probabilitytheory.

Acentralconceptintheirsolutionwaswhatwenowcall‘expectation’.Inagameofchance,thisisaplayer’saveragereturninthelongrun.Itwould,forexample,be92penceforcrownandanchorwitha£1stake.Afterhissecondconversion,Pascalputhisgamblingpastbehindhim,butheenlisteditsaidinafamousphilosophicalargument,Pascal’swager.2Pascalassumed,playingDevil’sadvocate,thatsomeonemightconsidertheexistenceofGodtobehighlyunlikely.InhisPensées(‘Thoughts’)of1669,Pascalanalysedtheconsequencesfromthepointofviewofprobabilities:

LetusweighthegainandthelossinwageringthatGodis[exists].Letusestimatethesetwochances.Ifyougain,yougainall;ifyoulose,youlosenothing.Wager,then,withouthesitationthatHeis…Thereishereaninfinityofaninfinitelyhappylifetogain,achanceofgainagainstafinitenumberofchancesofloss,andwhatyoustakeisfinite.Andsoourpropositionisofinfiniteforce,whenthereisthefinitetostakeinagamewherethereareequalrisksofgainandofloss,andtheinfinitetogain.

Probabilitytheoryarrivedasafullyfledgedareaofmathematicsin1713whenJacobBernoullipublishedhisArsConjectandi(‘ArtofConjecturing’).Hestartedwiththeusualworkingdefinitionoftheprobabilityofanevent:theproportionofoccasionsonwhichitwillhappen,inthelongrun,nearlyallthetime.Isay‘workingdefinition’becausethisapproachtoprobabilitiesrunsintotroubleifyoutrytomakeitfundamental.Forexample,supposethatIhaveafaircoinandkeeptossingit.MostofthetimeIgetarandom-lookingsequenceofheadsandtails,andifIkeeptossingforlongenoughIwillgetheadsroughlyhalfthetime.However,Iseldomgetheadsexactlyhalfthetime:thisisimpossibleonodd-numberedtosses,forexample.IfItrytomodifythedefinitionbytakinginspirationfromcalculus,sothattheprobabilityofthrowingheadsisthelimitoftheproportionofheadsasthenumberoftossestendstoinfinity,Ihavetoprovethatthislimitexists.Butsometimesitdoesn’t.Forexample,supposethatthesequenceofheadsandtailsgoes

THHTTTHHHHHHTTTTTTTTTTTT…

withonetail,twoheads,threetails,sixheads,twelvetails,andsoon–thenumbersdoublingateachstageafterthethreetails.Afterthreetossestheproportionofheadsis2/3,aftersixtossesitis1/3,aftertwelvetossesitisbackto2/3,aftertwenty-fouritis1/3,…sotheproportionoscillatestoandfro,between2/3and1/3,andthereforehasnowell-definedlimit.

Agreed,suchasequenceoftossesisveryunlikely,buttodefine‘unlikely’weneedtodefineprobability,whichiswhatthelimitissupposedtoachieve.Sothelogiciscircular.Moreover,evenifthelimitexists,itmightnotbethe‘correct’valueof1/2.Anextremecaseoccurswhenthecoinalwayslandsheads.Nowthelimitis1.Again,thisiswildlyimprobable,but…

Bernoullidecidedtoapproachthewholeissuefromtheoppositedirection.Startbysimplydefiningtheprobabilityofheadsandtailstobesomenumberpbetween0and1.Saythatthecoinisfairifp= ,andbiasedifnot.NowBernoulliprovesabasictheorem,thelawoflargenumbers.Introduceareasonableruleforassigningprobabilitiestoaseriesofrepeatedevents.Thelawoflargenumbersstatesthatinthelongrun,withtheexceptionofafractionoftrialsthatbecomesarbitrarilysmall,theproportionofheadsdoeshavealimit,andthatlimitisp.Philosophically,thistheoremshowsthatbyassigningprobabilities–thatis,numbers–inanaturalway,theinterpretation‘proportionofoccurrencesinthelongrun,ignoringrareexceptions’isvalid.SoBernoullitakesthepointofviewthatthenumbersassignedasprobabilitiesprovidea

consistentmathematicalmodeloftheprocessoftossingacoinoverandoveragain.

HisproofdependsonanumericalpatternthatwasveryfamiliartoPascal.ItisusuallycalledPascal’striangle,eventhoughhewasn’tthefirstpersontonoticeit.HistorianshavetraceditsoriginsbacktotheChandasShastra,aSanskittextattributedtoPingala,writtensometimebetween500BCand200BC.Theoriginalhasnotsurvived,buttheworkisknownthroughtenth-centuryHinducommentaries.Pascal’strianglelookslikethis:

whereallrowsstartandendin1,andeachnumberisthesumofthetwoimmediatelyaboveit.Wenowcallthesenumbersbinomialcoefficients,becausetheyariseinthealgebraofthebinomial(two-variable)expression(p+q)n.Namely,

andPascal’striangleisvisibleasthecoefficientsoftheseparateterms.Bernoulli’skeyinsightisthatifwetossacoinntimes,withaprobabilitypof

gettingheads,thentheprobabilityofaspecificnumberoftossesyieldingheadsisthecorrespondingtermof(p+q)n,whereq=1−p.Forexample,supposethatItossthecointhreetimes.Thentheeightpossibleresultsare:

whereI’vegroupedthesequencesaccordingtothenumberofheads.Sooutoftheeightpossiblesequences,thereare

1sequencewith3heads3sequenceswith2heads3sequenceswith1heads1sequencewith0heads

Thelinkwithbinomialcoefficientsisnocoincidence.Ifyouexpandthealgebraicformula(H+T)3butdon’tcollectthetermstogether,youget

HHH+HHT+HTH+THH+HTT+THT+TTH+TTT

CollectingtermsaccordingtothenumberofHsthengives

H3+3H2T+3HT2+T3

Afterthat,it’samatterofreplacingeachofHandTbyitsprobability,porqrespectively.

Eveninthiscase,eachextremeHHHandTTToccursonlyonceineighttrials,andmoreequitablenumbersoccurintheothersix.AmoresophisticatedcalculationusingstandardpropertiesofbinomialcoefficientsprovesBernoulli’slawoflargenumbers.

Advancesinmathematicsoftencomeaboutbecauseofignorance.Whenmathematiciansdon’tknowhowtocalculatesomethingimportant,theyfindawaytosneakuponitindirectly.Inthiscase,theproblemistocalculatethosebinomialcoefficients.There’sanexplicitformula,butif,forinstance,youwanttoknowtheprobabilityofgettingexactly42headswhentossingacoin100times,youhavetodo200multiplicationsandthensimplifyaverycomplicatedfraction.(Thereareshortcuts;it’sstillabigmess.)Mycomputertellsmeinasplitsecondthattheansweris

28,258,808,871,162,574,166,368,460,400p42q58

butBernoullididn’thavethatluxury.Noonediduntilthe1960s,andcomputeralgebrasystemsdidn’treallybecomewidelyavailableuntilthelate1980s.

Sincethiskindofdirectcalculationwasn’tfeasible,Bernoulli’simmediatesuccessorstriedtofindgoodapproximations.Around1730AbrahamDeMoivrederivedanapproximateformulafortheprobabilitiesinvolvedinrepeatedtossesofabiasedcoin.Thisledtotheerrorfunctionornormaldistribution,oftenreferredtoasthe‘bellcurve’becauseofitsshape.Whatheprovedwasthis.DefinethenormaldistributionΦ(x)withmeanμandvarianceσ2bytheformula

ThenforlargentheprobabilityofgettingmheadsinntossesofabiasedcoinisveryclosetoΦ(x)when

x=m/n−pμ=npσ=npq

Here‘mean’referstotheaverage,and‘variance’isameasureofhowfarthedataspreadout–thewidthofthebellcurve.Thesquarerootofthevariance,σitself,iscalledthestandarddeviation.Figure32(left)showshowthevalueofΦ(x)dependsonx.Thecurvelooksabitlikeabell,hencetheinformalname.Thebellcurveisanexampleofaprobabilitydistribution;thismeansthattheprobabilityofobtainingdatabetweentwogivenvaluesisequaltotheareaunderthecurveandbetweentheverticallinescorrespondingtothosevalues.Thetotalareaunderthecurveis1,thankstothatunexpectedfactor

Theideaismosteasilygraspedusinganexample.Figure32(right)showsagraphoftheprobabilitiesofgettingvariousnumbersofheadswhentossingafaircoin15times(rectangularbars)togetherwiththeapproximatingbellcurve.

Fig32Left:Bellcurve.Right:Howitapproximatesthenumberofheadsin15tossesofafaircoin.

Thebellcurvebegantoacquireiconicstatuswhenitstartedshowingupinempiricaldatainthesocialsciences,notjusttheoreticalmathematics.In1835AdolpheQuetelet,aBelgianwhoamongotherthingspioneeredquantitativemethodsinsociology,collectedandanalysedlargequantitiesofdataoncrime,thedivorcerate,suicide,births,deaths,humanheight,weight,andsoon–variablesthatnooneexpectedtoconformtoanymathematicallaw,becausetheirunderlyingcausesweretoocomplexandinvolvedhumanchoices.Consider,forexample,theemotionaltormentthatdrivessomeonetocommitsuicide.Itseemedridiculoustothinkthatthiscouldbereducedtoasimpleformula.

Theseobjectionsmakegoodsenseifyouwanttopredictexactlywhowillkillthemselves,andwhen.ButwhenQueteletconcentratedonstatisticalquestions,suchastheproportionofsuicidesinvariousgroupsofpeople,variouslocations,anddifferentyears,hestartedtoseepatterns.Theseprovedcontroversial:ifyoupredictthattherewillbesixsuicidesinParisnextyear,howcanthismakesensewheneachpersoninvolvedhasfreewill?Theycouldallchangetheirminds.Butthepopulationformedbythosewhodokillthemselvesisnotspecifiedbeforehand;itcomestogetherasaconsequenceofchoicesmadenotjustbythosewhocommitsuicide,butbythosewhothoughtaboutitanddidn’t.Peopleexercisefreewillinthecontextofmanyotherthings,whichinfluencewhattheyfreelydecide:heretheconstraintsincludefinancialproblems,relationshipproblems,mentalstate,religiousbackground…Inanycase,thebellcurvedoesnotmakeexactpredictions;itjuststateswhichfigureismostlikely.Fiveorsevensuicidesmightoccur,leavingplentyofroomforanyonetoexercisefreewillandchangetheirmind.

Thedataeventuallywontheday:forwhateverreason,peopleenmassebehavedmorepredictablythanindividuals.Perhapsthesimplestexamplewasheight.WhenQueteletplottedtheproportionsofpeoplewithagivenheight,heobtainedabeautifulbellcurve,Figure33.Hegotthesameshapeofcurveformanyothersocialvariables.

Queteletwassostruckbyhisresultsthathewroteabook,Surl’hommeetledéveloppementdesesfacultés(‘TreatiseonManandtheDevelopmentofHisFaculties’)publishedin1835.Init,heintroducedthenotionofthe‘averageman’,afictitiousindividualwhowasineveryrespectaverage.Ithaslongbeennotedthatthisdoesn’tentirelywork:theaverage‘man’–thatis,person,sothe

calculationincludesmalesandfemales–has(slightlylessthan)onebreast,onetesticle,2.3children,andsoon.NeverthelessQueteletviewedhisaveragemanasthegoalofsocialjustice,notjustasuggestivemathematicalfiction.It’snotquiteasabsurdasitsounds.Forexample,ifhumanwealthisspreadequallytoall,theneveryonewillhaveaveragewealth.It’snotapracticalgoal,barringenormoussocialchanges,butsomeonewithstrongegalitarianviewsmightdefenditasadesirabletarget.

Fig33Quetelet’sgraphofhowmanypeople(verticalaxis)haveagivenheight(horizontalaxis).

Thebellcurverapidlybecameaniconinprobabilitytheory,especiallyitsappliedarm,statistics.Thereweretwomainreasons:thebellcurvewasrelativelysimpletocalculate,andtherewasatheoreticalreasonforittooccurinpractice.Oneofthemainsourcesforthiswayofthinkingwaseighteenth-centuryastronomy.Observationaldataaresubjecttoerrors,causedbyslightvariationsinapparatus,humanmistakes,ormerelythemovementofaircurrentsintheatmosphere.Astronomersoftheperiodwantedtoobserveplanets,comets,andasteroids,andcalculatetheirorbits,andthisrequiredfindingwhicheverorbitfittedthedatabest.Thefitwouldneverbeperfect.

Thepracticalsolutiontothisproblemappearedfirst.Itboileddowntothis:runastraightlinethroughthedata,andchoosethislinesothatthetotalerrorisassmallaspossible.Errorsherehavetobeconsideredpositive,andtheeasywaytoachievethiswhilekeepingthealgebraniceistosquarethem.Sothetotalerroristhesumofthesquaresofthedeviationsofobservationsfromthestraight

linemodel,andthedesiredlineminimisesthis.In1805theFrenchmathematicianAdrien-MarieLegendrediscoveredasimpleformulaforthisline,makingiteasytocalculate.Theresultiscalledthemethodofleastsquares.Figure34illustratesthemethodonartificialdatarelatingstress(measuredbyaquestionnaire)andbloodpressure.Thelineinthefigure,foundusingLegendre’sformula,fitsthedatamostcloselyaccordingtothesquared-errormeasure.WithintenyearsthemethodofleastsquareswasstandardamongastronomersinFrance,Prussia,andItaly.WithinanothertwentyyearsitwasstandardinEngland.

Fig34Usingthemethodofleastsquarestorelatebloodpressureandstress.Dots:data.Solidline:best-fittingstraightline.

Gaussmadethemethodofleastsquaresacornerstoneofhisworkincelestialmechanics.Hegotintotheareain1801bysuccessfullypredictingthereturnoftheasteroidCeresafteritwashiddenintheglareoftheSun,whenmostastronomersthoughttheavailabledataweretoolimited.ThistriumphsealedhismathematicalreputationamongthepublicandsethimupforlifeasprofessorofastronomyattheUniversityofGöttingen.Gaussdidn’tuseleastsquaresforthisparticularprediction:hiscalculationsboileddowntosolvinganalgebraicequationoftheeighthdegree,whichhedidbyaspeciallyinventednumericalmethod.Butinhislaterwork,culminatinginhis1809TheoriaMotusCorporumCoelestiuminSectionibusConicisSolemAmbientum(‘TheoryofMotionoftheCelestialBodiesMovinginConicSectionsaroundtheSun’)heplacedgreatemphasisonthemethodofleastsquares.Healsostatedthathehaddevelopedtheidea,andusedit,tenyearsbeforeLegendre,whichcausedabitofafuss.It

wasverylikelytrue,however,andGauss’sjustificationofthemethodwasquitedifferent.Legendrehadvieweditasanexerciseincurve-fitting,whereasGausssawitasawaytofitaprobabilitydistribution.Hisjustificationoftheformulaassumedthattheunderlyingdata,towhichthestraightlinewasbeingfitted,followedabellcurve.

Itremainedtojustifythejustification.Whyshouldobservationalerrorsbenormallydistributed?In1810Laplacesuppliedanastonishinganswer,alsomotivatedbyastronomy.Inmanybranchesofscienceitisstandardtomakethesameobservationseveraltimes,independently,andthentaketheaverage.Soitisnaturaltomodelthisproceduremathematically.LaplaceusedtheFouriertransform,seeChapter9,toprovethattheaverageofmanyobservationsisdescribedbyabellcurve,eveniftheindividualobservationsarenot.Hisresult,thecentrallimittheorem,wasamajorturningpointinprobabilityandstatistics,becauseitprovidedatheoreticaljustificationforusingthemathematicians’favouritedistribution,thebellcurve,intheanalysisofobservationalerrors.3

Thecentrallimittheoremsingledoutthebellcurveastheprobabilitydistributionuniquelysuitedtothemeanofmanyrepeatedobservations.Itthereforeacquiredthename‘normaldistribution’,andwasseenasthedefaultchoiceforaprobabilitydistribution.Notonlydidthenormaldistributionhavepleasantmathematicalproperties,buttherewasalsoasolidreasonforassumingitmodelledrealdata.ThiscombinationofattributesprovedveryattractivetoscientistswishingtogaininsightsintothesocialphenomenathathadinterestedQuetelet,becauseitofferedawaytoanalysedatafromofficialrecords.In1865FrancisGaltonstudiedhowachild’sheightrelatestoitsparents’heights.Thiswaspartofawidergoal:understandingheredity–howhumancharacteristicspassfromparenttochild.Ironically,Laplace’scentrallimittheoreminitiallyledGaltontodoubtthatthiskindofinheritanceexisted.And,evenifitdid,provingthatwouldbedifficult,becausethecentrallimittheoremwasadouble-edgedsword.Quetelethadfoundabeautifulbellcurveforheights,butthatseemedtoimplyverylittleaboutthedifferentfactorsthataffectedheight,becausethecentrallimittheorempredictedanormaldistributionanyway,whateverthedistributionsofthosefactorsmightbe.Evenifcharacteristicsoftheparentswereamongthosefactors,theymightbeoverwhelmedbyalltheothers–suchasnutrition,health,socialstatus,andsoon.

By1889,however,Galtonhadfoundawayoutofthisdilemma.TheproofofLaplace’swonderfultheoremreliedonaveragingouttheeffectsofmanydistinctfactors,butthesehadtosatisfysomestringentconditions.In1875Galton

describedtheseconditionsas‘highlyartificial’,andnotedthattheinfluencesbeingaveraged

mustbe(1)allindependentintheireffects,(2)allequal[havingthesameprobabilitydistribution],(3)alladmittingofbeingtreatedassimplealternatives‘aboveaverage’or‘belowaverage’;and(4)…calculatedonthesuppositionthatthevariableinfluencesareinfinitelynumerous.

Noneoftheseconditionsappliedtohumanheredity.Condition(4)correspondstoLaplace’sassumptionthatthenumberoffactorsbeingaddedtendstoinfinity,so‘infinitelynumerous’isabitofanexaggeration;however,whatthemathematicsestablishedwasthattogetagoodapproximationtoanormaldistribution,youhadtocombinealargenumberoffactors.Eachofthesecontributedasmallamounttotheaverage:with,say,ahundredfactors,eachcontributedonehundredthofitsvalue.Galtonreferredtosuchfactorsas‘petty’.Eachonitsownhadnosignificanteffect.

Therewasapotentialwayout,andGaltonseizedonit.Thecentrallimittheoremprovidedasufficientconditionforadistributiontobenormal,notanecessaryone.Evenwhenitsassumptionsfailed,thedistributionconcernedmightstillbenormalforotherreasons.Galton’staskwastofindoutwhatthosereasonsmightbe.Tohaveanyhopeoflinkingtoheredity,theyhadtoapplytoacombinationofafewlargeanddisparateinfluences,nottoahugenumberofinsignificantinfluences.Heslowlygropedhiswaytowardsasolution,andfounditthroughtwoexperiments,bothdatingto1877.Onewasadevicehecalledaquincunx,inwhichballbearingsfelldownaslope,bouncingoffanarrayofpins,withanequalchanceofgoingleftorright.Intheorytheballsshouldpileupatthebottomaccordingtoabinomialdistribution,adiscreteapproximationtothenormaldistribution,sotheyshould–anddid–formaroughlybell-shapedheap,likeFigure32(right).Hiskeyinsightwastoimaginetemporarilyhaltingtheballswhentheywerepartwaydown.Theywouldstillformabellcurve,butitwouldbenarrowerthanthefinalone.Imaginereleasingjustonecompartmentofballs.Itwouldfalltothebottom,spreadingoutintoatinybellcurve.Thesamewentforanyothercompartment.Andthatmeantthatthefinal,largebellcurvecouldbeviewedasasumoflotsoftinyones.Thebellcurvereproducesitselfwhenseveralfactors,eachfollowingitsownseparatebellcurve,arecombined.

TheclincherarrivedwhenGaltonbredsweetpeas.In1875hedistributed

seedstosevenfriends.Eachreceived70seeds,butonereceivedverylightseeds,oneslightlyheavierones,andsoon.In1877hemeasuredtheweightsoftheseedsoftheresultingprogeny.Eachgroupwasnormallydistributed,butthemeanweightdifferedineachcase,beingcomparabletotheweightofeachseedintheoriginalgroup.Whenhecombinedthedataforallofthegroups,theresultswereagainnormallydistributed,butthevariancewasbigger–thebellcurvewaswider.Again,thissuggestedthatcombiningseveralbellcurvesledtoanotherbellcurve.Galtontrackeddownthemathematicalreasonforthis.Supposethattworandomvariablesarenormallydistributed,notnecessarilywiththesamemeansorthesamevariances.Thentheirsumisalsonormallydistributed;itsmeanisthesumofthetwomeans,anditsvarianceisthesumofthetwovariances.Obviouslythesamegoesforsumsofthree,four,ormorenormallydistributedrandomvariables.

Thistheoremworkswhenasmallnumberoffactorsarecombined,andeachfactorcanbemultipliedbyaconstant,soitactuallyworksforanylinearcombination.Thenormaldistributionisvalidevenwhentheeffectofeachfactorislarge.NowGaltoncouldseehowthisresultappliedtoheredity.Supposethattherandomvariablegivenbytheheightofachildissomecombinationofthecorrespondingrandomvariablesfortheheightsofitsparents,andthesearenormallydistributed.Assumingthatthehereditaryfactorsworkbyaddition,thechild’sheightwillalsobenormallydistributed.

Galtonwrotehisideasupin1889underthetitleNaturalInheritance.Inparticular,hediscussedanideahecalledregression.Whenonetallparentandoneshortonehavechildren,themeanheightofthechildrenshouldbeintermediate–infact,itshouldbetheaverageoftheparents’heights.Thevariancelikewiseshouldbetheaverageofthevariances,butthevariancesfortheparentsseemedtoberoughlyequal,sothevariancedidn’tchangemuch.Assuccessivegenerationspassed,themeanheightwould‘regress’toafixedmiddle-of-the-roadvalue,whilethevariancewouldstayprettymuchunchanged.SoQuetelet’sneatbellcurvecouldsurvivefromonegenerationtothenext.Itspeakwouldquicklysettletoafixedvalue,theoverallmean,whileitswidthwouldstaythesame.Soeachgenerationwouldhavethesamediversityofheights,despiteregressiontothemean.Diversitywouldbemaintainedbyrareindividualswhofailedtoregressandwasself-sustaininginasufficientlylargepopulation.

Withthecentralroleofthebellcurvefirmlycementedtowhatatthetimewereconsideredsolidfoundations,statisticianscouldbuildonGalton’sinsightsand

workersinotherfieldscouldapplytheresults.Socialsciencewasanearlybeneficiary,butbiologysoonfollowed,andthephysicalscienceswerealreadyaheadofthegamethankstoLegendre,Laplace,andGauss.Soonanentirestatisticaltoolboxwasavailableforanyonewhowantedtoextractpatternsfromdata.I’llfocusonjustonetechnique,becauseitisroutinelyusedtodeterminetheefficacyofdrugsandmedicalprocedures,alongwithmanyotherapplications.Itiscalledhypothesistesting,anditsgoalistoassessthesignificanceofapparentpatternsindata.Itwasfoundedbyfourpeople:theEnglishmenRonaldAylmerFisher,KarlPearson,andhissonEgon,togetherwithaRussian-bornPolewhospentmostofhislifeinAmerica,JerzyNeyman.I’llconcentrateonFisher,whodevelopedthebasicideaswhenworkingasanagriculturalstatisticianatRothamsteadExperimentalStation,analysingnewbreedsofplants.

Supposeyouarebreedinganewvarietyofpotato.Yourdatasuggestthatthisbreedismoreresistanttosomepest.Butallsuchdataaresubjecttomanysourcesoferror,soyoucan’tbefullyconfidentthatthenumberssupportthatconclusion–certainlynotasconfidentasaphysicistwhocanmakeveryprecisemeasurementsandeliminatemosterrors.Fisherrealisedthatthekeyissueistodistinguishagenuinedifferencefromonearisingpurelybychance,andthatthewaytodothisistoaskhowprobablethatdifferencewouldbeifonlychancewereinvolved.

Assume,forinstance,thatthenewbreedofpotatoappearstoconfertwiceasmuchresistance,inthesensethattheproportionofthenewbreedthatsurvivesthepestisdoubletheproportionfortheoldbreed.Itisconceivablethatthiseffectisduetochance,andyoucancalculateitsprobability.Infact,whatyoucalculateistheprobabilityofaresultatleastasextremeastheoneobservedinthedata.Whatistheprobabilitythattheproportionofthenewbreedthatsurvivesthepestisatleasttwicewhatitwasfortheoldbreed?Evenlargerproportionsarepermittedherebecausetheprobabilityofgettingexactlytwicetheproportionisboundtobeverysmall.Thewidertherangeofresultsyouinclude,themoreprobabletheeffectsofchancebecome,soyoucanhavegreaterconfidenceinyourconclusionifyourcalculationsuggestsitisnottheresultofchance.Ifthisprobabilityderivedbythiscalculationislow,say0.05,thentheresultisunlikelytobetheresultofchance;itissaidtobesignificantatthe95%level.Iftheprobabilityislower,say0.01,thentheresultisextremelyunlikelytobetheresultofchance,anditissaidtobesignificantatthe99%level.Thepercentagesindicatethatbychancealone,theresultwouldnotbeasextremeastheoneobservedin95%oftrials,orin99%ofthem.

Fisherdescribedhismethodasacomparisonbetweentwodistincthypotheses:thehypothesisthatthedataaresignificantatthestatedlevel,andtheso-callednullhypothesisthattheresultsareduetochance.Heinsistedthathismethodmustnotbeinterpretedasconfirmingthehypothesisthatthedataaresignificant;itshouldbeinterpretedasarejectionofthenullhypothesis.Thatis,itprovidesevidenceagainstthedatanotbeingsignificant.

Thismayseemaveryfinedistinction,sinceevidenceagainstthedatanotbeingsignificantsurelycountsasevidenceinfavourofitbeingsignificant.However,that’snotentirelytrue,andthereasonisthatthenullhypothesishasanextrabuilt-inassumption.Inordertocalculatetheprobabilitythataresultatleastasextremeisduetochance,youneedatheoreticalmodel.Thesimplestwaytogetoneistoassumeaspecificprobabilitydistribution.Thisassumptionappliesonlyinconnectionwiththenullhypothesis,becausethat’swhatyouusetodothesums.Youdon’tassumethedataarenormallydistributed.Butthedefaultdistributionforthenullhypothesisisnormal:thebellcurve.

Thisbuilt-inmodelhasanimportantconsequence,which‘rejectthenullhypothesis’tendstoconceal.Thenullhypothesisis‘thedataareduetochance’.Soitisalltooeasytoreadthatstatementas‘rejectthedatabeingduetochance’,whichinturnmeansyouacceptthatthey’renotduetochance.Actually,though,thenullhypothesisis‘thedataareduetochanceandtheeffectsofchancearenormallydistributed’,sotheremightbetworeasonstorejectthenullhypothesis:thedataarenotduetochance,ortheyarenotnormallydistributed.Thefirstsupportsthesignificanceofthedata,buttheseconddoesnot.Itsaysyoumightbeusingthewrongstatisticalmodel.

InFisher’sagriculturalwork,therewasgenerallyplentyofevidencefornormaldistributionsinthedata.SothedistinctionI’mmakingdidn’treallymatter.Inotherapplicationsofhypothesistesting,though,itmight.Sayingthatthecalculationsrejectthenullhypothesishasthevirtueofbeingtrue,butbecausetheassumptionofanormaldistributionisnotexplicitlymentioned,itisalltooeasytoforgetthatyouneedtochecknormalityofthedistributionofthedatabeforeyouconcludethatyourresultsarestatisticallysignificant.Asthemethodgetsusedbymoreandmorepeople,whohavebeentrainedinhowtodothesumsbutnotintheassumptionsbehindthem,thereisagrowingdangerofwronglyassumingthatthetestshowsyourdatatobesignificant.Especiallywhenthenormaldistributionhasbecometheautomaticdefaultassumption.

Inthepublicconsciousness,theterm‘bellcurve’isindeliblyassociatedwiththecontroversial1994bookTheBellCurvebytwoAmericans,thepsychologist

RichardJ.HerrnsteinandthepoliticalscientistCharlesMurray.Themainthemeofthebookisaclaimedlinkbetweenintelligence,measuredbyintelligencequotient(IQ),andsocialvariablessuchasincome,employment,pregnancyrates,andcrime.TheauthorsarguethatIQlevelsarebetteratpredictingsuchvariablesthanthesocialandeconomicstatusoftheparentsortheirlevelofeducation.Thereasonsforthecontroversy,andtheargumentsinvolved,arecomplex.Aquicksketchcannotreallydojusticetothedebate,buttheissuesgorightbacktoQueteletanddeservemention.

Controversywasinevitable,nomatterwhattheacademicmeritsordemeritsofthebookmighthavebeen,becauseittouchedasensitivenerve:therelationbetweenraceandintelligence.MediareportstendedtostresstheproposalthatdifferencesinIQhaveapredominantlygeneticorigin,butthebookwasmorecautiousaboutthislink,leavingtheinteractionbetweengenes,environment,andintelligenceopen.AnothercontroversialissuewasananalysissuggestingthatsocialstratificationintheUnitedStates(andindeedelsewhere)increasedsignificantlythroughoutthetwentiethcentury,andthatthemaincausewasdifferencesinintelligence.Yetanotherwasaseriesofpolicyrecommendationsfordealingwiththisallegedproblem.Onewastoreduceimmigration,whichthebookclaimedwasloweringaverageIQ.Perhapsthemostcontentiouswasthesuggestionthatsocialwelfarepoliciesallegedlyencouragingpoorwomentohavechildrenshouldbestopped.

Ironically,thisideagoesbacktoGaltonhimself.His1869bookHereditaryGeniusbuiltonearlierwritingstodeveloptheideathat‘aman’snaturalabilitiesarederivedbyinheritance,underexactlythesamelimitationsasaretheformandphysicalfeaturesofthewholeorganicworld.Consequently…itwouldbequitepracticabletoproduceahighly-giftedraceofmenbyjudiciousmarriagesduringseveralconsecutivegenerations.’Heassertedthatfertilitywashigheramongthelessintelligent,butavoidedanysuggestionofdeliberateselectioninfavourofintelligence.Instead,heexpressedthehopethatsocietymightchangesothatthemoreintelligentpeopleunderstoodtheneedtohaveplentyofchildren.

Tomany,HerrnsteinandMurray’sproposaltore-engineerthewelfaresystemwasuncomfortablyclosetotheeugenicsmovementoftheearlytwentiethcentury,inwhich60,000Americansweresterilised,allegedlybecauseofmentalillness.EugenicsbecamewidelydiscreditedwhenitbecameassociatedwithNaziGermanyandtheholocaust,andmanyofitspracticesarenowconsideredtobeviolationsofhumanrightslegislation,insomecasesamountingtocrimesagainsthumanity.Proposalstobreedhumansselectivelyarewidelyviewedas

inherentlyracist.Anumberofsocialscientistsendorsedthebook’sscientificconclusionsbutdisputedthechargeofracism;someofthemwerelesssureaboutthepolicyproposals.TheBellCurveinitiatedalengthydebateaboutthemethodsusedtocompile

data,themathematicalmethodsusedtoanalysethem,theinterpretationoftheresults,andthepolicysuggestionsbasedonthoseinterpretations.AtaskforcesetupbytheAmericanPsychologicalAssociationconcludedthatsomepointsmadeinthebookarevalid:IQscoresaregoodforpredictingacademicachievement,thiscorrelateswithemploymentstatus,andthereisnosignificantdifferenceintheperformanceofmalesandfemales.Ontheotherhand,thetaskforce’sreportreaffirmedthatbothgenesandenvironmentinfluenceIQanditfoundnosignificantevidencethatracialdifferencesinIQscoresaregeneticallydetermined.

Othercriticshavearguedthatthereareflawsinthescientificmethodology,suchasinconvenientdatabeingignored,andthatthestudyandsomeresponsestoitmaytosomeextenthavebeenpoliticallymotivated.Forexample,itistruethatsocialstratificationhasincreaseddramaticallyintheUnitedStates,butitcouldbearguedthatthemaincauseistherefusaloftherichtopaytaxes,ratherthandifferencesinintelligence.Therealsoseemstobeaninconsistencybetweentheallegedproblemandtheproposedsolution.Ifpovertycausespeopletohavemorechildren,andyoubelievethatthisisabadthing,whyonearthwouldyouwanttomakethemevenpoorer?

Animportantpartofthebackground,oftenignored,isthedefinitionofIQ.Ratherthanbeingsomethingdirectlymeasurable,suchasheightorweight,IQisinferredstatisticallyfromtests.Subjectsaresetquestions,andtheirscoresareanalysedusinganoffshootofthemethodofleastsquarescalledanalysisofvariance.Likethemethodofleastsquares,thistechniqueassumesthatthedataarenormallydistributed,anditseekstoisolatethosefactorsthatdeterminethelargestamountofvariabilityinthedata,andarethereforethemostimportantformodellingthedata.In1904thepsychologistCharlesSpearmanappliedthistechniquetoseveraldifferentintelligencetests.Heobservedthatthescoresthatsubjectsobtainedondifferenttestswerehighlycorrelated;thatis,ifsomeonedidwellononetest,theytendedtodowellonthemall.Intuitively,theyseemedtobemeasuringthesamething.Spearman’sanalysisshowedthatasinglecommonfactor–onemathematicalvariable,whichhecalledg,standingfor‘generalintelligence’–explainedalmostallofthecorrelation.IQisastandardisedversionofSpearman’sg.

Akeyquestioniswhethergisarealquantityoramathematicalfiction.TheansweriscomplicatedbythemethodsusedtochooseIQtests.Theseassumethatthe‘correct’distributionofintelligenceinthepopulationisnormal–theeponymousbellcurve–andcalibratethetestsbymanipulatingscoresmathematicallytostandardisethemeanandstandarddeviation.Apotentialdangerhereisthatyougetwhatyouexpectbecauseyoutakestepstofilteroutanythingthatwouldcontradictit.StephenJayGouldmadeanextensivecritiqueofsuchdangersin1981inTheMismeasureofMan,pointingoutamongotherthingsthatrawscoresonIQtestsareoftennotnormallydistributedatall.

Themainreasonforthinkingthatgrepresentsagenuinefeatureofhumanintelligenceisthatitisonefactor:mathematically,itdefinesasingledimension.Ifmanydifferenttestsallseemtobemeasuringthesamething,itistemptingtoconcludethatthethingconcernedmustbereal.Ifnot,whywouldtheresultsallbesosimilar?PartoftheanswercouldbethattheresultsofIQtestsarereducedtoasinglenumericalscore.Thissquashesamultidimensionalsetofquestionsandpotentialattitudesdowntoaone-dimensionalanswer.Moreover,thetesthasbeenselectedsothatthescorecorrelatesstronglywiththedesigner’sviewofintelligentanswers–ifnot,noonewouldconsiderusingit.

Byanalogy,imaginecollectingdataonseveraldifferentaspectsof‘size’intheanimalkingdom.Onemightmeasuremass,anotherheight,otherslength,width,diameteroflefthindleg,toothsize,andsoon.Eachsuchmeasurewouldbeasinglenumber.Theywouldingeneralbecloselycorrelated:tallanimalstendtoweightmore,havebiggerteeth,thickerlegs…Ifyouranthedatathroughananalysisofvarianceyouwouldveryprobablyfindthatasinglecombinationofthosedataaccountedforthevastmajorityofthevariability,justlikeSpearman’sgdoesfordifferentmeasurementsofthingsthoughttorelatetointelligence.Wouldthisnecessarilyimplythatallofthesefeaturesofanimalshavethesameunderlyingcause?Thatonethingcontrolsthemall?Possibly:agrowthhormonelevel,perhaps?Butprobablynot.Therichnessofanimalformdoesnotcomfortablycompressintoasinglenumber.Manyotherfeaturesdonotcorrelatewithsizeatall:abilitytofly,beingstripedorspotted,eatingfleshorvegetation.Thesinglespecialcombinationofmeasurementsthataccountsformostofthevariabilitycouldbeamathematicalconsequenceofthemethodsusedtofindit–especiallyifthosevariableswerechosen,ashere,tohavealotincommontobeginwith.

GoingbacktoSpearman,weseethathismuch-vauntedgmaybeone-dimensionalbecauseIQtestsareone-dimensional.IQisastatisticalmethodforquantifyingspecifickindsofproblem-solvingability,mathematicallyconvenient

butnotnecessarilycorrespondingtoarealattributeofthehumanbrain,andnotnecessarilyrepresentingwhateveritisthatwemeanby‘intelligence’.

Byfocusingononeissue,IQ,andusingthattosetpolicy,TheBellCurveignoresthewidercontext.Evenifitweresensibletogeneticallyengineeranation’spopulation,whyconfinetheprocesstothepoor?EvenifonaveragethepoorhavelowerIQsthantherich,abrightpoorchildwilloutperformadumbrichoneanyday,despitetheobvioussocialandeducationaladvantagesthatchildrenoftherichenjoy.Whyresorttowelfarecutswhenyoucouldaimmoreaccuratelyatwhatyouclaimtobetherealproblem:intelligenceitself?Whynotimproveeducation?Indeed,whyaimyourpolicyatincreasingintelligenceatall?Therearemanyotherdesirablehumantraits.Whynotreducegullibility,aggressiveness,orgreed?

Itisamistaketothinkaboutamathematicalmodelasifitwerethereality.Inthephysicalsciences,wherethemodeloftenfitsrealityverywell,thismaybeaconvenientwayofthinkingthatcauseslittleharm.Butinthesocialsciences,modelsareoftenlittlebetterthancaricatures.ThechoiceoftitleforTheBellCurvehintsatthistendencytoconflatemodelwithreality.TheideathatIQissomesortofprecisemeasureofhumanability,merelybecauseithasamathematicalpedigree,makesthesameerror.Itisnotsensibletobasesweepingandhighlycontentioussocialpolicyonsimplistic,flawedmathematicalmodels.TherealpointaboutTheBellCurve,onethatitmakesextensivelybutinadvertently,isthatcleverness,intelligence,andwisdomarenotthesame.

Probabilitytheoryiswidelyusedinmedicaltrialsofnewdrugsandtreatmentstotestthestatisticalsignificanceofdata.Thetestsareoften,butnotalways,basedontheassumptionthattheunderlyingdistributionisnormal.Atypicalexampleisthedetectionofcancerclusters.Acluster,forsomedisease,isagroupwithinwhichthediseaseoccursmorefrequentlythanexpectedintheoverallpopulation.Theclustermaybegeographical,oritmayrefermoremetaphoricallytopeoplewithaparticularlifestyle,oraspecificperiodoftime.Forexample,retiredprofessionalwrestlers,orboysbornbetween1960and1970.

Apparentclustersmaybedueentirelytochance.Randomnumbersareseldomspreadoutinaroughlyuniformway;instead,theyoftenclustertogether.InrandomsimulationsoftheUKNationalLottery,wheresixnumbersbetween1and49arerandomlydrawn,morethanhalfappeartoshowsomekindofregularpatternsuchastwonumbersbeingconsecutiveorthreenumbersseparatedbythesameamount,forexample5,9,13.Contrarytocommon

intuition,randomisclumped.Whenanapparentclusterisfound,themedicalauthoritiestrytoassesswhetheritisduetochanceorwhethertheremightbesomepossiblecausalconnection.Atonetime,mostchildrenofIsraelifighterpilotswereboys.Itwouldbeeasytothinkofpossibleexplanations–pilotsareveryvirileandvirilemensiremoreboys(nottrue,bytheway),pilotsareexposedtomoreradiationthannormal,theyexperiencehigherg-forces–butthisphenomenonwasshortlived,justarandomcluster.Inlaterdataitdisappeared.Inanypopulationofpeople,itisalwayslikelythattherewillbemorechildrenofonesexortheother;exactequalityisveryimprobable.Toassessthesignificanceofthecluster,youshouldkeepobservingandseewhetheritpersists.

However,thisprocrastinationcan’tbecontinuedindefinitely,especiallyiftheclusterinvolvesaseriousdisease.AIDSwasfirstdetectedasaclusterofpneumoniacasesinAmericanhomosexualmeninthe1980s,forinstance.Asbestosfibresasacauseofaformoflungcancer,mesothelioma,firstshowedupasaclusteramongformerasbestosworkers.Sostatisticalmethodsareusedtoassesshowprobablesuchaclusterwouldbeifitaroseforrandomreasons.Fisher’smethodsofsignificancetesting,andrelatedmethods,arewidelyusedforthatpurpose.

Probabilitytheoryisalsofundamentaltoourunderstandingofrisk.Thiswordhasaspecific,technicalmeaning.Itreferstothepotentialforsomeactiontoleadtoanundesirableoutcome.Forexample,flyinginanaircraftcouldresultinbeinginvolvedinacrash,smokingcigarettescouldleadtolungcancer,buildinganuclearpowerstationcouldleadtothereleaseofradiationinanaccidentoraterroristattack,buildingadamforhydroelectricpowercouldcausedeathsifthedamcollapses.‘Action’herecanrefertonotdoingsomething:failingtovaccinateachildmightleadtoitsdeathfromadisease,forexample.Inthiscasethereisalsoariskassociatedwithvaccinatingthechild,suchasanallergicreaction.Overthewholepopulationthisriskissmaller,butforspecificgroupsitcanbelarger.

Manydifferentconceptsofriskareemployedindifferentcontexts.Theusualmathematicaldefinitionisthattheriskassociatedwithsomeactionorinactionistheprobabilityofanadverseresult,multipliedbythelossthatwouldthenbeincurred.Bythisdefinitionaoneintenchanceofkillingtenpeoplehasthesamelevelofriskasaoneinamillionchanceofkillingamillionpeople.Themathematicaldefinitionisrationalinthesensethatthereisaspecificrationalebehindit,butthatdoesn’tmeanthatitisnecessarilysensible.We’vealreadyseenthat‘probability’referstothelongrun,butforrareeventsthelongrunis

verylongindeed.Humans,andtheirsocieties,canadapttorepeatedsmallnumbersofdeaths,butacountrythatsuddenlylostamillionpeopleatoncewouldbeinserioustrouble,becauseallpublicservicesandindustrywouldsimultaneouslycomeunderaseverestrain.Itwouldbelittlecomforttobetoldthatoverthenext10millionyears,thetotaldeathsinthetwocaseswouldbecomparable.Sonewmethodsarebeingdevelopedtoquantifyriskinsuchcases.

Statisticalmethods,derivedfromquestionsaboutgambling,haveahugevarietyofuses.Theyprovidetoolsforanalysingsocial,medical,andscientificdata.Likealltools,whathappensdependsonhowtheyareused.Anyoneusingstatisticalmethodsneedstobeawareoftheassumptionsbehindthosemethods,andtheirimplications.Blindlyfeedingnumbersintoacomputerandtakingtheresultsasgospel,withoutunderstandingthelimitationsofthemethodbeingused,isarecipefordisaster.Thelegitimateuseofstatistics,however,hasimprovedourworldoutofallrecognition.AnditallbeganwithQuetelet’sbellcurve.

8Goodvibrations

WaveEquation

Whatdoesitsay?

Theaccelerationofasmallsegmentofaviolinstringisproportionaltotheaveragedisplacementofneighbouringsegments.

Whyisthatimportant?

Itpredictsthatthestringwillmoveinwaves,anditgeneralisesnaturallytootherphysicalsystemsinwhichwavesoccur.

Whatdiditleadto?

Bigadvancesinourunderstandingofwaterwaves,soundwaves,lightwaves,elasticvibrations…SeismologistsusemodifiedversionsofittodeducethestructureoftheinterioroftheEarthfromhowitvibrates.Oilcompaniesusesimilarmethodstofindoil.InChapter11wewillseehowitpredictedtheexistenceofelectromagneticwaves,leadingtoradio,television,radar,andmoderncommunications.

Weliveinaworldofwaves.Ourearsdetectwavesofcompressionintheair:wecallthis‘hearing’.Oureyesdetectwavesofelectromagneticradiation:wecallthis‘seeing’.Whenanearthquakehitsatownoracity,thedestructioniscausedbywavesinthesolidbodyoftheEarth.Whenashipbobsupanddownontheocean,itisreactingtowavesinthewater.Surfersuseoceanwavesforrecreation;radio,television,andlargepartsofthemobiletelephonenetworkusewavesofelectromagneticradiation,similartothosethatweseeby,butofdifferingwavelengths.Microwaveovens…well,thenamegivesitaway,doesn’tit?

Withsomanypracticalinstancesofwavesimpingingondailylife,evencenturiesago,themathematicianswhodecidedtofollowupNewton’sepicdiscoverythatnaturehaslawscouldhardlyfailtostartthinkingaboutwaves.Whatgotthemstarted,though,camefromthearts:specifically,music.Howdoesaviolinstringcreatesound?Whatdoesitdo?

Therewasareasonforstartingwithviolins,thekindofreasonthatappealstomathematicians,thoughnottogovernmentsorbusinessmenconsideringinvestinginmathematiciansandexpectingaquickpayback.Aviolinstringcansensiblybemodelledasaninfinitelythinline,anditsmotion–whichisclearlythecauseofthesoundthattheinstrumentmakes–canbeassumedtotakeplaceinaplane.Thismakestheproblem‘low-dimensional’,whichmeansyouhaveachanceofsolvingit.Onceyouhaveunderstoodthissimpleexampleofwaves,there’sagoodchancethattheunderstandingcanbetransferred,ofteninsmallstages,tomorerealisticandmorepracticalinstancesofwaves.

Thealternative,toploughheadlongintohighlycomplexproblems,mayappearattractivetopoliticiansandcaptainsofindustry,butitusuallygetsboggeddownincomplexities.Mathematicsthrivesonsimplicities,andifnecessarymathematicianswillinventthemartificiallytoprovideanentryrouteintomorecomplexproblems.Theydeprecatinglyrefertosuchmodelsas‘toys’,butthesearetoyswithaseriouspurpose.Toymodelsofwavesledtotoday’sworldofelectronicsandhigh-speedglobalcommunications,wide-bodiedpassengerjetsandartificialsatellites,radio,television,tsunamiwarningsystems…butwe’dneverhaveachievedanyofthoseifafewmathematicianshadn’tstartedtopuzzleouthowaviolinworks,usingamodelthatwasn’trealistic,evenforaviolin.

ThePythagoreansbelievedthattheworldwasbasedonnumbers,bywhichtheymeantwholenumbersorratiosbetweenwholenumbers.Someoftheirbeliefstendedtowardsthemystical,investingspecificnumberswithhumanattributes:2wasmale,3female,5symbolisedmarriage,andsoon.Thenumber10wasveryimportanttothePythagoreansbecauseitwas1+2+3+4andtheybelievedtherewerefourelements:earth,air,fire,water.Thiskindofspeculationstrikesthemodernmindasslightlycrazy–well,mymind,atleast–butitwasreasonableinanagewhenhumanswereonlyjuststartingtoinvestigatetheworldaroundthem,seekingcrucialpatterns.Itjusttookawhiletoworkoutwhichpatternsweresignificantandwhichweredross.

OneofthegreattriumphsofthePythagoreanworldviewcamefrommusic.Variousstoriescirculate:accordingtoone,Pythagoraswaspassingablacksmith’sshopandhenoticedthathammersofdifferentsizesmadenoisesofdifferentpitch,andthathammersrelatedbysimplenumbers–onetwicethesizeoftheother,forinstance–madenoisesthatharmonised.Charmingthoughthistaleis,anyonewhoactuallytriesitoutwithrealhammerswilldiscoverthatablacksmith’soperationsarenotespeciallymusical,andhammersaretoocomplicatedashapetovibrateinharmony.Butthere’sagrainoftruth:onthewhole,smallobjectsmakehigher-pitchednoisesthanlargeones.

ThestoriesareonstrongergroundwhentheyrefertoaseriesofexperimentsthatthePythagoreansperformedusingastretchedstring,arudimentarymusicalinstrumentknownasacanon.WeknowabouttheseexperimentsbecausePtolemyreportedtheminhisHarmonicsaround150AD.Bymovingasupporttovariouspositionsalongthestring,thePythagoreansfoundthatwhentwostringsofequaltensionhadlengthsinasimpleratio,suchas2:1or3:2,theyproducedunusuallyharmoniousnotes.Morecomplexratioswerediscordantandunpleasanttotheear.Laterscientistspushedtheseideasmuchfurther,probablyabittoofar:whatseemspleasanttousdependsonthephysicsoftheear,whichismorecomplicatedthanthatofasinglestring,anditalsohasaculturaldimensionbecausetheearsofgrowingchildrenaretrainedbybeingexposedtothesoundsthatarecommonintheirsociety.Ipredictthattoday’schildrenwillbeunusuallysensitivetodifferencesinmobilephoneringtones.However,thereisasolidscientificstorybehindthesecomplexities,andalotofitconfirmsandexplainstheearlyPythagoreandiscoverieswiththeirsingle-stringedexperimentalinstrument.

Musiciansdescribepairsofnotesintermsoftheintervalbetweenthem,ameasureofhowmanystepsseparatetheminsomemusicalscale.Themostfundamentalintervalistheoctave,eightwhitenotesonapiano.Notesanoctave

apartsoundremarkablysimilar,exceptthatonenoteishigherthantheother,andtheyareextremelyharmonious.Somuchso,infact,thatharmoniesbasedontheoctavecanseemabitbland.Onaviolin,thewaytoplaythenoteoneoctaveaboveanopenstringistopressthemiddleofthatstringagainstthefingerboard.Astringhalfaslongplaysanoteoneoctavehigher.Sotheoctaveisassociatedwithasimplenumericalratioof2:1.

Otherharmoniousintervalsarealsoassociatedwithsimplenumericalratios.ThemostimportantforWesternmusicarethefourth,aratioof4:3,andthefifth,aratioof3:2.ThenamesmakesenseifyouconsideramusicalscaleofwholenotesCDEFGABC.WithCasbase,thenotecorrespondingtoafourthisF,thefifthisG,andtheoctaveC.Ifwenumberthenotesconsecutivelywiththebaseas1,thesearerespectivelythe4th,5th,and8thnotesalongthescale.Thegeometryisespeciallyclearonaninstrumentlikeaguitar,whichhassegmentsofwire,‘frets’,insertedattherelevantpositions.Thefretforthefourthisone-quarterofthewayalongthestring,thatforafifthisone-thirdofthewayalong,andtheoctaveishalfwayalong.Youcancheckthiswithatapemeasure.

Theseratiosprovideatheoreticalbasisforamusicalscaleandledtothescale(s)nowusedinmostWesternmusic.Thestoryiscomplex,soI’llgiveasimplifiedversion.ForlaterconvenienceI’llrewritearatiolike3:2asafraction3/2fromnowon.Startatabasenoteandascendinfifths,togetstringsoflengths

Multipliedout,thesefractionsbecome

Allofthesenotes,exceptthefirsttwo,aretoohigh-pitchedtoremainwithinanoctave,butwecanlowerthembyoneormoreoctaves,repeatedlydividingthefractionsby2untiltheresultliesbetween1and2.Thisyieldsthefractions

Finally,arrangetheseinascendingnumericalorder,obtaining

ThesecorrespondfairlycloselytothenotesCDEGABonapiano.NoticethatFismissing.Infact,totheear,thegapbetween81/64and3/2soundswiderthantheothers.Tofillthatgap,weinsert4/3,theratioforthefourth,whichisveryclosetoFonthepiano.ItisalsousefultocompletethescalewithasecondC,oneoctaveup,aratioof2.Nowweobtainamusicalscalebasedentirelyonfourths,fifths,andoctaves,withpitchesintheratios

Thelengthisinverselyproportionaltothepitch,sowewouldhavetoinvertthefractionstogetthecorrespondinglengths.

Wehavenowaccountedforallthewhitenotesonthepiano,buttherearealsoblacknotes.Theseappearbecausesuccessivenumbersinthescalebeartwodifferentratiostoeachother:9/8(calledatone)and256/243(semitone).Forexampletheratioof81/64to9/8is9/8,butthatof4/3to81/64is256/243.Thenames‘tone’and‘semitone’indicateanapproximatecomparisonoftheintervals.Numericallytheyare1.125and1.05.Thefirstislarger,soatonecorrespondstoabiggerchangeinpitchthanasemitone.Twosemitonesgivearatio1.052,whichisabout1.11;notfarfrom1.25.Sotwosemitonesareclosetoatone.Notveryclose,Iadmit.

Continuinginthisveinwecandivideeachtoneintotwointervals,eachclosetoasemitone,togeta12-notescale.Thiscanbedoneinseveraldifferentways,yieldingslightlydifferentresults.Howeveritisdone,therecanbesubtlebutaudibleproblemswhenchangingthekeyofapieceofmusic:theintervalschangeslightlyif,say,wemoveeverynoteupasemitone.Thiseffectcouldhavebeenavoidedifwehadchosenaspecificratioforasemitoneandarrangedforitstwelfthpowertoequal2.Thentwotoneswouldmakeanexactsemitone,12semitoneswouldmakeanoctave,andyoucouldchangescalebyshiftingallnotesupordownbyafixedamount.

Thereissuchanumber,namelythetwelfthrootof2,whichisabout1.059,anditleadstotheso-called‘equitemperedscale’.It’sacompromise;forexampleontheequitemperedscalethe4/3ratioforafourthis1.0595=1.335,insteadof4/3=1.333.Ahighlytrainedmusiciancandetectthedifference,but

it’seasytogetusedtoitandmostofusnevernotice.

ThePythagoreantheoryofharmonyinnature,then,isactuallybuiltintothebasisofWesternmusic.Toexplainwhysimpleratiosgohandinhandwithmusicalharmony,wehavetolookatthephysicsofavibratingstring.Thepsychologyofhumanperceptionalsocomesintothetale,butnotyet.

ThekeyisNewton’ssecondlawofmotion,relatingaccelerationtoforce.Youalsoneedtoknowhowtheforceexertedbyastringundertensionchangesasthestringmoves,stretchingorcontractingslightly.Forthis,weusesomethingthatNewton’sunwillingsparringpartnerHookediscoveredin1660,calledHooke’slaw:thechangeinlengthofaspringisproportionaltotheforceexertedonit.(Thisisnotamisprintforstring–aviolinstringiseffectivelyakindofspring,sothesamelawapplies.)Oneobstacleremains.WecanapplyNewton’slawstoasystemcomposedofafinitenumberofmasses:wegetoneequationpermass,andthendoourbesttosolvetheresultingsystem.Butaviolinstringisacontinuum,alinecomposedofinfinitelymanypoints.Sothemathematiciansoftheperiodthoughtofthestringasalargenumberofcloselyspacedpointmasses,linkedtogetherbyHooke’s-lawsprings.Theywrotedowntheequations,slightlysimplifiedtomakethemsoluble;solvedthem;finallytheyletthenumberofmassesbecomearbitrarilylarge,andworkedoutwhathappenedtothesolution.

JohnBernoullicarriedoutthisprogrammein1727,andtheoutcomewasextraordinarilypretty,consideringwhatdifficultieswerebeingsweptunderthecarpet.Toavoidconfusioninthedescriptionsthatfollow,imaginethattheviolinislyingonitsbackwiththestringhorizontal.Ifyoupluckthestringitvibratesupanddownatrightanglestotheviolin.Thisistheimagetobearinmind.Usingthebowcausesthestringtovibratesideways,andthepresenceofthebowisconfusing.Inthemathematicalmodel,allwehaveisonestring,fixedatitsends,andnoviolin;thestringvibratesupanddowninaplane.Inthisset-upBernoullifoundthattheshapeofthevibratingstring,atanyinstantoftime,wasasinecurve.Theamplitudeofthevibration–themaximumheightofthiscurve–alsofollowedasinecurve,intimeratherthanspace.Insymbols,hissolutionlookedlikesinctsinx,wherecisaconstant,Figure35.Thespatialpartsinxtellsustheshape,butthisisscaledbyafactorsinctattimet.Theformulasaysthatthestringvibratesupanddown,repeatingthesamemotionoverandoveragain.Theperiodofoscillation,thetimebetweensuccessiverepeats,is2π/c.

Fig35Successivesnapshotsofavibratingstring.Theshapeisasinecurveateachinstant.Theamplitudealsovariessinusoidallywithtime.

ThiswasthesimplestsolutionthatBernoulliobtained,buttherewereothers;allofthemsinecurves,different‘modes’ofvibration,with1,2,3,ormorewavesalongthelengthofthestring,Figure36.Again,thesinecurvewasasnapshotoftheshapeatanyinstant,anditsamplitudewasmultipliedbyatime-dependentfactor,whichalsovariedsinusoidally.Theformulasweresin2ctsin2x,sin3ctsin3x,andsoon.Thevibrationalperiodswere2π/2c,2π/3c,andsoon;sothemorewavestherewere,thefasterthestringvibrated.

Fig36Snapshotsofmodes1,2,3ofavibratingstring.Ineachcase,thestringvibratesupanddown,anditsamplitudevariessinusoidallywithtime.Themorewavesthereare,thefasterthevibration.

Thestringisalwaysatrestatitsends,bytheconstructionoftheinstrumentandtheassumptionsofthemathematicalmodel.Inallmodesexceptthefirst,thereareadditionalpointswherethestringisnotvibrating;theseoccurwherethecurvecrossesthehorizontalaxis.These‘nodes’arethemathematicalreasonfortheoccurrenceofsimplenumericalratiosinthePythagoreanexperiments.Forexample,sincevibrationalmodes2and3occurinthesamestring,thegapbetweensuccessivenodesinthemode-2curveis3/2timesthecorrespondinggapinthemode-3curve.Thisexplainswhyratioslike3:2arisenaturallyfromthedynamicsofthevibratingspring,butnotwhytheseratiosareharmonious

whileothersarenot.Beforetacklingthisquestion,weintroducethemaintopicofthischapter:thewaveequation.

ThewaveequationemergesfromNewton’ssecondlawofmotionifweapplyBernoulli’sapproachatthelevelofequationsratherthansolutions.In1746JeanLeRondd’Alembertfollowedstandardprocedure,treatingavibratingviolinstringasacollectionofpointmasses,butinsteadofsolvingtheequationsandlookingforapatternwhenthenumberofmassestendedtoinfinity,heworkedoutwhathappenedtotheequationsthemselves.Hederivedanequationthatdescribedhowtheshapeofthestringchangesovertime.ButbeforeIshowyouwhatitlookslike,weneedanewidea,calleda‘partialderivative’.

Imagineyourselfinthemiddleoftheocean,watchingwavesofvariousshapesandsizespassby.Astheydoso,youbobupanddown.Physically,youcandescribehowyoursurroundingsarechanginginseveraldifferentways.Inparticular,youcanfocusonchangesintimeorchangesinspace.Astimepassesatyourlocation,therateatwhichyourheightchanges,withrespecttotime,isthederivative(inthesenseofcalculus,Chapter3)ofyourheight,alsowithrespecttotime.Butthisdoesn’tdescribetheshapeoftheoceannearyou,justhowhighthewavesareastheypassyou.Todescribetheshape,youcanfreezetime(conceptually)andworkouthowhighthewavesare:notjustatyourlocation,butatnearbyones.Thenyoucanusecalculustoworkouthowsteeplythewaveslopesatyourlocation.Areyouatapeakortrough?Ifso,theslopeiszero.Areyouhalfwaydownthesideofawave?Ifso,theslopeisquitelarge.Intermsofcalculus,youcanputanumbertothatslopebyworkingoutthederivativeofthewave’sheightwithrespecttospace.

Ifafunctionudependsonjustonevariable,callitx,wewritethederivativeasdu/dx:‘smallchangeinudividedbysmallchangeinx’.Butinthecontextofoceanwavesthefunctionu,thewaveheight,dependsnotjustonspacexbutalsoontimet.Atanyfixedinstantoftime,wecanstillworkoutdu/dx;ittellsusthelocalslopeofthewave.Butinsteadoffixingtimeandlettingspacevary,wecanalsofixspaceandlettimevary;thistellsustherateatwhichwearebobbingupanddown.Wecouldusethenotationdu/dtforthis‘timederivative’andinterpretitas‘smallchangeinudividedbysmallchangeint’.Butthisnotationhidesanambiguity:thesmallchangeinheight,du,maybe,andusuallyis,differentinthetwocases.Ifyouforgetthat,youarelikelytogetyoursumswrong.Whenwearedifferentiatingwithrespecttospace,weletthespacevariablechangealittlebitandseehowtheheightchanges;whenwearedifferentiatingwithrespecttotime,weletthetimevariablechangealittlebit

andseehowtheheightchanges.Thereisnoreasonwhychangesovertimeshouldequalchangesoverspace.

Somathematiciansdecidedtoremindthemselvesofthisambiguitybychangingthesymboldtosomethingthatdidnot(directly)makethemthink‘smallchange’.Theysettledonaverycutecurlyd,written∂.Thentheywrotethetwoderivativesas∂u/∂xand∂u/∂t.Youcouldarguethatthisisn’tabigadvance,becauseit’sjustaseasytoconfusetwodifferentmeaningsof∂u.Therearetwoanswerstothiscriticism.Oneisthatinthiscontextyouarenotsupposedtothinkof∂uasaspecificsmallchangeinu.Theotheristhatusingafancynewsymbolremindsyounottogetconfused.Thesecondanswerdefinitelyworks:assoonasyousee∂,ittellsyouthatyouwillbelookingatratesofchangewithrespecttoseveraldifferentvariables.Theseratesofchangearecalledpartialderivatives,becauseconceptuallyyouchangeonlypartofthesetofvariables,keepingtherestfixed.

Whend’Alembertworkedouthisequationforthevibratingstring,hefacedjustthissituation.Theshapeofthestringdependsonspace–howfaralongthestringyoulook–andontime.Newton’ssecondlawofmotiontoldhimthattheaccelerationofasmallsegmentofstringisproportionaltotheforcethatactsonit.Accelerationisa(second)timederivative.Buttheforceiscausedbyneighbouringsegmentsofthestringpullingontheonewe’reinterestedin,and‘neighbouring’meanssmallchangesinspace.Whenhecalculatedthoseforces,hewasledtotheequation

whereu(x,t)istheverticalpositionatlocationxonthestringattimet,andcisaconstantrelatedtothetensioninthestringandhowspringyitis.ThecalculationswereactuallyeasierthanBernoulli’s,becausetheyavoidedintroducingspecialfeaturesofparticularsolutions.1

D’Alembert’selegantformulaisthewaveequation.LikeNewton’ssecondlaw,itisadifferentialequation–itinvolves(second)derivativesofu.Sincethesearepartialderivatives,itisapartialdifferentialequation.Thesecondspacederivativerepresentsthenetforceactingonthestring,andthesecondtimederivativeistheacceleration.Thewaveequationsetaprecedent:mostofthekeyequationsofclassicalmathematicalphysics,andalotofthemodernonesforthatmatter,arepartialdifferentialequations.

Onced’Alemberthadwrittendownhiswaveequation,hewasinapositiontosolveit.Thistaskwasmademucheasierbecauseitturnedouttobealinearequation.Partialdifferentialequationshavemanysolutions,typicallyinfinitelymany,becauseeachinitialstateleadstoadistinctsolution.Forexample,theviolinstringcaninprinciplebebentintoanyshapeyoulikebeforeitisreleasedandthewaveequationtakesover.‘Linear’meansthatifu(x,t)andv(x,t)aresolutions,thensoisanylinearcombinationau(x,t)+bv(x,t),whereaandbareconstants.Anothertermis‘superposition’.ThelinearityofthewaveequationstemsfromtheapproximationthatBernoulliandd’Alemberthadtomaketogetsomethingtheycouldsolve:alldisturbancesareassumedtobesmall.Nowtheforceexertedbythestringcanbecloselyapproximatedbyalinearcombinationofthedisplacementsoftheindividualmasses.Abetterapproximationwouldleadtoanonlinearpartialdifferentialequation,andlifewouldbefarmorecomplicated.Inthelongrun,thesecomplicationshavetobetackledhead-on,butthepioneershadenoughtocontendwithalready,sotheyworkedwithanapproximatebutveryelegantequationandconfinedtheirattentiontosmall-amplitudewaves.Itworkedverywell.Infact,itoftenworkedprettywellforwavesoflargeramplitudetoo,aluckybonus.

D’Alembertknewhewasontherighttrackbecausehefoundsolutionsinwhichafixedshapetravelledalongthestring,justlikeawave.2Thespeedofthewaveturnedouttobetheconstantcintheequation.Thewavecouldtraveleithertotheleftortotheright,andherethesuperpositionprinciplecameintoplay.D’Alembertprovedthateverysolutionisasuperpositionoftwowaves,onetravellingleftwardsandtheotherrightwards.Moreover,eachseparatewavecouldhaveanyshapewhatsoever.3Thestandingwavesfoundintheviolinstring,withfixedends,turnouttobeacombinationoftwowavesofthesameshape,onebeingupsidedowncomparedtotheother,withonetravellingtotheleftandtheother(upsidedown)travellingtotheright.Attheends,thetwowavesexactlycanceleachotherout:peaksofonecoincidewithtroughsoftheother.Sotheycomplywiththephysicalboundaryconditions.

Mathematiciansnowhadanembarrassmentofriches.Thereweretwowaystosolvethewaveequation:Bernoulli’s,whichledtosinesandcosines,andd’Alembert’s,whichledtowaveswithanyshapewhatsoever.Atfirstitlookedasthoughd’Alembert’ssolutionmustbemoregeneral:sinesandcosinesarefunctions,butmostfunctionsarenotsinesandcosines.However,thewaveequationislinear,soyoucouldcombineBernoulli’ssolutionsbyaddingconstantmultiplesofthemtogether.Tokeepitsimpleconsiderjustasnapshotat

afixedtime,gettingridofthetime-dependence.Figure37shows5sinx+4sin2x−2cos6x,forexample.Ithasafairlyirregularshape,anditwigglesalot,butit’sstillsmoothandwavy.

Fig37Typicalcombinationofsinesandcosineswithvariousamplitudesandfrequencies.

Whatbotheredthemorethoughtfulmathematicianswasthatsomefunctionsareveryroughandjagged,andyoucan’tgetthoseasalinearcombinationofsinesandcosines.Well,notifyouusefinitelymanyterms–andthatsuggestedawayout.Aconvergentinfiniteseriesofsinesandcosines(onewhosesumtoinfinitymakessense)alsosatisfiesthewaveequation.Doesitallowjaggedfunctionsaswellassmoothones?Theleadingmathematiciansarguedaboutthisquestion,whichfinallycametoaheadwhenthesameissueturnedupinthetheoryofheat.Problemsaboutheatflownaturallyinvolveddiscontinuousfunctions,withsuddenjumps,whichwasevenworsethanjaggedones.I’lltellthatstoryinChapter9,buttheupshotisthatmost‘reasonable’waveshapescanberepresentedbyaninfiniteseriesofsinesandcosines,sotheycanbeapproximatedascloselyasyouwishbyfinitecombinationsofsinesandcosines.

SinesandcosinesexplaintheharmoniousratiosthatsoimpressedthePythagoreans.Thesespecialshapesofwavesareimportantinthetheoryofsoundbecausetheyrepresent‘pure’tones–singlenotesonanidealinstrument,sotospeak.Anyrealinstrumentproducesmixturesofpurenotes.Ifyoupluckaviolinstring,themainnoteyouhearisthesinxwave,butsuperposedonthatisabitofsin2x,maybesomesin3x,andsoon.Themainnoteiscalledthefundamentalandtheothersareitsharmonics.Thenumberinfrontofxiscalled

thewavenumber.Bernoulli’scalculationstellusthatthewavenumberisproportionaltothefrequency:howmanytimesthestringvibrates,forthatparticularsinewave,duringasingleoscillationofthefundamental.

Inparticular,sin2xhastwicethefrequencyofsinx.Whatdoesitsoundlike?Itisthenoteoneoctavehigher.Thisisthenotethatsoundsmostharmoniouswhenplayedalongsidethefundamental.Ifyoulookattheshapeofthestringforthesecondmode(sin2x)inFigure36,you’llnoticethatitcrossestheaxisatitsmidpointaswellasthetwoends.Atthatpoint,aso-callednode,itremainsfixed.Ifyouplacedyourfingeratthatpoint,thetwohalvesofthestringwouldstillbeabletovibrateinthesin2xpattern,butnotinthesinxone.ThisexplainsthePythagoreandiscoverythatastringhalfaslongproducedanoteoneoctavehigher.Asimilarexplanationdealswiththeothersimpleratiosthattheydiscovered:theyareallassociatedwithsinecurveswhosefrequencieshavethatratio,andsuchcurvesfittogetherneatlyonastringoffixedlengthwhoseendsarenotallowedtomove.

Whydotheseratiossoundharmonious?Partoftheexplanationisthatsinewaveswithfrequenciesthatarenotinsimpleratiosproduceaneffectcalled‘beats’whentheyaresuperposed.Forinstance,aratiolike11:23correspondstosin11x+sin23x,whichlookslikeFigure38,withlotsofsuddenchangesinshape.Anotherpartisthattheearrespondstoincomingsoundsinroughlythesamewayastheviolinstring.Theear,too,vibrates.Whentwonotesbeat,thecorrespondingsoundislikeabuzzingnoisethatrepeatedlygetslouderandsofter.Soitdoesn’tsoundharmonious.However,thereisathirdpartoftheexplanation:theearsofbabiesbecomeattunedtothesoundsthattheyhearmostoften.Therearemorenerveconnectionsfromthebraintotheearthanthereareintheotherdirection.Sothebrainadjuststheear’sresponsetoincomingsounds.Inotherwords,whatweconsidertobeharmonioushasaculturaldimension.Butthesimplestratiosarenaturallyharmonious,somostculturesusethem.

Fig38Beats.

Mathematiciansfirstderivedthewaveequationinthesimplestsettingtheycouldthinkof:avibratingline,aone-dimensionalsystem.Realisticapplicationsrequiredamoregeneraltheory,modellingwavesintwoandthreedimensions.Evenstayingwithinmusic,adrumrequirestwodimensionstomodelthepatternsinwhichthedrumskinvibrates.Thesamegoesforwaterwavesonthesurfaceoftheocean.Whenanearthquakestrikes,thewholeEarthringslikeabell,andourplanetisthree-dimensional.Manyotherareasofphysicsinvolvemodelswithtwoorthreedimensions.Extendingthewaveequationtohigherdimensionsturnedouttobestraightforward;allyouhadtodowasrepeatthesamekindsofcalculationthathadworkedfortheviolinstring.Havinglearnedtoplaythegameinthissimplesetting,itwasn’thardtoplayitforreal.

Inthreedimensions,forexample,weusethreespacecoordinates(x,y,z)andtimet.Thewaveisdescribedbyafunctionuthatdependsonthesefourcoordinates.Forinstance,thismightdescribethepressureinabodyofairassoundwavespassthroughit.Makingthesameassumptionsasd’Alembert,inparticularthattheamplitudeofthedisturbanceissmall,thesameapproachleadstoanequallyprettyequation:

TheformulainsidethebracketsiscalledtheLaplacian,anditcorrespondstotheaveragedifferencebetweenthevalueofuatthepointinquestion,anditsvaluenearby.Thisexpressionarisessoofteninmathematicalphysicsthatithasitsownspecialsymbol:∇2u.TogettheLaplacianintwodimensions,wejustomit

theterminvolvingz,leadingtothewaveequationinthatsetting.Themainnoveltyinhigherdimensionsisthattheshapewithinwhichthe

wavesarise,calledthedomainoftheequation,canbecomplicated.Inonedimensiontheonlyconnectedshapeisaninterval,asegmentoftheline.Intwodimensions,however,itcanbeanyshapeyoucandrawintheplane,andinthreedimensions,anyshapeinspace.Youcanmodelasquaredrum,arectangulardrum,acirculardrum,4oradrumshapedlikethesilhouetteofacat.Forearthquakes,youmightemployasphericaldomain,orforgreateraccuracy,anellipsoidsquashedslightlyatthepoles.Ifyouaredesigningacarandwanttoeliminateunwantedvibrations,yourdomainshouldbecar-shaped–orwhateverpartofthecartheengineerswanttofocuson.

Fig39Left:Snapshotofonemodeofavibratingrectangulardrum,withwavenumbers2and3.Right:Snapshotofonemodeofavibratingcirculardrum.

Foranychosenshapeofdomain,therearefunctionsanalogoustoBernoulli’ssinesandcosines:thesimplestpatternsofvibration.Thesepatternsarecalledmodes,ornormalmodesifyouwanttomakeitabsolutelyclearwhatyou’retalkingabout.Allotherwavescanbeobtainedbysuperposingnormalmodes,againusinganinfiniteseriesifnecessary.Thefrequenciesofthenormalmodesrepresentthenaturalvibrationalfrequenciesofthedomain.Ifthedomainisarectangle,thesearetrigonometricfunctionsoftheformsinmxcosny,forintegersmandn,producingwavesshapedlikeFigure39(left).Ifitisacircle,theyaredeterminedbynewfunctions,calledBesselfunctions,withmoreinterestingshapes,Figure39(right).Theresultingmathematicsappliesnotonlytodrums,buttowaterwaves,soundwaves,electromagneticwavessuchaslight(Chapter11),evenquantumwaves(Chapter14).Itisfundamentaltoalloftheseareas.TheLaplacianalsoturnsupinequationsforotherphysicalphenomena;inparticular,electric,magnetic,andgravitationalfields.Themathematician’s

favouritetrickofstartingwithatoyproblem,onesosimplethatitcannotpossiblyberealistic,paysoffbigtimeforwaves.

Thisisonereasonwhyitisunwisetojudgeamathematicalideabythecontextinwhichitfirstarises.Modellingaviolinstringmayseempointlesswhenwhatyouwanttounderstandisearthquakes.Butifyoujumpinatthedeepend,andtrytocopewithallofthecomplexitiesofrealearthquakes,you’lldrown.Youshouldstartoutpaddlingintheshallowendandgainconfidencetoswimafewlengthsofthepool.Thenyou’llbereadyforthehighdivingboard.

Thewaveequationwasaspectacularsuccess,andinsomeareasofphysicsitdescribesrealityveryclosely.However,itsderivationrequiresseveralsimplifyingassumptions.Whenthoseassumptionsareunrealistic,thesamephysicalideascanbemodifiedtosuitthecontext,leadingtodifferentversionsofthewaveequation.

Earthquakesareatypicalexample.Herethemainproblemisnotd’Alembert’sassumptionthattheamplitudeofthewaveissmall,butchangesinthephysicalpropertiesofthedomain.Thesepropertiescanhaveastrongeffectonseismicwaves,vibrationsthattravelthroughtheEarth.Byunderstandingthoseeffects,wecanlookdeepinsideourplanetandfindoutwhatitismadeof.

Therearetwomainkindsofseismicwave:pressurewavesandshearwaves,usuallyabbreviatedtoP-wavesandS-waves.(Therearemanyothers:thisisasimplifiedaccount,coveringsomeofthebasics.)Bothcanoccurinasolidmedium,butS-wavesdon’toccurinfluids.P-wavesarewavesofpressure,analogoustosoundwavesinair,andthechangesinpressurepointinthedirectionalongwhichthewavepropagates.Suchwavesaresaidtobelongitudinal.S-wavesaretransversewaves,changingatrightanglestothedirectionoftravel,likethewavesonaviolinstring.Theycausesolidstoshear,thatis,deformlikeapackofcardspushedsideways,sothatthecardsslidealongoneanother.Fluidsdon’tbehavelikepacksofcards.

Whenanearthquakehappens,itsendsoutbothkindsofwave.TheP-wavestravelfaster,soaseismologistsomewhereelseontheEarth’ssurfaceobservesthosefirst.ThentheslowerS-wavesarrive.In1906theEnglishgeologistRichardOldhamexploitedthisdifferencetomakeamajordiscoveryaboutourplanet’sinterior.Roughlyspeaking,theEarthhasanironcore,surroundedbyarockymantle,andthecontinentsfloatontopofthemantle.Oldhamsuggestedthattheouterlayersofthecoremustbeliquid.Ifso,S-wavescan’tpassthroughthoseregions,butP-wavescan.SothereisakindofS-waveshadow,andyou

canworkoutwhereitisbyobservingsignalsfromearthquakes.TheEnglishmathematicianHaroldJeffreyssortedoutthedetailsin1926andconfirmedthatOldhamwasright.

Iftheearthquakeisbigenough,itcancausetheentireplanettovibrateinoneofitsnormalmodes–theanaloguesfortheEarthofsinesandcosinesforaviolin.Thewholeplanetringslikeabell,inasensethatwouldbeliteralifonlywecouldheartheverylowfrequenciesinvolved.Instrumentssensitiveenoughtorecordthesemodesappearedinthe1960s,andtheywereusedtoobservethetwomostpowerfulearthquakesyetrecordedscientifically.TheseweretheChileanearthquakeof1960(magnitude9.5)andtheAlaskanearthquakeof1964(magnitude9.2).Thefirstkilledaround5000people;thesecondkilledabout130thankstoitsremotelocation.Bothcausedtsunamisanddidahugeamountofdamage.BothofferedanunprecedentedviewoftheEarth’sdeepinterior,byexcitingtheEarth’sbasicvibrationalmodes.

Sophisticatedversionsofthewaveequationhavegivenseismologiststheabilitytoseewhat’shappeninghundredsofkilometresbeneathourfeet.TheycanmaptheEarth’stectonicplatesasoneslidesbeneathanother,knownassubduction.Subductioncausesearthquakes,especiallyso-calledmegathrustearthquakeslikethetwojustmentioned.Italsogivesrisetomountainchainsalongtheedgesofcontinents,suchastheAndes,andvolcanoes,wheretheplategetssodeepthatitstartstomeltandmagmarisestothesurface.Arecentdiscoveryisthattheplatesneednotsubductasawhole,butcanbreakupintogiganticslabs,sinkingbackintothemantleatdifferentdepths.

Thebiggestprizeinthisareawouldbeareliablewaytopredictearthquakesandvolcaniceruptions.Thisisprovingelusive,becausetheconditionsthattriggersucheventsarecomplexcombinationsofmanyfactorsinmanylocations.However,someprogressisbeingmade,andtheseismologists’versionofthewaveequationunderpinsmanyofthemethodsbeinginvestigated.

Thesameequationshavemorecommercialapplications.Oilcompaniesprospectforliquidgold,afewkilometresunderground,bysettingoffexplosionsatthesurfaceandusingreturningechoesfromtheseismicwavestheygeneratetomapouttheunderlyinggeology.Themainmathematicalproblemhereistoreconstructthegeologyfromthesignalsreceived,whichisabitlikeusingthewaveequationbackwards.Insteadofsolvingtheequationinaknowndomaintoworkoutwhatthewavesdo,mathematiciansusetheobservedwavepatternstoreconstructthegeologicalfeaturesofthedomain.Asisoftenthecase,workingbackwardslikethis–solvingtheinverseproblem,inthejargon–isharderthan

goingtheotherway.Butpracticalmethodsexist.Oneofthemajoroilcompaniesperformssuchcalculationsaquarterofamilliontimeseveryday.

Drillingforoilhasitsownproblems,astheblowoutattheDeepwaterHorizonoilrigin2010madeclear.Butatthemoment,humansocietyisheavilydependentonoil,anditwouldtakedecadestoreducethissignificantly,evenifeveryonewantedto.Nexttimeyoufillupyourtank,giveathoughttothemathematicalpioneerswhowantedtoknowhowaviolinproducesitssounds.Itwasn’tapracticalproblemthen,anditstillisn’ttoday.Butwithouttheirdiscoveries,yourcarwouldtakeyounowhere.

9Ripplesandblips

FourierTransform

Whatdoesitsay?

Anypatterninspaceandtimecanbethoughtofasasuperpositionofsinusoidalpatternswithdifferentfrequencies.

Whyisthatimportant?

Thecomponentfrequenciescanbeusedtoanalysethepatterns,createthemtoorder,extractimportantfeatures,andremoverandomnoise.

Whatdiditleadto?

Fourier’stechniqueisverywidelyused,forexampleinimageprocessingandquantummechanics.ItisusedtofindthestructureoflargebiologicalmoleculeslikeDNA,tocompressimagedataindigitalphotography,tocleanupoldordamagedaudiorecordings,andtoanalyseearthquakes.Modernvariantsareusedtostorefingerprintdataefficientlyandtoimprovemedicalscanners.

Newton’sPrincipiaopenedthedoortothemathematicalstudyofnature,buthisfellowcountrymenweretooobsessedwiththeprioritydisputeovercalculustofindoutwhatlaybeyond.WhileEngland’sfinestwereseethingoverwhattheyperceivedtobedisgracefulallegationsaboutthecountry’sgreatestlivingmathematician–muchofitprobablyhisownfaultforlisteningtowell-intentionedbutfoolishfriends–theircontinentalcolleagueswereextendingNewton’sideasaboutlawsofnaturetomostofthephysicalsciences.Thewaveequationwasquicklyfollowedbyremarkablysimilarequationsforgravitation,electrostatics,elasticity,andheatflow.Manyborethenamesoftheirinventors:Laplace’sequation,Poisson’sequation.Theequationforheatdoesnot;itbearstheunimaginativeandnotentirelyaccuratename‘heatequation’.ItwasintroducedbyJosephFourier,andhisideasledtothecreationofanewareaofmathematicswhoseramificationsweretospreadfarbeyonditsoriginalsource.Thoseideascouldhavebeentriggeredbythewaveequation,wheresimilarmethodswerefloatingaroundinthecollectivemathematicalconsciousness,buthistoryplumpedforheat.

Thenewmethodhadapromisingbeginning:in1807FouriersubmittedanarticleonheatflowtotheFrenchAcademyofSciences,basedonanewpartialdifferentialequation.Althoughthatprestigiousbodydeclinedtopublishthework,itencouragedFouriertodevelophisideasfurtherandtryagain.AtthattimetheAcademyofferedanannualprizeforresearchonwhatevertopictheyfeltwassufficientlyinteresting,andtheymadeheatthetopicofthe1812prize.Fourierdulysubmittedhisrevisedandextendedarticle,andwon.Hisheatequationlookslikethis:

Hereu(x,t)isthetemperatureofametalrodatpositionxandtimet,consideringtherodtobeinfinitelythin,andaisaconstant,thethermaldiffusivity.Soitreallyoughttobecalledthetemperatureequation.Healsodevelopedahigher-dimensionalversion,

validonanyspecifiedregionoftheplaneorspace.

Theheatequationbearsanuncannyresemblancetothewaveequation,withonecrucialdifference.Thewaveequationusesthesecondtimederivative∂2u/∂t2,butintheheatequationthisisreplacedbythefirstderivative∂u/∂t.Thischangemayseemsmall,butitsphysicalmeaningishuge.Heatdoesnotpersistindefinitely,inthewaythatavibratingviolinstringcontinuestovibrateforever(accordingtothewaveequation,whichassumesnofrictionorotherdamping).Instead,heatdissipates,diesaway,astimepasses,unlessthereissomeheatsourcethatcantopitup.Soatypicalproblemmightbe:heatoneendofarodtokeepitstemperaturesteady,cooltheotherendtodothesame,andfindouthowthetemperaturevariesalongtherodwhenitsettlestoasteadystate.Theansweristhatitfallsoffexponentially.Anothertypicalproblemistospecifytheinitialtemperatureprofilealongtherod,andthenaskhowitchangesastimepasses.Perhapsthelefthalfstartsatahightemperatureandtherighthalfatacoolerone;theequationthentellsushowtheheatfromthehotpartdiffusesintothecoolerpart.

ThemostintriguingaspectofFourier’sprizewinningmemoirwasnottheequation,buthowhesolvedit.Whentheinitialprofileisatrigonometricfunction,suchassinx,itiseasy(tothosewithexperienceinsuchmatters)tosolvetheequation,andtheanswerise−αtsinx.Thisresemblesthefundamentalmodeofthewaveequation,buttheretheformulawassinctsinx.Theeternaloscillationofaviolinstring,correspondingtothesinctfactor,hasbeenreplacedbyanexponential,andtheminussignintheexponent−αttellsusthattheentiretemperatureprofilediesawayatthesamerate,allalongtherod.(Thephysicaldifferencehereisthatwavesconserveenergy,butheatflowdoesnot.)Similarly,foraprofilesin5x,say,thesolutionise−25αtsin5x,whichalsodiesout,butatamuchfasterrate.The25is52,andthisisanexampleofageneralpattern,applicabletoinitialprofilesoftheformsinnxorcosnx.1Tosolvetheheatequation,justmultiplybye−n

2αt.Nowthestoryfollowsthesamegeneraloutlineasthewaveequation.The

heatequationislinear,sowecansuperposesolutions.Iftheinitialprofileis

u(x;0)=sinx+sin5x

thenthesolutionis

andeachmodedieswayatadifferentrate.Butinitialprofileslikethisareabit

artificial.TosolvetheproblemImentionedearlier,wewantaninitialprofilewhereu(x,0)=1forhalftherodbut−1fortheotherhalf.Thisprofileisdiscontinuous,asquarewaveinengineeringterminology.Butsineandcosinecurvesarecontinuous.Sonosuperpositionofsineandcosinecurvescanrepresentasquarewave.

Nofinitesuperposition,certainly.But,again,whatifweallowedinfinitelymanyterms?Thenwecantrytoexpresstheinitialprofileasaninfiniteseries,oftheform

forsuitableconstantsa0,a1,a2,a3,…,b1,b2,b3,….(Thereisnob0becausesin0x=0.)Nowitdoesseempossibletogetasquarewave(seeFigure40).Infact,mostcoefficientscanbesettozero.Onlythebnfornoddareneeded,andthenbn=8/nπ.

Fig40Howtogetasquarewavefromsinesandcosines.Left:Thecomponentsinusoidalwaves.Right:Theirsumandasquarewave.HereweshowthefirstfewtermsoftheFourierseries.Additionaltermsmaketheapproximationtoasquarewaveeverbetter.

Fourierevenhadgeneralformulasforthecoefficientsanandbnforageneralprofilef(x),intermsofintegrals:

Afteralengthytrekthroughpowerseriesexpansionsoftrigonometricfunctions,herealisedthattherewasamuchsimplerwaytoderivetheseformulas.Ifyoutaketwodifferenttrigonometricfunctions,saycos2xandsin5x,multiplythemtogether,andintegratefrom0to2π,yougetzero.Thisiseventhecasewhentheylooklikecos5xandsin5x.Butiftheyarethesame–saybothequaltosin5x–theintegraloftheirproductisnotzero.Infact,itisπ.Ifyoustartby

assumingthatf(x)isthesumofatrigonometricseries,multiplyeverythingbysin5x,andintegrate,allofthetermsdisappearexceptfortheonecorrespondingtosin5x,namelyb5sin5x.Heretheintegralisπ.Dividebythat,andyouhaveFourier’sformulaforb5.Thesamegoesforalltheothercoefficients.

Althoughitwontheacademy’sprize,Fourier’smemoirwasroundlycriticisedforbeinginsufficientlyrigorous,andtheacademydeclinedtopublishit.ThiswashighlyunusualanditgreatlyirritatedFourier,buttheacademyhelditsground.Fourierwasincensed.Physicalintuitiontoldhimhewasright,andifyoupluggedhisseriesintothisequationitwasclearlyasolution.Itworked.Therealproblemwasthatunwittinglyhehadreopenedanoldwound.AswesawinChapter8,EulerandBernoullihadbeenarguingforagesaboutasimilarissueforthewaveequation,whereFourier’sexponentialdissipationovertimewasreplacedbyanunendingsinusoidaloscillationinthewaveamplitude.Theunderlyingmathematicalissueswereidentical.Infact,Eulerhadalreadypublishedtheintegralformulasforthecoefficientsinthecontextofthewaveequation.

However,Eulerhadneverclaimedthattheformulaworkedfordiscontinuousfunctionsf(x),themostcontroversialfeatureofFourier’swork.Theviolin-stringmodeldidn’tinvolvediscontinuousinitialconditionsanyway–thosewouldmodelabrokenstring,whichwouldnotvibrateatall.Butforheat,itwasnaturaltoconsiderholdingoneregionofarodatonetemperatureandanadjacentregionatadifferentone.Inpracticethetransitionwouldbesmoothandverysteep,butadiscontinuousmodelwasreasonableandmoreconvenientforcalculations.Infact,thesolutiontotheheatequationexplainedwhythetransitionwouldrapidlybecomesmoothandverysteep,astheheatdiffusedsideways.SoanissuethatEulerhadn’tneededtoworryaboutwasbecomingunavoidable,andFouriersufferedfromthefallout.

Mathematicianswerestartingtorealisethatinfiniteseriesweredangerousbeasts.Theydidn’talwaysbehavelikenicefinitesums.Eventually,thesetangledcomplexitiesgotsortedout,butittookanewviewofmathematicsandahundredyearsofhardworktodothat.InFourier’sday,everyonethoughttheyalreadyknewwhatintegrals,functions,andinfiniteserieswere,butinrealityitwasallrathervague–‘IknowonewhenIseeone.’SowhenFouriersubmittedhisepoch-makingpaper,thereweregoodreasonsfortheacademyofficialstobewary.Theyrefusedtobudge,soin1822Fouriergotroundtheirobjectionsbypublishinghisworkasabook,Théorieanalytiquedelachaleur(‘AnalyticTheoryofHeat’).In1824hegothimselfappointedsecretaryoftheacademy,

thumbedhisnoseatallthecritics,andpublishedhisoriginal1811memoir,unchanged,intheacademy’sprestigiousjournal.

WenowknowthatalthoughFourierwasrightinspirit,hiscriticshadgoodreasonsforworryingaboutrigour.Theproblemsaresubtleandtheanswersarenotterriblyintuitive.Fourieranalysis,aswenowcallit,worksverywell,butithashiddendepthsofwhichFourierwasunaware.

Thequestionseemedtobe:whendoestheFourierseriesconvergetothefunctionitallegedlyrepresents?Thatis,ifyoutakemoreandmoreterms,doestheapproximationtothefunctiongeteverbetter?EvenFourierknewthattheanswerwasnot‘always’.Itseemedtobe‘usually,butwithpossibleproblemsatdiscontinuities’.Forinstanceatitsmidpoint,wherethetemperaturejumps,thesquarewave’sFourierseriesconverges–buttothewrongnumber.Thesumis0,butthesquarewavetakesvalue1.

Formostphysicalpurposes,itdoesn’tgreatlymatterifyouchangethevalueofafunctionatoneisolatedpoint.Thesquarewave,thusmodified,stilllookssquare.Itjustdoessomethingslightlydifferentatthediscontinuity.ToFourier,thiskindofissuedidn’treallymatter.Hewasmodellingtheflowofheat,andhedidn’tmindifthemodelwasabitartificial,orneededtechnicalchangesthathadnoimportanteffectontheendresult.Buttheconvergenceissuecouldnotbedismissedsolightly,becausefunctionscanhavefarmorecomplicateddiscontinuitiesthanasquarewave.

However,Fourierwasclaimingthathismethodworkedforanyfunction,soitoughttoapplyeventofunctionssuchas:f(x)=0whenxisrational,1whenxisirrational.Thisfunctionisdiscontinuouseverywhere.Forsuchfunctions,atthattime,itwasn’tevenclearwhattheintegralmeant.Andthatturnedouttobetherealcauseofthecontroversy.Noonehaddefinedwhatanintegralwas,notforstrangefunctionslikethisone.Worse,noonehaddefinedwhatafunctionwas.Andevenifyoucouldtidyupthoseomissions,itwasn’tjustamatterofwhethertheFourierseriesconverged.Therealdifficultywastosortoutinwhatsenseitconverged.

Resolvingtheseissueswastricky.Itrequiredanewtheoryofintegration,suppliedbyHenriLebesgue,areformulationofthefoundationsofmathematicsintermsofsettheory,startedbyGeorgCantorandopeningupseveralentirelynewcansofworms,majorinsightsfromsuchtoweringfiguresasRiemann,andadoseoftwentieth–centuryabstractiontosortouttheconvergenceissues.Thefinalverdictwasthat,withtherightinterpretations,Fourier’sideacouldbemade

rigorous.Itworkedforaverybroad,thoughnotuniversal,classoffunctions.Whethertheseriesconvergedtof(x)foreveryvalueofxwasn’tquitetherightquestion;everythingwasfineprovidedtheexceptionalvaluesofxwhereitdidn’tconvergeweresufficientlyrare,inaprecisebuttechnicalsense.Ifthefunctionwascontinuous,theseriesconvergedforanyx.Atajumpdiscontinuity,likethechangefrom1to–1inthesquarewave,theseriesconvergedverydemocraticallytotheaverageofthevaluesimmediatelytoeithersideofthejump.Buttheseriesalwaysconvergedtothefunctionwiththerightinterpretationof‘converge’.Itconvergedasawhole,ratherthanpointbypoint.Statingthisrigorouslydependedonfindingtherightwaytomeasurethedistancebetweentwofunctions.Withallthisinplace,Fourierseriesdidindeedsolvetheheatequation.Buttheirrealsignificancewasmuchbroader,andthemainbeneficiaryoutsidepuremathematicswasnotthephysicsofheatbutengineering.Especiallyelectronicengineering.

InitsmostgeneralformFourier’smethodrepresentsasignal,determinedbyafunctionf,asacombinationofwavesofallpossiblefrequencies.ThisiscalledtheFouriertransformofthewave.Itreplacestheoriginalsignalbyitsspectrum:alistofamplitudesandfrequenciesforthecomponentsinesandcosines,encodingthesameinformationinadifferentway–engineerstalkoftransformingfromthetimedomaintothefrequencydomain.Whendataarerepresentedindifferentways,operationsthataredifficultorimpossibleinonerepresentationmaybecomeeasyintheother.Forexample,youcanstartwithatelephoneconversation,formitsFouriertransform,andstripoutallpartsofthesignalwhoseFouriercomponentshavefrequenciestoohighortoolowforthehumaneartohear.Thismakesitpossibletosendmoreconversationsoverthesamecommunicationchannels,andit’sonereasonwhytoday’sphonebillsare,relativelyspeaking,sosmall.Youcan’tplaythisgameontheoriginal,untransformedsignal,becausethatdoesn’thave‘frequency’asanobviouscharacteristic.Youdon’tknowwhattostripout.

Oneapplicationofthistechniqueistodesignbuildingsthatwillsurviveearthquakes.TheFouriertransformofthevibrationsproducedbyatypicalearthquakereveals,amongotherthings,thefrequenciesatwhichtheenergyimpartedbytheshakinggroundisgreatest.Abuildinghasitsownnaturalmodesofvibration,whereitwillresonatewiththeearthquake,thatis,respondunusuallystrongly.Sothefirstsensiblesteptowardsearthquake-proofingabuildingistomakesurethatthebuilding’spreferredfrequenciesaredifferentfromtheearthquake’s.Theearthquake’sfrequenciescanbeobtainedfrom

observations;thoseofthebuildingcanbecalculatedusingacomputermodel.Thisisjustoneofmanywaysinwhich,tuckedawaybehindthescenes,the

Fouriertransformaffectsourlives.Peoplewholiveorworkinbuildingsinearthquakezonesdon’tneedtoknowhowtocalculateaFouriertransform,buttheirchanceofsurvivinganearthquakeisconsiderablyimprovedbecausesomepeopledo.TheFouriertransformhasbecomearoutinetoolinscienceandengineering;itsapplicationsincluderemovingnoisefromoldsoundrecordings,suchasclickscausedbyscratchesonvinylrecords,findingthestructureoflargebiochemicalmoleculessuchasDNAusingX-raydiffraction,improvingradioreception,tidyingupphotographstakenfromtheair,sonarsystemssuchasthoseusedbysubmarines,andpreventingunwantedvibrationsincarsatthedesignstage.I’llfocushereonjustoneofthethousandsofeverydayusesofFourier’smagnificentinsight,onethatmostofusunwittinglytakeadvantageofeverytimewegoonholiday:digitalphotography.

OnarecenttriptoCambodiaItookabout1400photographs,usingadigitalcamera,andtheyallwentona2GBmemorycardwithroomforabout400more.Now,Idon’ttakeparticularlyhigh-resolutionphotographs,soeachphotofileisabout1.1MB.Butthepicturesarefullcolour,andtheydon’tshowanynoticeablepixellationona27-inchcomputerscreen,sothelossinqualityisn’tobvious.Somehow,mycameramanagestocramintoasingle2GBcardabouttentimesasmuchdataasthecardcanpossiblyhold.It’slikepouringalitreofmilkintoaneggcup.Yetitallfitsin.Thequestionis:how?

Theanswerisdatacompression.Theinformationthatspecifiestheimageisprocessedtoreduceitsquantity.Someofthisprocessingis‘lossless’,meaningthattheoriginalrawinformationcanifnecessaryberetrievedfromthecompressedversion.Thisispossiblebecausemostreal-worldimagescontainredundantinformation.Bigblocksofsky,forinstance,areoftenthesameshadeofblue(well,theyarewherewetendtogo).Insteadofrepeatingthecolourandbrightnessinformationforabluepixeloverandoveragain,youcouldstorethecoordinatesoftwooppositecornersofarectangleandashortcodethatmeans‘colourthisentireregionblue’.That’snotquitehowit’sdone,ofcourse,butitshowswhylosslesscompressionissometimespossible.Whenit’snot,‘lossy’compressionisoftenacceptable.Thehumaneyeisnotespeciallysensitivetocertainfeaturesofimages,andthesefeaturescanberecordedonacoarserscalewithoutmostofusnoticing,especiallyifwedon’thavetheoriginalimagetocomparewith.Compressinginformationthiswayislikescramblinganegg:it’seasyinonedirection,anddoestherequiredjob,butit’snotpossibletoreverseit.

Non-redundantinformationislost.Itwasjustinformationthatdidn’tdoalottobeginwith,givenhowhumanvisionworks.

Mycamera,likemostpoint-and-clickones,savesitsimagesinfileswithlabelslikeP1020339.JPG.ThesuffixreferstoJPEG,theJointPhotographicExpertsGroup,anditindicatesthataparticularsystemofdatacompressionhasbeenused.Softwareformanipulatingandprintingphotos,suchasPhotoshoporiPhoto,iswrittensothatitcandecodetheJPEGformatandturnthedatabackintoapicture.MillionsofususeJPEGfilesregularly,fewerareawarethatthey’recompressed,andfewerstillwonderhowit’sdone.Thisisnotacriticism:youdon’thavetoknowhowitworkstouseit,that’sthepoint.Thecameraandsoftwarehandleitallforyou.Butit’softensensibletohavearoughideaofwhatsoftwaredoes,andhow,ifonlytodiscoverhowcunningsomeofitis.Youcanskipthedetailshereifyouwish:I’dlikeyoutoappreciatejusthowmuchmathematicsgoesintoeachimageonyourcamera’smemorycard,butexactlywhatmathematicsislessimportant.

TheJPEGformat2combinesfivedifferentcompressionsteps.Thefirstconvertsthecolourandbrightnessinformation,whichstartsoutasthreeintensitiesforred,green,andblue,intothreedifferentmathematicallyequivalentonesthataremoresuitedtothewaythehumanbrainperceivesimages.One(luminance)representstheoverallbrightness–whatyouwouldseewithablack-and-whiteor‘greyscale’versionofthesameimage.Theothertwo(chrominance)arethedifferencesbetweenthisandtheamountsofblueandredlight,respectively.

Next,thechrominancedataarecoarsened:reducedtoasmallerrangeofnumericalvalues.Thisstepalonehalvestheamountofdata.Itdoesnoperceptibleharmbecausethehumanvisualsystemismuchlesssensitivetocolourdifferencesthanthecamerais.

ThethirdstepusesavariantoftheFouriertransform.Thisworksnotwithasignalthatchangesovertime,butwithapatternintwodimensionsofspace.Themathematicsisvirtuallyidentical.Thespaceconcernedisan8×8sub-blockofpixelsfromtheimage.Forsimplicitythinkjustoftheluminancecomponent:thesameideaappliestothecolourinformationaswell.Westartwithablockof64pixels,andforeachofthemweneedtostoreonenumber,theluminancevalueforthatpixel.Thediscretecosinetransform,aspecialcaseoftheFouriertransform,decomposestheimageintoasuperpositionofstandard‘striped’imagesinstead.Inhalfofthemthestripesrunhorizontally;intheotherhalftheyarevertical.Theyarespacedatdifferentintervals,likethevariousharmonicsin

theusualFouriertransform,andtheirgreyscalevaluesareacloseapproximationtoacosinecurve.Incoordinatesontheblocktheyarediscreteversionsofcosmxcosnyforvariousintegersmandn,seeFigure41.

Fig41The64basicpatternsfromwhichanyblockof8×8pixelscanbeobtained.

Thissteppavesthewaytostepfour,asecondexploitationofthedeficienciesofhumanvision.Wearemoresensitivetovariationsinbrightness(orcolour)overlargeregionsthanwearetocloselyspacedvariations.Sothepatternsinthefigurecanberecordedlessaccuratelyasthespacingofthestripesbecomesfiner.Thiscompressesthedatafurther.Thefifthandfinalstepusesa‘Huffmancode’toexpressthelistofstrengthsofthe64basicpatternsinamoreefficientmanner.

EverytimeyoutakeadigitalimageusingJPEG,theelectronicsinyourcameradoesallofthesethings,exceptperhapsstepone.(ProfessionalsarenowmovingovertoRAWfiles,whichrecordtheactualdatawithoutcompression,togetherwiththeusual‘metadata’suchasdate,time,exposure,andsoon.Filesinthisformattakeupmorememory,butmemorygetsbiggerandcheaperbythemonth,sothatnolongermatters.)AtrainedeyecanspotthelossofimagequalitycreatedbyJPEGcompressionwhenthequantityofdataisreducedtoabout10%oftheoriginal,andanuntrainedeyecanseeitclearlybythetimethefilesizeisdownto2–3%.Soyourcameracanrecordabouttentimesasmanyimagesonamemorycard,comparedwiththerawimagedata,beforeanyoneotherthananexpertwouldnotice.

Becauseofapplicationslikethese,Fourieranalysishasbecomeareflexamongengineersandscientists,butforsomepurposesthetechniquehasonemajorfault:sinesandcosinesgoonforever.Fourier’smethodrunsintoproblemswhenittriestorepresentacompactsignal.Ittakeshugenumbersofsinesandcosinestomimicalocalisedblip.Theproblemisnotgettingthebasicshapeoftheblipright,butmakingeverythingoutsidetheblipequaltozero.Youhavetokillofftheinfinitelylongripplingtailsofallthosesinesandcosines,whichyoudobyaddingonevenmorehigh-frequencysinesandcosinesinadesperateefforttocancelouttheunwantedjunk.SotheFouriertransformishopelessforblip-likesignals:thetransformedversionismorecomplicated,andneedsmoredatatodescribeit,thantheoriginal.

WhatsavesthedayisthegeneralityofFourier’smethod.Sinesandcosinesworkbecausetheysatisfyonesimplecondition:theyaremathematicallyindependent.Formally,thismeansthattheyareorthogonal:inanabstractbutmeaningfulsense,theyareatrightanglestoeachother.ThisiswhereEuler’strick,eventuallyrediscoveredbyFourier,comesin.Multiplyingtwoofthebasicsinusoidalwaveformstogetherandintegratingoveroneperiodisawaytomeasurehowcloselyrelatedtheyare.Ifthisnumberislarge,theyareverysimilar;ifitiszero(theconditionfororthogonality),theyareindependent.Fourieranalysisworksbecauseitsbasicwaveformsarebothorthogonalandcomplete:theyareindependentandthereareenoughofthemtorepresentanysignaliftheyaresuitablysuperposed.Ineffect,theyprovideacoordinatesystemonthespaceofallsignals,justliketheusualthreeaxesofordinaryspace.Themainnewfeatureisthatwenowhaveinfinitelymanyaxes:oneforeachbasicwaveform.Butthisdoesn’tcausemanydifficultiesmathematically,onceyougetusedtoit.Itjustmeansyouhavetoworkwithinfiniteseriesinsteadoffinitesums,andworryalittleaboutwhentheseriesconverge.

Eveninfinite-dimensionalspaces,therearemanydifferentcoordinatesystems;theaxescanberotatedtopointinnewdirections,forexample.It’snotsurprisingtofindthatinaninfinite-dimensionalspaceofsignals,therearealternativecoordinatesystemsthatdifferwildlyfromFourier’s.Oneofthemostimportantdiscoveriesinthewholearea,inrecentyears,isanewcoordinatesysteminwhichthebasicwaveformsareconfinedtoalimitedregionofspace.Theyarecalledwavelets,andtheycanrepresentblipsveryefficientlybecausetheyareblips.

Onlyrecentlydidanyonerealisethatblip-likeFourieranalysiswaspossible.Gettingstartedisstraightforward:chooseaparticularshapeofblip,themotherwavelet(Figure42).Thengeneratedaughterwavelets(andgranddaughters,

great-granddaughters,whatever)byslidingthemotherwaveletsidewaysintovariouspositions,andexpandingherorcompressingherbyachangeofscale.Inthesameway,Fourier’sbasicsineandcosinecurvesare‘mothersinelets’,andthehigher-frequencysinesandcosinesaredaughters.Beingperiodic,thesecurvescannotbeblip-like.

Fig42Daubechieswavelet.

Waveletsaredesignedtodescribebliplikedataefficiently.Moreover,becausethedaughterandgranddaughterwaveletsarejustrescaledversionsofmother,itispossibletofocusonparticularlevelsofdetail.Ifyoudon’twanttoseesmall-scalestructure,youjustremoveallthegreat-granddaughterwaveletsfromthewavelettransform.Torepresentaleopardbywavelets,youneedafewbigonestogetthebodyright,smalleronesfortheeyes,nose,andofcoursethespots,andverytinyonesforindividualhairs.Tocompressthedatarepresentingtheleopard,youmightdecidethattheindividualhairsdon’tmatter,soyoujustremovethoseparticularcomponentwavelets.Thegreatthingis,theimagestilllookslikealeopard,anditstillhasspots.IfyoutrytodothiswiththeFouriertransformofaleopardthenthelistofcomponentsishuge,it’snotclearwhichitemsyoushouldremove,andyouprobablywon’trecognisetheresultasaleopard.

Allverywellandgood,butwhatshapeshouldthemotherwaveletbe?Foralongtimenobodycouldworkthatout,orevenshowthatagoodshapeexists.Butintheearly1980sgeophysicistJeanMorletandmathematicalphysicistAlexanderGrossmannfoundthefirstsuitablemotherwavelet.In1985YvesMeyerfoundabettermotherwavelet,andin1987IngridDaubechies,amathematicianatBellLaboratories,blewthewholefieldwideopen.Althoughthepreviousmotherwaveletslookedsuitablybliplike,theyallhadaverytiny

mathematicaltailthatwiggledofftoinfinity.Daubechiesfoundamotherwaveletwithnotailatall:outsidesomeinterval,motherwasalwaysexactlyzero–agenuineblip,confinedentirelytoafiniteregionofspace.

Thebliplikefeaturesofwaveletsmakethemespeciallygoodforcompressingimages.Oneoftheirfirstlarge-scalepracticaluseswastostorefingerprints,andthecustomerwastheFederalBureauofInvestigation.TheFBI’sfingerprintdatabasecontains300millionrecords,eachofeightfingerprintsandtwothumbprints,whichwereoriginallystoredasinkedimpressionsonpapercards.Thisisnotaconvenientstoragemedium,sotherecordshavebeenmodernisedbydigitisingtheimagesandstoringtheresultsonacomputer.Obviousadvantagesincludebeingabletomountarapidautomatedsearchforprintsthatmatchthosefoundatthesceneofacrime.

Thecomputerfileforeachfingerprintcardis10megabyteslong:80millionbinarydigits.Sotheentirearchiveoccupies3000terabytesofmemory:24quadrillionbinarydigits.Tomakemattersworse,thenumberofnewsetsoffingerprintsgrowsby30,000everyday,sothestoragerequirementwouldgrowby2.4trillionbinarydigitseveryday.TheFBIsensiblydecidedthattheyneededsomemethodfordatacompression.JPEGwasn’tsuitable,forvariousreasons,soin2002theFBIdecidedtodevelopanewsystemofcompressionusingwavelets,thewavelet/scalarquantization(WSQ)method.WSQreducesthedatato5%ofitssizebyremovingfinedetailthroughouttheimage.Thisisirrelevanttotheeye’sability,aswellasacomputer’s,torecognisethefingerprint.

Therearealsomanyrecentapplicationsofwaveletstomedicalimaging.Hospitalsnowemployseveraldifferentkindsofscanner,whichassembletwo-dimensionalcross-sectionsofthehumanbodyorimportantorganssuchasthebrain.ThetechniquesincludeCT(computerisedtomography),PET(positronemissiontomography),andMRI(magneticresonanceimaging).Intomography,themachineobservesthetotaltissuedensity,orasimilarquantity,inasingledirectionthroughthebody,ratherlikewhatyouwouldseefromafixedpositionifallthetissueweretobecomeslightlytransparent.Atwo-dimensionalpicturecanbereconstructedbyapplyingsomeclevermathematicstoawholeseriesofsuch‘projections’,takenatmanydifferentangles.InCT,eachprojectionrequiresanX-rayexposure,sotherearegoodreasonstolimittheamountofdataacquired.Inallsuchscanningmethods,lessdatatakeslesstimetoacquire,somorepatientscanusethesameamountofequipment.Ontheotherhand,goodimagesneedmoredatasothatthereconstructionmethodcanworkmoreeffectively.Waveletsprovideacompromise,inwhichreducingtheamountof

dataleadstoequallyacceptableimages.Bytakingawavelettransform,removingunwantedcomponents,and‘detransforming’backtoanimageagain,apoorimagecanbesmoothedandcleanedup.Waveletsalsoimprovethestrategiesbywhichthescannersacquiretheirdatainthefirstplace.

Infact,waveletsareturningupalmosteverywhere.Researchersinareasaswideapartasgeophysicsandelectricalengineeringaretakingthemonboardandputtingthemtoworkintheirownfields.RonaldCoifmanandVictorWickerhauserhaveusedthemtoremoveunwantednoisefromrecordings:arecenttriumphwasaperformanceofBrahmsplayingoneofhisownHungarianDances.Itwasoriginallyrecordedonawaxcylinderin1889,whichpartiallymelted;itwasre-recordedontoa78rpmdisc.Coifmanstartedfromaradiobroadcastofthedisc,bywhichtimethemusicwasvirtuallyinaudibleamidthesurroundingnoise.Afterwaveletcleansing,youcouldhearwhatBrahmswasplaying–notperfectly,butatleastitwasaudible.It’sanimpressivetrackrecordforanideathatfirstaroseinthephysicsofheatflow200yearsago,andwasrejectedforpublication.

10Theascentofhumanity

Navier–StokesEquation

Whatdoesitsay?

It’sNewton’ssecondlawofmotionindisguise.Theleft-handsideistheaccelerationofasmallregionoffluid.Theright-handsideistheforcesthatactonit:pressure,stress,andinternalbodyforces.

Whyisthatimportant?

Itprovidesareallyaccuratewaytocalculatehowfluidsmove.Thisisakeyfeatureofinnumerablescientificandtechnologicalproblems.

Whatdiditleadto?

Modernpassengerjets,fastandquietsubmarines,Formula1racingcarsthatstayonthetrackathighspeeds,andmedicaladvancesonbloodflowinveinsandarteries.Computermethodsforsolvingtheequations,knownascomputationalfluiddynamics(CFD),arewidelyusedbyengineerstoimprovetechnologyinsuchareas.

Seenfromspace,theEarthisabeautifulglowingblue-and-whitespherewithpatchesofgreenandbrown,quiteunlikeanyotherplanetintheSolarSystem–oranyofthe500-plusplanetsnowknowntobecirclingotherstars,forthatmatter.Theveryword‘Earth’instantlybringsthisimagetomind.Yetalittleoverfiftyyearsago,thealmostuniversalimageforthesamewordwouldhavebeenahandfulofdirt,earthinthegardeningsense.Beforethetwentiethcentury,peoplelookedattheskyandwonderedaboutthestarsandplanets,buttheydidsofromgroundlevel.Humanflightwasnothingmorethanadream,thesubjectofmythsandlegends.Hardlyanyonethoughtabouttravellingtoanotherworld.

Afewintrepidpioneersbegantheslowclimbintothesky.TheChinesewerethefirst.Around500BCLuBaninventedawoodenbird,whichmighthavebeenaprimitiveglider.In559ADtheupstartGaoYangstrappedYuanHuangtou,theemperor’sson,toakite–againsthiswill–tospyontheenemyfromabove.Yuansurvivedtheexperiencebutwaslaterexecuted.Withtheseventeeth-centurydiscoveryofhydrogentheurgetoflyspreadtoEurope,inspiringafewbraveindividualstoascendintothelowerreachesofEarth’satmosphereinballoons.Hydrogenisexplosive,andin1783theFrenchbrothersJoseph-MichelandJacques-ÉtienneMontgolfiergaveapublicdemonstrationoftheirnewandmuchsaferidea,thehot-airballoon–firstwithanunmannedtestflight,thenwithÉtienneaspilot.

Thepaceofprogress,andtheheightstowhichhumanscouldascend,begantoincreaserapidly.In1903OrvilleandWilburWrightmadethefirstpoweredflightinanaeroplane.Thefirstairline,DELAG(DeutscheLuftschiffahrts-Aktiengesellschaft),beganoperationsin1910,flyingpassengersfromFrankfurttoBaden-BadenandDüsseldorfusingairshipsmadebytheZeppelinCorporation.By1914theStPetersburg–TampaAirboatLinewasflyingpassengerscommerciallybetweenthetwoFloridacities,ajourneythattook23minutesinTonyJannus’sflyingboat.Commercialairtravelquicklybecamecommonplace,andjetaircraftarrived:theDeHavillandCometbeganregularflightsin1952,butmetalfatiguecausedseveralcrashes,andtheBoeing707becamethemarketleaderfromitslaunchin1958.

Ordinaryindividualscouldnowroutinelybefoundatanaltitudeof8kilometres,theirlimittothisday,atleastuntilVirginGalacticstartslow-orbitalflights.Militaryflightsandexperimentalaircraftrosetogreaterheights.Spaceflight,hithertothedreamofafewvisionaries,startedtobecomeaplausible

proposition.In1961theSovietcosmonautYuriGagarinmadethefirstmannedorbitoftheEarthinVostok1.In1969NASA’sApollo11missionlandedtwoAmericanastronauts,NeilArmstrongandBuzzAldrin,ontheMoon.Thespaceshuttlebeganoperationalflightsin1982,andwhilebudgetconstraintspreventeditachievingtheoriginalaims–areusablevehiclewitharapidturnaround–itbecameoneoftheworkhorsesoflow-orbitspaceflight,alongwithRussia’sSoyuzspacecraft.Atlantishasnowmadethefinalflightofthespaceshuttleprogramme,butnewvehiclesarebeingplanned,mainlybyprivatecompanies.Europe,India,China,andJapanhavetheirownspaceprogrammesandagencies.

Thisliteralascentofhumanityhaschangedourviewofwhoweareandwherewelive–themainreasonwhy‘Earth’nowmeansablue–whiteglobe.Thosecoloursholdacluetoournewfoundabilitytofly.Theblueiswater,andthewhiteiswatervapourintheformofclouds.Earthisawaterworld,withoceans,seas,rivers,lakes.Whatwaterdoesbestistoflow,oftentoplaceswhereit’snotwanted.Theflowmightberaindrippingfromarooforthemightytorrentofawaterfall.Itcanbegentleandsmooth,orroughandturbulent–thesteadyflowoftheNileacrosswhatwouldotherwisebedesert,orthefrothywhitewaterofitssixcataracts.

Itwasthepatternsformedbywater,ormoregenerallyanymovingfluid,thatattractedtheattentionofmathematiciansinthenineteenthcentury,whentheyderivedthefirstequationsforfluidflow.Thevitalfluidforflightislessvisiblethanwater,butjustasubiquitous:air.Theflowofairismorecomplexmathematically,becauseaircanbecompressed.Bymodifyingtheirequationssothattheyappliedtoacompressiblefluid,mathematiciansinitiatedthesciencethatwouldeventuallygettheAgeofFlightofftheground:aerodynamics.Earlypioneersmightflybyruleofthumb,butcommercialairlinersandthespaceshuttleflybecauseengineershavedonethecalculationsthatmakethemsafeandreliable(barringoccasionalaccidents).Aircraftdesignrequiresadeepunderstandingofthemathematicsoffluidflow.AndthepioneeroffluiddynamicswastherenownedmathematicianLeonhardEuler,whodiedintheyeartheMontgolfiersmadetheirfirstballoonflight.

TherearefewareasofmathematicstowardswhichtheprolificEulerdidnotturnhisattention.Ithasbeensuggestedthatonereasonforhisprodigiousandversatileoutputwaspolitics,ormoreprecisely,itsavoidance.HeworkedinRussiaformanyyears,atthecourtofCatherinetheGreat,andaneffectivewaytoavoidbeingcaughtupinpoliticalintrigue,withpotentiallydisastrousconsequences,wastobesobusywithhismathematicsthatnoonewouldbelieve

hehadanytimetospareforpolitics.Ifthisiswhathewasdoing,wehaveCatherine’scourttothankformanywonderfuldiscoveries.ButI’minclinedtothinkthatEulerwasprolificbecausehehadthatsortofmind.Hecreatedhugequantitiesofmathematicsbecausehecoulddonoother.

Therewerepredecessors.Archimedesstudiedthestabilityoffloatingbodiesover2200yearsago.In1738theDutchmathematicianDanielBernoullipublishedHydrodynamica(‘Hydrodynamics’),containingtheprinciplethatfluidsflowfasterinregionswherethepressureislower.Bernoulli’sprincipleisofteninvokedtodaytoexplainwhyaircraftcanfly:thewingisshapedsothattheairflowsfasteracrossthetopsurface,loweringthepressureandcreatinglift.Thisexplanationisabittoosimplistic,andmanyotherfactorsareinvolvedinflight,butitdoesillustratethecloserelationshipbetweenbasicmathematicalprinciplesandpracticalaircraftdesign.Bernoulliembodiedhisprincipleinanalgebraicequationrelatingvelocityandpressureinanincompressiblefluid.

In1757Eulerturnedhisfertilemindtofluidflow,publishinganarticle‘Principesgénérauxdumouvementdesfluides’(Generalprinciplesofthemovementoffluids)intheMemoirsoftheBerlinAcademy.Itwasthefirstseriousattempttomodelfluidflowusingapartialdifferentialequation.Tokeeptheproblemwithinreasonablebounds,Eulermadesomesimplifyingassumptions:inparticular,heassumedthefluidwasincompressible,likewaterratherthanair,andhadzeroviscosity–nostickiness.Theseassumptionsallowedhimtofindsomesolutions,buttheyalsomadehisequationsratherunrealistic.Euler’sequationisstillinusetodayforsometypesofproblem,butonthewholeitistoosimpletobeofmuchpracticaluse.

Twoscientistscameupwithamorerealisticequation.Claude-LouisNavierwasaFrenchengineerandphysicist;GeorgeGabrielStokeswasanIrishmathematicianandphysicist.Navierderivedasystemofpartialdifferentialequationsfortheflowofaviscousfluidin1822;Stokesstartedpublishingonthetopictwentyyearslater.TheresultingmodeloffluidflowisnowcalledtheNavier–Stokesequation(oftenthepluralisusedbecausetheequationisstatedintermsofavector,soithasseveralcomponents).Thisequationissoaccuratethatnowadaysengineersoftenusecomputersolutionsinsteadofperformingphysicaltestsinwindtunnels.Thistechnique,knownascomputationalfluiddynamics(CFD),isnowstandardinanyprobleminvolvingfluidflow:theaerodynamicsofthespaceshuttle,thedesignofFormula1racingcarsandeverydayroadcars,andbloodcirculatingthroughthehumanbodyoranartificialheart.

Therearetwowaystolookatthegeometryofafluid.Oneistofollowthe

movementsofindividualtinyparticlesoffluidandseewheretheygo.Theotheristofocusonthevelocitiesofsuchparticles:howfast,andinwhichdirection,theyaremovingatanyinstant.Thetwoareintimatelyrelated,buttherelationshipisdifficulttodisentangleexceptinnumericalapproximations.OneofthegreatinsightsofEuler,Navier,andStokeswastherealisationthateverythinglooksalotsimplerintermsofthevelocities.Theflowofafluidisbestunderstoodintermsofavelocityfield:amathematicaldescriptionofhowthevelocityvariesfrompointtopointinspaceandfrominstanttoinstantintime.SoEuler,Navier,andStokeswrotedownequationsdescribingthevelocityfield.Theactualflowpatternsofthefluidcanthenbecalculated,atleasttoagoodapproximation.

TheNavier–Stokesequationlookslikethis:

wherepisthedensityofthefluid,visitsvelocityfield,pispressure,Tdeterminesthestresses,andfrepresentsbodyforces–forcesthatactthroughouttheentireregion,notjustatitssurface.Thedotisanoperationonvectors,and∇isanexpressioninpartialderivatives,namely

Theequationisderivedfrombasicphysics.Aswiththewaveequation,acrucialfirststepistoapplyNewton’ssecondlawofmotiontorelatethemovementofafluidparticletotheforcesthatactonit.Themainforceiselasticstress,andthishastwomainconstituents:frictionalforcescausedbytheviscosityofthefluid,andtheeffectsofpressure,eitherpositive(compression)ornegative(rarefaction).Therearealsobodyforces,whichstemfromtheaccelerationofthefluidparticleitself.CombiningallthisinformationleadstotheNavier–Stokesequation,whichcanbeseenasastatementofthelawofconservationofmomentuminthisparticularcontext.Theunderlyingphysicsisimpeccable,andthemodelisrealisticenoughtoincludemostofthesignificantfactors;thisiswhyitfitsrealitysowell.Likeallofthetraditionalequationsofclassicalmathematicalphysicsitisacontinuummodel:itassumesthatthefluidisinfinitelydivisible.

ThisisperhapsthemainplacewheretheNavier–Stokesequationpotentiallylosestouchwithreality,butthediscrepancyshowsuponlywhenthemotioninvolvesrapidchangesonthescaleofindividualmolecules.Suchsmall-scale

motionsareimportantinonevitalcontext:turbulence.Ifyouturnonatapandletthewaterflowoutslowly,itarrivesinasmoothtrickle.Turnthetaponfull,however,andyouoftengetasurging,frothy,foaminggushofwater.Similarfrothyflowsoccurinrapidsonariver.Thiseffectisknownasturbulence,andthoseofuswhoflyregularlyarewellawareofitseffectswhenitoccursinair.Itfeelsasthoughtheaircraftisdrivingalongaverybumpyroad.

SolvingtheNavier–Stokesequationishard.Untilreallyfastcomputerswereinvented,itwassohardthatmathematicianswerereducedtoshortcutsandapproximations.Butwhenyouthinkaboutwhatarealfluidcando,itoughttobehard.Youonlyhavetolookatwaterflowinginastream,orwavesbreakingonabeach,toseethatfluidscanflowinextremelycomplexways.Thereareripplesandeddies,wavepatternsandwhirlpools,andfascinatingstructuresliketheSevernbore,awallofwaterthatracesuptheestuaryoftheRiverSeverninsouth-westEnglandwhenthetidecomesin.Thepatternsoffluidflowhavebeenthesourceofinnumerablemathematicalinvestigations,yetoneofthebiggestandmostbasicquestionsinthearearemainsunanswered:isthereamathematicalguaranteethatsolutionsoftheNavier–Stokesequationactuallyexist,validforallfuturetime?Thereisamillion-dollarprizeforanyonewhocansolveit,oneofthesevenClayInstituteMillenniumPrizeproblems,chosentorepresentthemostimportantunsolvedmathematicalproblemsofourage.Theansweris‘yes’intwo-dimensionalflow,butnooneknowsforthree-dimensionalflow.

Despitethis,theNavier–Stokesequationprovidesausefulmodelofturbulentflowbecausemoleculesareextremelysmall.Turbulentvorticesafewmillimetresacrossalreadycapturemanyofthemainfeaturesofturbulence,whereasamoleculeisfarsmaller,soacontinuummodelremainsappropriate.Themainproblemthatturbulencecausesispractical:itmakesitvirtuallyimpossibletosolvetheNavier–Stokesequationnumerically,becauseacomputercan’thandleinfinitelycomplexcalculations.Numericalsolutionsofpartialdifferentialequationsuseagrid,dividingspaceintodiscreteregionsandtimeintodiscreteintervals.Tocapturethevastrangeofscalesonwhichturbulenceoperates–itsbigvortices,middle-sizedones,rightdowntothemillimetre-scaleones–youneedanimpossiblyfinecomputationalgrid.Forthisreason,engineersoftenusestatisticalmodelsofturbulenceinstead.

TheNavier–Stokesequationhasrevolutionisedmoderntransport.Perhapsitsgreatestinfluenceisonthedesignofpassengeraircraft,becausenotonlydothesehavetoflyefficiently,buttheyhavetofly,stablyandreliably.Shipdesign

alsobenefitsfromtheequation,becausewaterisafluid.Butevenordinaryhouseholdcarsarenowdesignedonaerodynamicprinciples,notjustbecauseitmakesthemlooksleekandcool,butbecauseefficientfuelconsumptionreliesonminimisingdragcausedbytheflowofairpastthevehicle.Onewaytoreduceyourcarbonfootprintistodriveanaerodynamicallyefficientcar.Ofcoursethereareotherways,rangingfromsmaller,slowercarstoelectricmotors,orjustdrivingless.Someofthebigimprovementsinfuelconsumptionfigureshavecomefromimprovedenginetechnology,somefrombetteraerodynamics.

Intheearliestdaysofaircraftdesign,pioneersputtheiraeroplanestogetherusingback-of-the-envelopecalculations,physicalintuition,andtrialanderror.Whenyouraimwastoflymorethanahundredmetresnomorethanthreemetresofftheground,thatwasgoodenough.ThefirsttimethatWrightFlyerIgotproperlyofftheground,insteadofstallingandcrashingafterthreesecondsintheair,ittravelled120feetataspeedjustbelow7mph.Orville,thepilotonthatoccasion,managedtokeepitaloftforastaggering12seconds.Butthesizeofpassengeraircraftquicklygrew,foreconomicreasons:themorepeopleyoucancarryinoneflight,themoreprofitableitwillbe.Soonaircraftdesignhadtobebasedonamorerationalandreliablemethod.Thescienceofaerodynamicswasborn,anditsbasicmathematicaltoolswereequationsforfluidflow.Sinceairisbothviscousandcompressible,theNavier–Stokesequation,orsomesimplificationthatmakessenseforagivenproblem,tookcentrestageasfarastheorywent.

However,solvingthoseequations,intheabsenceofmoderncomputers,wasvirtuallyimpossible.Sotheengineersresortedtoananaloguecomputer:placingmodelsoftheaircraftinawindtunnel.Usingafewgeneralpropertiesoftheequationstoworkouthowvariableschangeasthescaleofthemodelchanges,thismethodprovidedbasicinformationquicklyandreliably.MostFormula1teamstodayusewindtunnelstotesttheirdesignsandevaluatepotentialimprovements,butcomputerpowerisnowsogreatthatmostalsouseCFD.Forexample,Figure43showsaCFDcalculationofairflowpastaBMWSaubercar.AsIwrite,oneteam,VirginRacing,usesonlyCFD,buttheywillbeusingawindtunnelaswellnextyear.

Fig43ComputedairflowpastaFormula1car.

Windtunnelsarenotterriblyconvenient;theyareexpensivetobuildandrun,andtheyneedlotsofscalemodels.Perhapsthebiggestdifficultyistomakeaccuratemeasurementsoftheflowofairwithoutaffectingit.Ifyouputaninstrumentinthewindtunneltomeasure,say,airpressure,thentheinstrumentitselfdisturbstheflow.PerhapsthebiggestpracticaladvantageofCFDisthatyoucancalculatetheflowwithoutaffectingit.Anythingyoumightwishtomeasureiseasilyavailable.Moreover,youcanmodifythedesignofthecar,oracomponent,insoftware,whichisalotquickerandcheaperthanmakinglotsofdifferentmodels.Modernmanufacturingprocessesofteninvolvecomputermodelsatthedesignstageanyway.

Supersonicflight,wheretheaircraftgoesfasterthansound,isespeciallytrickytostudyusingmodelsinawindtunnel,becausethewindspeedsaresogreat.Atsuchspeeds,theaircannotmoveawayfromtheaircraftasquicklyastheaircraftpushesitselfthroughtheair,andthiscausesshockwaves–suddendiscontinuitiesinairpressure,heardonthegroundasasonicboom.ThisenvironmentalproblemwasonereasonwhythejointAnglo-FrenchairlinerConcorde,theonlysupersoniccommercialaircraftevertogointoservice,hadlimitedsuccess:itwasnotallowedtoflyatsupersonicspeedsexceptoveroceans.CFDiswidelyusedtopredicttheflowofairpastasupersonicaircraft.

Thereareabout600millioncarsontheplanetandtensofthousandsofcivilaircraft,soeventhoughtheseapplicationsofCFDmayseemhightech,theyaresignificantineverydaylife.OtherwaystouseCFDhaveamorehumandimension.Itiswidelyusedbymedicalresearcherstounderstandbloodflowinthehumanbody,forexample.Heartmalfunctionisoneoftheleadingcausesof

deathinthedevelopedworld,anditcanbetriggeredeitherbyproblemswiththeheartitselforbycloggedarteries,whichdisruptthebloodflowandcancauseclots.Themathematicsofbloodflowinthehumanbodyisespeciallyintractableanalyticallybecausethewallsofthearteriesareelastic.It’sdifficultenoughtocalculatethemovementoffluidthrougharigidtube;it’smuchharderifthetubecanchangeitsshapedependingonthepressurethatthefluidexerts,becausenowthedomainforthecalculationdoesn’tstaythesameastimepasses.Theshapeofthedomainaffectstheflowpatternofthefluid,andsimultaneouslytheflowpatternofthefluidaffectstheshapeofthedomain.Pen-and-papermathematicscan’thandlethatsortoffeedbackloop.

CFDisidealforthiskindofproblembecausecomputerscanperformbillionsofcalculationseverysecond.Theequationhastobemodifiedtoincludetheeffectsofelasticwalls,butthat’smostlyamatterofextractingthenecessaryprinciplesfromelasticitytheory,anotherwell-developedpartofclassicalcontinuummechanics.Forexample,aCFDcalculationofhowbloodflowsthroughtheaorta,themainarteryenteringtheheart,hasbeencarriedoutattheÉcolePolytechniqueFéderaledeLausanneinSwitzerland.Theresultsprovideinformationthatcanhelpdoctorsgetabetterunderstandingofcardiovascularproblems.

Theyalsohelpengineerstodevelopimprovedmedicaldevicessuchasstents–smallmetal-meshtubesthatkeepthearteryopen.SuncicaCanichasusedCFDandmodelsofelasticpropertiestodesignbetterstents,derivingamathematicaltheoremthatcausedonedesigntobeabandonedandsuggestedbetterdesigns.ModelsofthistypehavebecomesoaccuratethattheUSFoodandDrugsAdministrationisconsideringrequiringanygroupdesigningstentstocarryoutmathematicalmodellingbeforeperformingclinicaltrials.MathematiciansanddoctorsarejoiningforcestousetheNavier–Stokesequationtoobtainbetterpredictionsof,andbettertreatmentsfor,themaincausesofheartattacks.

Another,related,applicationistoheartbypassoperations,inwhichaveinisremovedfromelsewhereinthebodyandgraftedintothecoronaryartery.Thegeometryofthegrafthasastrongeffectonthebloodflow.Thisinturnaffectsclotting,whichismorelikelyiftheflowhasvorticesbecausebloodcanbecometrappedinavortexandfailtocirculateproperly.Sohereweseeadirectlinkbetweenthegeometryoftheflowandpotentialmedicalproblems.

TheNavier–Stokesequationhasanotherapplication:climatechange,otherwiseknownasglobalwarming.Climateandweatherarerelated,butdifferent.Weatheriswhathappensatagivenplace,atagiventime.Itmayberainingin

London,snowinginNewYork,orbakingintheSahara.Weatherisnotoriouslyunpredictable,andtherearegoodmathematicalreasonsforthis:seeChapter16onchaos.However,muchoftheunpredictabilityconcernssmall-scalechanges,bothinspaceandtime:thefinedetails.IftheTVweathermanpredictsshowersinyourtowntomorrowafternoonandtheyhappensixhourslaterand20kilometresaway,hethinkshedidagoodjobandyouarewildlyunimpressed.Climateisthelong-term‘texture’ofweather–howrainfallandtemperaturebehavewhenaveragedoverlongperiods,perhapsdecades.Becauseclimateaveragesoutthesediscrepancies,itisparadoxicallyeasiertopredict.Thedifficultiesarestillconsiderable,andmuchofthescientificliteratureinvestigatespossiblesourcesoferror,tryingtoimprovethemodels.

Climatechangeisapoliticallycontentiousissue,despiteaverystrongscientificconsensusthathumanactivityoverthepastcenturyorsohascausedtheaveragetemperatureoftheEarthtorise.Theincreasetodatesoundssmall,about0.75degreesCelsiusduringthetwentiethcentury,buttheclimateisverysensitivetotemperaturechangesonaglobalscale.Theytendtomaketheweathermoreextreme,withdroughtsandfloodsbecomingmorecommon.

‘Globalwarming’doesnotimplythatthetemperatureeverywhereischangingbythesametinyamount.Onthecontrary,therearelargefluctuationsfromplacetoplaceandfromtimetotime.In2010Britainexperienceditscoldestwinterfor31years,promptingtheDailyExpresstoprinttheheadline‘andstilltheyclaimit’sglobalwarming’.Asithappens,2010tiedwith2005asthehottestyearonrecord,acrosstheglobe.1So‘they’wereright.Infact,thecoldsnapwascausedbythejetstreamchangingposition,pushingcoldairsouthfromtheArctic,andthishappenedbecausetheArcticwasunusuallywarm.TwoweeksoffrostincentralLondondoesnotdisproveglobalwarming.Oddly,thesamenewspaperreportedthatEasterSunday2011wasthehottestonrecord,butmadenoconnectiontoglobalwarming.Onthatoccasiontheycorrectlydistinguishedweatherfromclimate.I’mfascinatedbytheselectiveapproach.

Similarly,‘climatechange’doesnotsimplymeanthattheclimateischanging.Ithasdonethatwithouthumanassistancerepeatedly,mainlyonlongtimescales,thankstovolcanicashandgases,long-termvariationsintheEarth’sorbitaroundtheSun,evenIndiacollidingwithAsiatocreatetheHimalayas.Inthecontextcurrentlyunderdebate,‘climatechange’isshortfor‘anthropogenicclimatechange’–changesinglobalclimatecausedbyhumanactivity.Themaincausesaretheproductionoftwogases:carbondioxideandmethane.Therearegreenhousegases:theytrapincomingradiation(heat)fromtheSun.Basicphysicsimpliesthatthemoreofthesegasestheatmospherecontains,themore

heatittraps;althoughtheplanetdoesradiatesomeheataway,onbalanceitwillgetwarmer.Globalwarmingwaspredicted,onthisbasis,inthe1950s,andthepredictedtemperatureincreaseisinlinewithwhathasbeenobserved.

Theevidencethatcarbondioxidelevelshaveincreaseddramaticallycomesfrommanysources.Themostdirectisicecores.Whensnowfallsinthepolarregions,itpackstogethertoformice,withthemostrecentsnowatthetopandtheoldestatthebottom.Airistrappedintheice,andtheconditionsthatprevailthereleaveitvirtuallyunchangedforverylongperiodsoftime,keepingtheoriginalairinandmorerecentairout.Withcare,itispossibletomeasurethecompositionofthetrappedairandtodeterminethedatewhenitwastrapped,veryaccurately.MeasurementsmadeintheAntarcticshowthattheconcentrationofcarbondioxideintheatmospherewasprettymuchconstantoverthepast100,000years–exceptforthelast200,whenitshotupby30%.Thesourceoftheexcesscarbondioxidecanbeinferredfromtheproportionsofcarbon-13,oneoftheisotopes(differentatomicforms)ofcarbon.Humanactivityisbyfarthemostlikelyexplanation.

Themainreasonwhytheskepticshaveevenfaintglimmeringsofacaseisthecomplexityofclimateforecasting.Thishastobedoneusingmathematicalmodels,becauseit’saboutthefuture.Nomodelcanincludeeverysinglefeatureoftherealworld,and,ifitdid,youcouldneverworkoutwhatitpredicted,becausenocomputercouldeversimulateit.Everydiscrepancybetweenmodelandreality,howeverinsignificant,ismusictotheskeptics’ears.Thereiscertainlyroomfordifferencesofopinionaboutthelikelyeffectsofclimatechange,orwhatweshoulddotomitigateit.Butburyingourheadsinthesandisn’tasensibleoption.

Twovitalaspectsofclimatearetheatmosphereandtheoceans.Botharefluids,andbothcanbestudiedusingtheNavier–Stokesequation.In2010theUK’smainsciencefundingbody,theEngineeringandPhysicalSciencesResearchCouncil,publishedadocumentonclimatechange,singlingoutmathematicsasaunifyingforce:‘Researchersinmeteorology,physics,geographyandahostofotherfieldsallcontributetheirexpertise,butmathematicsistheunifyinglanguagethatenablesthisdiversegroupofpeopletoimplementtheirideasinclimatemodels.’Thedocumentalsoexplainedthat‘ThesecretsoftheclimatesystemarelockedawayintheNavier–Stokesequation,butitistoocomplextobesolveddirectly.’Instead,climatemodellersusenumericalmethodstocalculatethefluidflowatthepointsofathree-dimensionalgrid,coveringtheglobefromtheoceandepthstotheupperreachesoftheatmosphere.Thehorizontalspacingofthegridis100kilometres–anythingsmallerwould

makethecomputationsimpractical.Fastercomputerswon’thelpmuch,sothebestwayforwardistothinkharder.MathematiciansareworkingonmoreefficientwaystosolvetheNavier–Stokesequationnumerically.

TheNavier–Stokesequationisonlypartoftheclimatepuzzle.Otherfactorsincludeheatflowwithinandbetweentheoceansandtheatmosphere,theeffectofclouds,non-humancontributionssuchasvolcanoes,evenaircraftemissionsinthestratosphere.Skepticsliketoemphasisesuchfactorstosuggestthemodelsarewrong,butmostofthemareknowntobeirrelevant.Forexample,everyyearvolcanoescontributeamere0.6%ofthecarbondioxideproducedbyhumanactivity.Allofthemainmodelssuggestthatthereisaseriousproblem,andhumanshavecausedit.Themainquestionisjusthowmuchtheplanetwillwarmup,andwhatlevelofdisasterwillresult.Sinceperfectforecastingisimpossible,itisineverybody’sintereststomakesurethatourclimatemodelsarethebestwecandevise,sothatwecantakeappropriateaction.Astheglaciersmelt,theNorthwestPassageopensupasArcticiceshrinks,andAntarcticiceshelvesarebreakingoffandslidingintotheocean,wecannolongertaketheriskofbelievingthatwedon’tneedtodoanythinganditwillallsortitselfout.

11Wavesintheether

Maxwell’sEquations

Whatdotheysay?

Electricityandmagnetismcan’tjustleakaway.Aspinningregionofelectricfieldcreatesamagneticfieldatrightanglestothespin.Aspinningregionofmagneticfieldcreatesanelectricfieldatrightanglestothespin,butintheoppositedirection.

Whyisthatimportant?

Itwasthefirstmajorunificationofphysicalforces,showingthatelectricityandmagnetismareintimatelyinterrelated.

Whatdiditleadto?

Thepredictionthatelectromagneticwavesexist,travellingatthespeedoflight,solightitselfissuchawave.Thismotivatedtheinventionofradio,radar,television,wirelessconnectionsforcomputerequipment,andmostmoderncommunications.

Atthestartofthenineteenthcenturymostpeoplelittheirhousesusingcandlesandlanterns.Gaslighting,whichdatesfrom1790,wasoccasionallyusedinhomesandbusinesspremises,mainlybyinventorsandentrepreneurs.GasstreetlightingcameintouseinParisin1820.Atthattime,thestandardwaytosendmessageswastowritealetterandsenditbyhorse-drawncarriage;forurgentmessages,keepthehorsebutomitthecarriage.Themainalternative,mostlyrestrictedtomilitaryandofficialcommunications,wastheopticaltelegraph.Thisusedsemaphore:mechanicaldevicesplacedontowers,whichcouldrepresentlettersorwordsincodebyarrangingrigidarmsatvariousangles.Theseconfigurationscouldbeseenthroughatelescopeandrelayedtothenexttowerinline.Thefirstextensivesystemofthiskinddatesfrom1792,whentheFrenchengineerClaudeChappebuilt556towerstocreatea4800kilometrenetworkacrossmostofFrance.Itremainedinuseforsixtyyears.

Withinahundredyears,homesandstreetshadelectriclighting,electrictelegraphyhadcomeandgone,andpeoplecouldtalktoeachotherbytelephone.Physicistshaddemonstratedradiocommunicationsintheirlaboratories,andoneentrepreneurhadalreadysetupafactoryselling‘wirelesses’–radiosets–tothepublic.Twoscientistsmadethemaindiscoveriesthattriggeredthissocialandtechnologicalrevolution.OnewastheEnglishmanMichaelFaraday,whoestablishedthebasicphysicsofelectromagnetism–atightly-knitcombinationofthepreviouslyseparatephenomenaofelectricityandmagnetism.TheotherwasaScotsman,JamesClerkMaxwell,whoturnedFaraday’smechanicaltheoriesintomathematicalequationsandusedthemtopredicttheexistenceofradiowavestravellingatthespeedoflight.

TheRoyalInstitutioninLondonisanimposingbuilding,frontedbyclassicalcolumns,tuckedawayonasidestreetnearPiccadillyCircus.Todayitsmainactivityistohostpopularscienceeventsforthepublic,butwhenitwasfoundedin1799itsbriefalsoincluded‘diffusingtheknowledge,andfacilitatingthegeneralintroduction,ofusefulmechanicalinventions’.WhenJohn‘MadJack’FullerestablishedaChairinChemistryattheRoyalInstitution,itsfirstincumbentwasnotanacademic.Hewasthesonofawould-beblacksmith,andhehadtrainedasabookseller’sapprentice.Thepositionallowedhimtoreadvoraciously,despitehisfamily’slackofcash,andJaneMarcet’sConversationsonChemistryandIsaacWatts’sTheImprovementoftheMindinspiredadeepinterestinscienceingeneralandelectricityinparticular.

TheyoungmanwasMichaelFaraday.HehadattendedlecturesattheRoyalInstitutiongivenbytheeminentchemistHumphryDavy,andhesentthelecturer300pagesofnotes.ShortlyafterwardsDavyhadanaccidentthatdamagedhiseyesight,andaskedFaradaytobecomehissecretary.ThenanassistantattheRoyalInstitutiongotthesack,andDavysuggestedFaradayasareplacement,settinghimtoworkonthechemistryofchlorine.

TheRoyalInstitutionallowedFaradaytopursuehisownscientificinterestsaswell,andhecarriedoutinnumerableexperimentsonthenewlydiscoveredtopicofelectricity.In1821helearnedoftheworkoftheDanishscientistHansChristianØrsted,linkingelectricitytothemucholderphenomenonofmagnetism.Faradayexploitedthislinktoinventanelectricmotor,butDavygotupsetwhenhedidn’tgetanycredit,andtoldFaradaytoworkonotherthings.Davydiedin1831,andtwoyearslaterFaradaybeganaseriesofexperimentsonelectricityandmagnetismthatsealedhisreputationasoneofthegreatestscientistsevertohavelived.Hisextensiveinvestigationswerepartlymotivatedbytheneedtocomeupwithlargenumbersofnovelexperimentstoedifythemaninthestreetandentertainthegreatandthegood,aspartoftheRoyalInstitution’sbrieftoencouragethepublicunderstandingofscience.

AmongFaraday’sinventionsweremethodsforturningelectricityintomagnetismandbothintomotion(amotor)andforturningmotionintoelectricity(agenerator).Theseexploitedhisgreatestdiscovery,electromagneticinduction.Ifmaterialthatcanconductelectricitymovesthroughamagneticfield,anelectricalcurrentwillflowthroughit.Faradaydiscoveredthisin1831.FrancescoZantedeschihadalreadynoticedtheeffectin1829,andJosephHenryalsospotteditalittlelater.ButHenrydelayedpublishinghisdiscovery,andFaradaytooktheideamuchfurtherthanZantedeschihaddone.Faraday’sworkwentfarbeyondtheRoyalInstitution’sbrieftofacilitateusefulmechanicalinventions,bycreatinginnovativemachinesthatexploitedfrontierphysics.Thisled,fairlydirectly,toelectricpower,lighting,andathousandothergadgets.Whenotherstookupthebaton,thewholepanoplyofmodernelectricalandelectronicequipmentburstuponthescene,startingwithradio,movingontotelevision,radar,andlong-distancecommunications.ItwasFaraday,morethananyothersingleindividual,whocreatedthemoderntechnologicalworld,withthehelpofvitalnewideasfromhundredsofgiftedengineers,scientists,andbusinessmen.

Beingworkingclassandlackingthenormaleducationofagentleman,Faradaytaughthimselfsciencebutnotmathematics.Hedevelopedhisowntheoriestoexplainandguidehisexperiments,buttheyrestedonmechanical

analogiesandconceptualmachines,notonformulasandequations.HisworktookitsdeservedplaceinbasicphysicsthroughtheinterventionofoneofScotland’sgreatestscientificintellects,JamesClerkMaxwell.

MaxwellwasbornthesameyearthatFaradayannouncedthediscoveryofelectromagneticinduction.Oneapplication,theelectromagnetictelegraph,quicklyfollowed,thankstoGaussandhisassistantWilhelmWeber.GausswantedtousewirestocarryelectricalsignalsbetweenGöttingenObservatory,wherehehungout,totheInstituteofPhysicsakilometreaway,whereWeberworked.Presciently,Gausssimplifiedtheprevioustechniquefordistinguishinglettersofthealphabet–onewireperletter–byintroducingabinarycodeusingpositiveandnegativecurrent,seeChapter15.By1839theGreatWesternRailwaycompanywassendingmessagesbytelegraphfromPaddingtontoWestDrayton,adistanceof21kilometres.InthesameyearSamuelMorseindependentlyinventedhisownelectrictelegraphintheUSA,employingMorsecode(inventedbyhisassistantAlfredVail)andsendingitsfirstmessagein1838.

In1876,threeyearsbeforeMaxwelldied,AlexanderGrahamBelltookoutthefirstpatentonanewgadget,theacoustictelegraph.Itwasadevicethatturnedsound,especiallyspeech,intoelectricalimpulses,andtransmittedthemalongawiretoareceiver,whichturnedthembackintosound.Wenowknowitasthetelephone.Hewasn’tthefirstpersontoconceiveofsuchathing,oreventobuildone,butheheldthemasterpatent.ThomasEdisonimprovedthedesignwithhiscarbonmicrophoneof1878.Ayearlater,Edisondevelopedthecarbonfilamentelectriclightbulb,andcementedhimselfinthepopularmindastheinventorofelectriclighting.Inpointoffact,hewasprecededbyatleast23inventors,thebestknownbeingJosephSwan,whohadpatentedhisversionin1878.In1880,oneyearafterMaxwell’sdeath,thecityofWabash,Illinoisbecamethefirsttouseelectriclightingforitsstreets.

TheserevolutionsincommunicationandlightingowedalottoFaraday;electricalpowergenerationalsoowedalottoMaxwell.ButMaxwell’smostfar-reachinglegacywastomakethetelephoneseemlikeachild’stoy.Anditstemmed,directlyandinevitably,fromhisequationsforelectromagnetism.

MaxwellwasbornintoatalentedbuteccentricEdinburghfamily,whichincludedlawyers,judges,musicians,politicians,poets,miningspeculators,andbusinessmen.Asateenagerhebegantosuccumbtothecharmsofmathematics,winningaschoolcompetitionwithanessayonhowtoconstructovalcurves

usingpinsandthread.At16hewenttoEdinburghUniversity,wherehestudiedmathematicsandexperimentedinchemistry,magnetism,andoptics.HepublishedpapersinpureandappliedmathematicsintheRoyalSocietyofEdinburgh’sjournal.In1850hismathematicalcareertookamoreseriousturnandhemovedtoCambridgeUniversity,wherehewasprivatelycoachedforthemathematicaltriposexaminationbyWilliamHopkins.Thetriposinthosedaysconsistedofsolvingcomplicatedproblems,ofteninvolvingclevertricksandextensivecalculations,againsttheclock.LaterGodfreyHaroldHardy,oneofEngland’sbestmathematiciansandaCambridgeprofessor,wouldhavestrongviewsabouthowtodocreativemathematics,andcrammingforatrickyexaminationwasn’tit.In1926heremarkedthathisaimwas‘not…toreformthetripos,buttodestroyit’.ButMaxwellcrammed,andthrived,inthecompetitiveatmosphere,probablybecausehehadthatsortofmind.

Healsocontinuedhisweirdexperiments,amongotherthingstryingtoworkouthowacatalwayslandsonitsfeet,evenwhenitisheldupsidedownonlyafewcentimetresaboveabed.ThedifficultyisthatthisappearstoviolateNewtonianmechanics;thecathastorotatethrough180degrees,buthasnothingtopushagainst.Theprecisemechanismeludedhim,andwasnotworkedoutuntiltheFrenchdoctorJulesMareymadeaseriesofphotographsofafallingcatin1894.Thesecretisthatthecatisnotrigid:ittwistsitsfrontandbackinoppositedirectionsandbackagain,whileextendingandretractingitspawstostopthesemotionscancellingout.1

Maxwellgothismathematicsdegree,andcontinuedasapostgraduateatTrinityCollege.TherehereadFaraday’sExperimentalResearchesandworkedonelectricityandmagnetism.HetookupachairofNaturalPhilosophyinAberdeen,investigatingSaturn’sringsandthedynamicsofthemoleculesingases.In1860hemovedtoKing’sCollegeLondon,andherehecouldsometimesmeetwithFaraday.NowMaxwellembarkedonhismostinfluentialquest:toformulateamathematicalbasisforFaraday’sexperimentsandtheories.

Atthetime,mostphysicistsworkingonelectricityandmagnetismwerelookingforanalogieswithgravity.Itseemedsensible:oppositeelectricalchargesattracteachotherwithaforcewhich,likegravity,isproportionaltotheinversesquareofthedistanceseparatingthem.Likechargesrepeleachotherwithasimilarlyvaryingforce,andthesamegoesformagnetism,wherechargesarereplacedbymagneticpoles.Thestandardwayofthinkingwasthatgravitywasaforcewherebyonebodymysteriouslyactedonanotherdistantbody,withoutanythingpassingbetweenthetwo;electricityandmagnetismwereassumedtoactinthe

samemanner.Faradayhadadifferentidea:theyareboth‘fields’,phenomenathatpervadespaceandcanbedetectedbytheforcestheyproduce.

Whatisafield?Maxwellcouldmakelittleprogressuntilhecoulddescribetheconceptmathematically.ButFaraday,lackingmathematicaltraining,hadposedhistheoriesintermsofgeometricstructures,suchas‘linesofforce’alongwhichthefieldspulledandpushed.Maxwell’sfirstgreatbreakthroughwastoreformulatetheseideasbyanalogywiththemathematicsoffluidflow,wherethefieldineffectisthefluid.Linesofforcewerethenanalogoustothepathsfollowedbythemoleculesofthefluid;thestrengthoftheelectricormagneticfieldwasanalogoustothevelocityofthefluid.Informally,afieldwasaninvisiblefluid;mathematically,itbehavedexactlylikethat,whateveritreallywas.Maxwellborrowedideasfromthemathematicsoffluidsandmodifiedthemtodescribemagnetism.Hismodelaccountedforthemainpropertiesobservedinelectricity.

Notcontentwiththisinitialattempt,hewentontoincludenotjustmagnetism,butitsrelationtoelectricity.Astheelectricalfluidflowed,itaffectedthemagneticone,andviceversa.FormagneticfieldsMaxwellusedthementalimageoftinyvorticesspinninginspace.Electricfieldsweresimilarlycomposedoftinychargedspheres.Followingthisanalogyandtheresultingmathematics,Maxwellbegantounderstandhowachangeintheelectricforcecouldcreateamagneticfield.Asthespheresofelectricitymove,theycausethemagneticvorticestospin,likeafootballfanpassingthroughaturnstile.Thefanmoveswithoutspinning;theturnstilespinswithoutmoving.

Maxwellwasslightlydissatisfiedwiththisanalogy,saying‘Idonotbringitforward…asamodeofconnectionexistinginnature…Itis,however…mechanicallyconceivableandeasilyinvestigated,anditservestobringouttheactualmechanicalconnectionsbetweentheknownelectromagneticphenomena.’Toshowwhathemeant,heusedthemodeltoexplainwhyparallelwirescarryingoppositeelectricalcurrentsrepeleachother,andhealsoexplainedFaraday’scrucialdiscoveryofelectromagneticinduction.

Thenextstepwastoretainthemathematicswhilegettingridofthemechanicalgadgetrythatpropelledtheanalogy.Thisamountedtowritingdownequationsforthebasicinteractionsbetweentheelectricalandmagneticfields,derivedfromthemechanicalmodel,butdivorcedfromthisorigin.Maxwellachievedthisgoalin1864inhisfamouspaper‘Adynamicaltheoryoftheelectromagneticfield’.

Wenowinterprethisequationsusingvectors,whicharequantitiesthat

possessnotjustasize,butadirection.Themostfamiliarisvelocity:thesizeisthespeed,howfasttheobjectismoving;thedirectionistheonealongwhichitmoves.Thedirectionreallydoesmatter:abodymovingverticallyupwardsat10kpsbehavesverydifferentlyfromonemovingverticallydownwardsat10kps.Mathematically,avectorisrepresentedbyitsthreecomponents:itseffectalongthreeaxesatrightanglestoeachother,suchasnorth/south,east/west,andup/down.Thebarebonesarethusthatavectorisatriple(x,y,z)composedofthreenumbers,Figure44.Thevelocityofafluidatagivenpoint,forinstance,isavector.Incontrast,thepressureatagivenpointisasinglenumber:thefancytermusedtodistinguishthisfromavectoris‘scalar’.

Fig44Athree-dimensionalvector.

Intheseterms,whatistheelectricfield?FromFaraday’sperspectiveitisdeterminedbylinesofelectricalforce.InMaxwell’sanalogy,theseareflow-linesoftheelectricalfluid.Aflow-linetellsusinwhichdirectionthefluidisflowing,andasamoleculemovesalongtheflow-line,wecanalsoobserveitsspeed.Foreachpointinspace,theflow-linepassingthroughthatpointthereforedeterminesavector,whichdescribesthespeedanddirectionoftheelectricfluid,thatis,thestrengthanddirectionoftheelectricfieldatthatpoint.Conversely,ifweknowthesespeedsanddirections,foreverypointinspace,wecandeducewhattheflow-lineslooklike,soinprincipleweknowtheelectricfield.

Inshort:theelectricfieldisasystemofvectors,oneforeachpointinspace.Eachvectorprescribesthestrengthanddirectionoftheelectricalforce(exertedonatinychargedtestparticle)atthatpoint.Mathematicianscallsuchaquantityavectorfield:itisafunctionthatassignstoeachpointinspacethecorrespondingvector.Similarly,themagneticfieldisdeterminedbythemagneticlinesofforce;itisthevectorfieldcorrespondingtotheforcesthatwouldbeexertedonatinymagnetictestparticle.

Havingsortedoutwhatelectricandmagneticfieldswere,Maxwellcould

writedownequationsdescribingwhattheydid.Wenowexpresstheseequationsusingtwovectoroperators,knownasdivergenceandcurl.Maxwellusedspecificformulasinvolvingthethreecomponentsoftheelectricandmagneticfields.Inthespecialcaseinwhichtherearenoconductingwiresormetalplates,nomagnets,andeverythinghappensinavacuum,theequationstakeaslightlysimplerform,andIwillrestrictthediscussiontothiscase.

Twooftheequationstellusthattheelectricandmagneticfluidsareincompressible–thatis,electricityandmagnetismcannotjustleakaway,theyhavetogosomewhere.Thistranslatesas‘thedivergenceiszero’,leadingtotheequations

∇.E=0∇.H=0

wheretheupside-downtriangleandthedotarethenotationforthedivergence.Twomoreequationstellusthatwhenaregionofelectricfieldspinsinasmallcircle,itcreatesamagneticfieldatrightanglestotheplaneofthatcircle,andsimilarlyaspinningregionofmagneticfieldcreatesanelectricfieldatrightanglestotheplaneofthatcircle.Thereisacurioustwist:theelectricandmagneticfieldspointinoppositedirectionsforagivendirectionofspin.Theequationsare

wherenowtheupside-downtriangleandthecrossarethenotationforthecurl.Thesymboltstandsfortime,and∂/∂tistherateofchangewithrespecttotime.Noticethatthefirstequationhasaminussign,buttheseconddoesnot:thisrepresentstheoppositeorientationsthatImentioned.

Whatisc?Itisaconstant,theratioofelectromagnetictoelectrostaticunits.Experimentallythisratioisjustunder300,000,inunitsofkilometresdividedbyseconds.Maxwellimmediatelyrecognisedthisnumber:itisthespeedoflightinavacuum.Whydidthatquantityappear?Hedecidedtofindout.Oneclue,datingbacktoNewton,anddevelopedbyothers,wasthediscoverythatlightwassomekindofwave.Butnooneknewwhatthewaveconsistedof.

Asimplecalculationprovidedtheanswer.Onceyouknowtheequationsforelectromagnetism,youcansolvethemtopredicthowtheelectricandmagneticfieldsbehaveindifferentcircumstances.Youcanalsoderivegeneralmathematicalconsequences.Forinstance,thesecondpairofequationsrelatesE

toH;anymathematicianwillimmediatelytrytoderiveequationsthatcontainonlyEandonlyH,becausethatletsusconcentrateoneachfieldseparately.Consideringitsepicconsequences,thistaskturnsouttobeabsurdlysimple–ifyouhavesomefamiliaritywithvectorcalculus.I’veputthedetailedworkingintheNotes,2buthere’saquicksummary.Followingournoses,westartwiththethirdequation,whichrelatesthecurlofEtothetime-derivativeofH.Wedon’thaveanyotherequationsinvolvingthetime-derivativeofH,butwedohaveonethatinvolvesthecurlofH,namely,thefourthequation.Thissuggeststhatweshouldtakethethirdequationandformthecurlofbothsides.Thenweapplythefourthequation,simplify,andemergewith

whichisthewaveequation!ThesametrickappliedtothecurlofHproducesthesameequationwithHin

placeofE.(Theminussignisappliedtwice,soitdisappears.)Soboththeelectricandmagneticfields,inavacuum,obeythewaveequation.Sincethesameconstantcoccursineachwaveequation,theybothtravelatthesamespeed,namelyc.Sothislittlecalculationpredictsthatboththeelectricfieldandthemagneticfieldcansimultaneouslysupportawave–makingitanelectromagneticwave,inwhichbothfieldsvaryinconcertwitheachother.Andthespeedofthatwaveis…thespeedoflight.

It’sanotherofthosetrickquestions.Whattravelsatthespeedoflight?Thistimetheansweriswhatyou’dexpect:light.Butthereisamomentousimplication:lightisanelectromagneticwave.

Thiswasstupendousnews.Therewasnoreason,priortoMaxwell’sderivationofhisequations,toimaginesuchafundamentallinkbetweenlight,electricity,andmagnetism.Buttherewasmore.Lightcomesinmanydifferentcolours,andonceyouknowthatlightisawave,youcanworkoutthatthesecorrespondtowaveswithdifferentwavelengths–distancebetweenssuccessivepeaks.Thewaveequationimposesnoconditionsonthewavelength,soitcanbeanything.Thewavelengthsofvisiblelightarerestrictedtoasmallrange,becauseofthechemistryoftheeye’slight-detectingpigments.Physicistsalreadyknewof‘invisiblelight’,ultravioletandinfrared.Those,ofcourse,hadwavelengthsjustoutsidethevisiblerange.NowMaxwell’sequationsledtoadramaticprediction:electromagneticwaveswithotherwavelengthsshouldalsoexist.Conceivably,anywavelength–longorshort–couldoccur,Figure45.

Fig45Theelectromagneticspectrum.

Noonehadexpectedthis,butassoonastheorysaiditoughttohappen,experimentalistscouldgoandlookforit.OneofthemwasaGerman,HeinrichHertz.In1886heconstructedadevicethatcouldgenerateradiowavesandanotherthatcouldreceivethem.Thetransmitterwaslittlemorethanamachinethatcouldproduceahigh-voltagespark;theoryindicatedthatsuchasparkwouldemitradiowaves.Thereceiverwasacircularloopofcopperwire,whosesizewaschosentoresonatewiththeincomingwaves.Asmallgapintheloop,afewhundredthsofamillimetreacross,wouldrevealthosewavesbyproducingtinysparks.In1887Hertzdidtheexperiment,anditwasasuccess.Hewentontoinvestigatemanydifferentfeaturesofradiowaves.Healsomeasuredtheirspeed,gettingananswerclosetothespeedoflight,whichconfirmedMaxwell’spredictionandconfirmedthathisapparatusreallywasdetectingelectromagneticwaves.

Hertzknewthathisworkwasimportantasphysics,andhepublisheditinElectricWaves:beingresearchesonthepropagationofelectricactionwithfinitevelocitythroughspace.Butitneveroccurredtohimthattheideamighthavepracticaluses.Whenasked,hereplied‘It’sofnousewhatsoever…justanexperimentthatprovesMaestroMaxwellwasright–wejusthavethesemysteriouselectromagneticwavesthatwecannotseewiththenakedeye.Buttheyarethere.’Pressedforhisviewoftheimplications,hesaid‘Nothing,Iguess.’

Wasitafailureofimagination,orjustalackofinterest?It’shardtotell.ButHertz’s‘useless’experiment,confirmingMaxwell’spredictionofelectromagneticradiation,wouldquicklyleadtoaninventionthatmadethe

telephonelooklikeachildren’stoy.Radio.

Radiomakesuseofanespeciallyintriguingrangeofthespectrum:waveswithwavelengthsmuchlongerthanlight.Suchwaveswouldbelikelytoretaintheirstructureoverlongdistances.Thekeyidea,theonethatHertzmissed,issimple:ifyoucouldsomehowimpressasignalonawaveofthatkind,youcouldtalktotheworld.

Otherphysicists,engineers,andentrepreneursweremoreimaginative,andquicklyspottedradio’spotential.Torealisethatpotential,however,theyhadtosolveanumberoftechnicalproblems.Theyneededatransmitterthatcouldproduceasufficientlypowerfulsignal,andsomethingtoreceiveit.Hertz’sapparatuswasrestrictedtoadistanceofafewfeet;youcanunderstandwhyhedidn’tsuggestcommunicationasapossibleapplication.Anotherproblemwashowtoimposeasignal.Athirdwashowfarthesignalcouldbesent,whichmightwellbelimitedbythecurvatureoftheEarth.Ifastraightlinebetweentransmitterandreceiverhitstheground,thiswouldpresumablyblockthesignal.Lateritturnedoutthatnaturehasbeenkindtous,andtheEarth’sionospherereflectsradiowavesinawiderangeofwavelengths,butbeforethiswasdiscoveredtherewereobviouswaysroundthepotentialproblemanyway.Youcouldbuildtalltowersandputthetransmittersandreceiversonthose.Byrelayingsignalsfromonetowertoanother,youcouldsendmessagesroundtheglobe,veryfast.

Therearetworelativelyobviouswaystoimpressasignalonaradiowave.Youcanmaketheamplitudevaryoryoucanmakethefrequencyvary.Thesemethodsarecalledamplitude-modulationandfrequency-modulation:AMandFM.Bothwereusedandbothstillexist.Thatwasoneproblemsolved.By1893theSerbianengineerNikolaTeslahadinventedandbuiltallofthemaindevicesneededforradiotransmission,andhehaddemonstratedhismethodstothepublic.In1894OliverLodgeandAlexanderMuirheadsentaradiosignalfromtheClarendonlaboratoryinOxfordtoanearbylecturetheatre.AyearlatertheItalianinventorGuglielmoMarconitransmittedsignalsoveradistanceof1.5kilometresusingnewapparatushehadinvented.TheItaliangovernmentdeclinedtofinancefurtherwork,soMarconimovedtoEngland.WiththesupportoftheBritishPostOfficehesoonimprovedtherangeto16kilometres.FurtherexperimentsledtoMarconi’slaw:thedistanceoverwhichsignalscanbesentisroughlyproportionaltothesquareoftheheightofthetransmittingantenna.Makethetowertwiceastallandthesignalgoesfourtimesasfar.This,

too,wasgoodnews:itsuggestedthatlong-rangetransmissionshouldbepractical.MarconisetupatransmittingstationontheIsleofWightintheUKin1897,andopenedafactorythenextyear,makingwhathecalled‘wirelesses’.Westillcalledthemthatin1952,whenIlistenedtotheGoonShowandDanDareonthewirelessinmybedroom,buteventhenwealsoreferredtothedeviceas‘theradio’.Theword‘wireless’hasofcoursecomebackintovogue,butnowitisthelinksbetweenyourcomputeranditskeyboard,mouse,modem,andInternetrouterthatarewireless,ratherthanthelinkfromyourreceivertoadistanttransmitter.It’sstilldonebyradio.

InitiallyMarconiownedthemainpatentstoradio,buthelostthemtoTeslain1943inacourtbattle.Technologicaladvancesquicklymadethosepatentsobsolete.From1906tothe1950s,thevitalelectroniccomponentofaradiowasthevacuumtube,likeasmallishlightbulb,soradioshadtobebigandbulky.Thetransistor,amuchsmallerandmorerobustdevice,wasinventedin1947atBellLaboratoriesbyanengineeringteamthatincludedWilliamShockley,WalterBrattain,andJohnBardeen(seeChapter14).By1954transistorradioswereonthemarket,butradiowasalreadylosingitsprimacyasanentertainmentmedium.

By1953,I’dalreadyseenthefuture.ItwasthecoronationofQueenElizabethII,andmyauntinTonbridgehad…atelevisionset!Sowepiledintomyfather’sricketycaranddrove40milestowatchtheevent.IwasmoreimpressedbyBillandBentheFlowerpotMenthanbythecoronation,tobehonest,butfromthatmomentradiowasnolongertheepitomeofmodernhouseholdentertainment.Soonwe,too,possessedatelevisionset.Anyonewhohasgrownupwith48-inchflatscreencolourTVswithhighdefinitionandathousandchannelswillbeappalledtohearthatinthosedaysthepicturewasblack-and-white,about12inchesacross,and(intheUK)therewasexactlyonechannel,theBBC.Whenwewatched‘thetelevision’itreallymeantthetelevision.

Entertainmentwasjustoneapplicationofradiowaves.Theywerealsovitaltothemilitary,forcommunicationsandotherpurposes.Theinventionofradar(radiodetectionandranging)maywellhavewonWorldWarIIfortheAllies.Thistop-secretdevicemadeitpossibletodetectaircraft,especiallyenemyaircraft,bybouncingradiosignalsoffthemandobservingthereflectedwaves.Theurbanmyththatcarrotsaregoodforyoureyesightoriginatedinwartimedisinformation,intendedtostoptheNaziswonderingwhytheBritishweregettingsogoodatspottingraidingbombers.Radarhaspeacetimeusesaswell.Itishowairtrafficcontrollerskeeptabsonwherealltheplanesare,toprevent

collisions;itguidespassengerjetstotherunwayinfog;itwarnspilotsofimminentturbulence.Archaeologistsuseground-penetratingradartolocatelikelysitesfortheremainsoftombsandancientstructures.

X-rays,firststudiedsystematicallybyWilhelmRöntgenin1875,havemuchshorterwavelengthsthanlight.Thismakesthemmoreenergetic,sotheycanpassthroughopaqueobjects,notablythehumanbody.DoctorscoulduseX-raystodetectbrokenbonesandotherphysiologicalproblems,andstilldo,althoughmodernmethodsaremoresophisticatedandsubjectthepatienttofarlessdamagingradiation.X-rayscannerscannowcreatethree-dimensionalimagesofahumanbody,orsomepartofit,inacomputer.Otherkindsofscannercandothesamethingusingdifferentphysics.

Microwavesareefficientwaystosendtelephonesignals,andtheyalsoturnupinthekitcheninmicrowaveovens,quickwaystoheatfood.Oneofthelatestapplicationstoemergeisinairportsecurity.Terahertzradiation,otherwiseknownasT-waves,canpenetrateclothingandevenbodycavities.Customsofficialscanusethemtospotdrugsmugglersandterrorists.Theiruseisalittlecontroversial,sincetheyamounttoanelectronicstrip-search,butmostofusseemtothinkthat’sasmallpricetopayifitstopsaplanebeingblownuporcocainehittingthestreets.T-wavesarealsousefultoarthistorians,becausetheycanrevealmuralscoveredinlayersofplaster.ManufacturersandcommercialcarrierscanuseT-wavestoinspectproductswithouttakingthemoutoftheirboxes.

Theelectromagneticspectrumissoversatile,andsoeffective,thatitsinfluenceisnowfeltinvirtuallyallspheresofhumanactivity.Itmakesthingspossiblethattoanypreviousgenerationwouldappearmiraculous.Ittookavastnumberofpeople,fromeveryprofession,toturnthepossibilitiesinherentinthemathematicalequationsintorealgadgetsandcommercialsystems.Butnoneofthiswaspossibleuntilsomeonerealisedthatelectricityandmagnetismcanjoinforcestocreateawave.Thewholepanoplyofmoderncommunications,fromradioandtelevisiontoradarandmicrowavelinksformobilephones,wastheninevitable.Anditallstemmedfromfourequationsandacoupleoflinesofbasicvectorcalculus.

Maxwell’sequationsdidn’tjustchangetheworld.Theyopenedupanewone.

12Lawanddisorder

SecondLawofThermodynamics

Whatdoesitsay?

Theamountofdisorderinathermodynamicsystemalwaysincreases.

Whyisthatimportant?

Itplaceslimitsonhowmuchusefulworkcanbeextractedfromheat.

Whatdiditleadto?

Bettersteamengines,estimatesoftheefficiencyofrenewableenergy,the‘heatdeathoftheuniverse’scenario,proofthatmatterismadeofatoms,andparadoxicalconnectionswiththearrowoftime.

InMay1959thephysicistandnovelistC.P.SnowdeliveredalecturewiththetitleTheTwoCultures,whichprovokedwidespreadcontroversy.TheresponseoftheprominentliterarycriticF.R.Leaviswastypicaloftheothersideoftheargument;hesaidbluntlythattherewasonlyoneculture:his.Snowsuggestedthatthesciencesandthehumanitieshadlosttouchwitheachother,andarguedthatthiswasmakingitverydifficulttosolvetheworld’sproblems.Weseethesametodaywithclimatechangedenialandattacksonevolution.Themotivationmaybedifferent,butculturalbarriershelpsuchnonsensetothrive–thoughitispoliticsthatdrivesit.

Snowwasparticularlyunhappyaboutwhathesawasdecliningstandardsofeducation,saying:

AgoodmanytimesIhavebeenpresentatgatheringsofpeoplewho,bythestandardsofthetraditionalculture,arethoughthighlyeducatedandwhohavewithconsiderablegustobeenexpressingtheirincredulityattheilliteracyofscientists.OnceortwiceIhavebeenprovokedandhaveaskedthecompanyhowmanyofthemcoulddescribetheSecondLawofThermodynamics,thelawofentropy.Theresponsewascold:itwasalsonegative.YetIwasaskingsomethingwhichisaboutthescientificequivalentof:‘HaveyoureadaworkofShakespeare’s?’

Perhapshesensedhewasaskingtoomuch–manyqualifiedscientistscan’tstatethesecondlawofthermodynamics.Sohelateradded:

InowbelievethatifIhadaskedanevensimplerquestion–suchas,Whatdoyoumeanbymass,oracceleration,whichisthescientificequivalentofsaying,‘Canyouread?’–notmorethanoneintenofthehighlyeducatedwouldhavefeltthatIwasspeakingthesamelanguage.Sothegreatedificeofmodernphysicsgoesup,andthemajorityofthecleverestpeopleinthewesternworldhaveaboutasmuchinsightintoitastheirNeolithicancestorswouldhavehad.

TakingSnowliterally,myaiminthischapteristotakeusoutoftheNeolithicage.Theword‘thermodynamics’containsaclue:itappearstomeanthedynamicsofheat.Canheatbedynamic?Yes:heatcanflow.Itcanmovefrom

onelocationtoanother,fromoneobjecttoanother.Gooutsideonawinter’sdayandyousoonfeelcold.Fourierhadwrittendownthefirstseriousmodelofheatflow,Chapter9anddonesomebeautifulmathematics.Butthemainreasonscientistswerebecominginterestedinheatflowwasanewfangledandhighlyprofitableitemoftechnology:thesteamengine.

Thereisanoft-repeatedstoryofJamesWattasaboy,sittinginhismother’skitchenwatchingboilingsteamliftthelidoffakettle,andhissuddenflashofinspiration:steamcanperformwork.So,whenhegrewup,heinventedthesteamengine.It’sinspirationalstuff,butlikemanysuchtalesthisoneisjusthotair.Wattdidn’tinventthesteamengine,andhedidn’tlearnaboutthepowerofsteamuntilhewasanadult.Thestory’sconclusionaboutthepowerofsteamistrue,buteveninWatt’sdayitwasoldhat.

Around50BCtheRomanarchitectandengineerVitruviusdescribedamachinecalledanaeolipileinhisDeArchitectura(‘OnArchitecture’),andtheGreekmathematicianandengineerHeroofAlexandriabuiltoneacenturylater.Itwasahollowspherewithsomewaterinside,andtwotubespokedout,bentatanangleasinFigure46.Heatthesphereandthewaterturnstosteam,escapesthroughtheendsofthetubes,andthereactionmakesthespherespin.Itwasthefirststeamengine,anditprovedthatsteamcoulddowork,butHerodidnothingwithitbeyondentertainingpeople.Hedidmakeasimilarmachineusinghotairinanenclosedchambertopullaropethatopenedthedoorsofatemple.Thismachinehadapracticalapplication,producingareligiousmiracle,butitwasn’tasteamengine.

Fig46Hero’saeolipile.

Wattlearnedthatsteamcouldbeasourceofpowerin1762whenhewas26yearsold.Hedidn’tdiscoveritwatchingakettle:hisfriendJohnRobison,aprofessorofnaturalphilosophyattheUniversityofEdinburgh,toldhimaboutit.Butpracticalsteampowerwasmucholder.ItsdiscoveryisoftencreditedtotheItalianengineerandarchitectGiovanniBranca,whoseLeMachine(‘Machine’)of1629contained63woodcutsofmechanicalgadgets.Oneshowsapaddlewheelthatwouldspinonitsaxlewhensteamfromapipecollidedwithitsvanes.Brancaspeculatedthatthismachinemightbeusefulforgrindingflour,liftingwater,andcuttingupwood,butitwasprobablyneverbuilt.Itwasmoreofathoughtexperiment,amechanicalpipedreamlikeLeonardodaVinci’sflyingmachine.

Inanycase,BrancawasanticipatedbyTaqial-DinMuhammadibnMa’rufal-Shamial-Asadi,wholivedaround1550intheOttomanEmpireandwaswidelyheldtobethegreatestscientistofhisage.Hisachievementsareimpressive.Heworkedineverythingfromastrologytozoology,includingclock-making,medicine,philosophy,andtheology,andhewroteover90books.Inhis1551Al-turuqal-samiyyafial-alatal-ruhaniyya(‘TheSublimeMethodsofSpiritualMachines’),al-Dindescribedaprimitivesteamturbine,sayingthatitcouldbeusedtoturnroastingmeatonaspit.

ThefirsttrulypracticalsteamenginewasawaterpumpinventedbyThomasSaveryin1698.Thefirsttomakecommercialprofits,builtbyThomasNewcomenin1712,triggeredtheIndustrialRevolution.ButNewcomen’senginewasveryinefficient.Watt’scontributionwastointroduceaseparatecondenserforthesteam,reducingheatloss.DevelopedusingmoneyprovidedbytheentrepreneurMatthewBolton,thisnewtypeofengineusedonlyaquarterasmuchcoal,leadingtohugesavings.BoultonandWatt’smachinewentintoproductionin1775,morethan220yearsafteral-Din’sbook.By1776,threewereupandrunning:oneinacoalmineatTipton,oneinaShropshireironworks,andoneinLondon.

Steamenginesperformedavarietyofindustrialtasks,butbyfarthecommonestwaspumpingwaterfrommines.Itcostalotofmoneytodevelopamine,butastheupperlayersbecameworkedoutandoperatorswereforcedtodigdeeperintotheground,theyhitthewatertable.Itwasworthspendingquitealotofmoneytopumpthewaterout,sincethealternativewastoclosethemineandstartagainsomewhereelse–andthatmightnotevenbefeasible.Butnoonewantedtopaymorethantheyhadto,soamanufacturerwhocoulddesignand

buildamoreefficientsteamenginewouldcornerthemarket.Sothebasicquestionofhowefficientasteamenginecouldbecriedoutforattention.Itsanswerdidmorethanjustdescribethelimitstosteamengines:itcreatedanewbranchofphysics,whoseapplicationswerealmostboundless.Thenewphysicsshedlightoneverythingfromgasestothestructureoftheentireuniverse;itappliednotjusttothedeadmatterofphysicsandchemistry,butperhapsalsotothecomplexprocessesoflifeitself.Itwascalledthermodynamics:themotionofheat.And,justasthelawofconservationofenergyinmechanicsruledoutmechanicalperpetualmotionmachines,thelawsofthermodynamicsruledoutsimilarmachinesusingheat.

Oneofthoselaws,thefirstlawofthermodynamics,revealedanewformofenergyassociatedwithheat,andextendedthelawofconservationofenergy(Chapter3)intothenewrealmofheatengines.Another,withoutanypreviousprecedent,showedthatsomepotentialwaystoexchangeheat,whichdidnotconflictwithconservationofenergy,wereneverthelessimpossiblebecausetheywouldhavetocreateorderfromdisorder.Thiswasthesecondlawofthermodynamics.

Thermodynamicsisthemathematicalphysicsofgases.Itexplainshowlarge-scalefeaturesliketemperatureandpressurearisefromthewaythegasmoleculesinteract.Thesubjectbeganwithaseriesoflawsofnaturerelatingtemperature,pressure,andvolume.Thisversioniscalledclassicalthermodynamics,anddidnotinvolvemolecules–atthattimefewscientistsbelievedinthem.Later,thegaslawswereunderpinnedbyafurtherlayerofexplanation,basedonasimplemathematicalmodelexplicitlyinvolvingmolecules.Thegasmoleculeswerethoughtofastinyspheresthatbouncedoffeachotherlikeperfectlyelasticbilliardballs,withnoenergybeinglostinthecollision.Althoughmoleculesarenotspherical,thismodelprovedtoberemarkablyeffective.Itiscalledthekinetictheoryofgases,anditledtoexperimentalproofthatmoleculesexist.

Theearlygaslawsemergedinfitsandstartsoveraperiodofnearlyfiftyyears,andaremainlyattributedtotheIrishphysicistandchemistRobertBoyle,theFrenchmathematicianandballoonpioneerJacquesAlexandreCésarCharles,andtheFrenchphysicistandchemistJosephLouisGay-Lussac.However,manyofthediscoveriesweremadebyothers.In1834,theFrenchengineerandphysicistÉmileClapeyroncombinedalloftheselawsintoone,theidealgaslaw,whichwenowwriteas

pV=RT

Herepispressure,Visvolume,Tisthetemperature,andRisaconstant.Theequationstatesthatpressuretimesvolumeisproportionaltotemperature.Ittookalotofworkwithmanydifferentgasestoconfirmeachseparatelaw,andClapeyron’soverallsynthesis,experimentally.Theword‘ideal’appearsbecauserealgasesdonotobeythelawinallcircumstances,especiallyathighpressureswhereinteratomicforcescomeintoplay.Buttheidealversionwasgoodenoughfordesigningsteamengines.

Thermodynamicsisencapsulatedinanumberofmoregenerallaws,notreliantonthepreciseformofthegaslaw.However,itdoesrequiretheretobesomesuchlaw,becausetemperature,pressure,andvolumearenotindependent.Therehastobesomerelationbetweenthem,butitdoesn’tgreatlymatterwhat.

Thefirstlawofthermodynamicsstemsfromthemechanicallawofconservationofenergy.InChapter3wesawthattherearetwodistinctkindsofenergyinclassicalmechanics:kineticenergy,determinedbymassandspeed,andpotentialenergy,determinedbytheeffectofforcessuchasgravity.Neitherofthesetypesofenergyisconservedonitsown.Ifyoudropaball,itspeedsup,therebygainingkineticenergy.Italsofalls,losingpotentialenergy.Newton’ssecondlawofmotionimpliesthatthesetwochangescanceleachotheroutexactly,sothetotalenergydoesnotchangeduringthemotion.

However,thisisnotthefullstory.Ifyouputabookonatableandgiveitapush,itspotentialenergydoesn’tchangeprovidedthetableishorizontal.Butitsspeeddoeschange:afteraninitialincreaseproducedbytheforcewithwhichyoupushedit,thebookquicklyslowsdownandcomestorest.Soitskineticenergystartsatanonzeroinitialvaluejustafterthepush,andthendropstozero.Thetotalenergythereforealsodecreases,soenergyisnotconserved.Wherehasitgone?Whydidthebookstop?AccordingtoNewton’sfirstlaw,thebookshouldcontinuetomove,unlesssomeforceopposesit.Thatforceisfrictionbetweenthebookandthetable.Butwhatisfriction?

Frictionoccurswhenroughsurfacesrubtogether.Theroughsurfaceofthebookhasbitsthatstickoutslightly.Thesecomeintocontactwithpartsofthetablethatalsostickoutslightly.Thebookpushesagainstthetable,andthetable,obeyingNewton’sthirdlaw,resists.Thiscreatesaforcethatopposesthemotionofthebook,soitslowsdownandlosesenergy.Sowheredoestheenergygo?Perhapsconservationsimplydoesnotapply.Alternatively,theenergyisstilllurkingsomewhere,unnoticed.Andthat’swhatthefirstlawofthermodynamics

tellsus:themissingenergyappearsasheat.Bothbookandtableheatupslightly.Humanshaveknownthatfrictioncreatesheatevensincesomebrightsparkdiscoveredhowtorubtwostickstogetherandstartafire.Ifyouslidedownaropetoofast,yourhandsgetropeburnsfromthefriction.Therewereplentyofclues.Thefirstlawofthermodynamicsstatesthatheatisaformofenergy,andenergy–thusextended–isconservedinthermodynamicprocesses.

Thefirstlawofthermodynamicsplaceslimitsonwhatyoucandowithaheatengine.Theamountofkineticenergythatyoucangetout,intheformofmotion,cannotbemorethantheamountofenergyyouputinasheat.Butitturnedoutthatthereisafurtherrestrictiononhowefficientlyaheatenginecanconvertheatenergyintokineticenergy;notjustthepracticalpointthatsomeoftheenergyalwaysgetslost,butatheoreticallimitthatpreventsalloftheheatenergybeingconvertedtomotion.Onlysomeofit,the‘free’energy,canbesoconverted.Thesecondlawofthermodynamicsturnedthisideaintoageneralprinciple,butitwilltakeawhilebeforewegettothat.ThelimitationwasdiscoveredbyNicolasLéonardSadiCarnotin1824,inasimplemodelofhowasteamengineworks:theCarnotcycle.

TounderstandtheCarnotcycleitisimportanttodistinguishbetweenheatandtemperature.Ineverydaylife,wesaythatsomethingishotifitstemperatureishigh,andsoconfusethetwoconcepts.Inclassicalthermodynamics,neitherconceptisstraightforward.Temperatureisapropertyofafluid,butheatmakessenseonlyasameasureofthetransferofenergybetweenfluids,andisnotanintrinsicpropertyofthestate(thatis,thetemperature,pressure,andvolume)ofthefluid.Inthekinetictheory,thetemperatureofafluidistheaveragekineticenergyofitsmolecules,andtheamountofheattransferredbetweenfluidsisthechangeinthetotalkineticenergyoftheirmolecules.Inasenseheatisabitlikepotentialenergy,whichisdefinedrelativetoanarbitraryreferenceheight;thisintroducesanarbitraryconstant,so‘the’potentialenergyofabodyisnotuniquelydefined.Butwhenthebodychangesheight,thedifferenceinpotentialenergiesisthesamewhateverreferenceheightisused,becausetheconstantcancelsout.Inshort,heatmeasureschanges,buttemperaturemeasuresstates.Thetwoarelinked:heattransferispossibleonlywhenthefluidsconcernedhavedifferenttemperatures,andthenitistransferredfromthehotteronetothecoolerone.ThisisoftencalledtheZerothlawofthermodynamicsbecauselogicallyitprecedesthefirstlaw,buthistoricallyitwasrecognisedlater.

Temperaturecanbemeasuredusingathermometer,whichexploitstheexpansionofafluid,suchasmercury,causedbyincreasedtemperature.Heat

canbemeasuredbyusingitsrelationtotemperature.Inastandardtestfluid,suchaswater,every1-degreeriseintemperatureof1gramoffluidcorrespondstoafixedincreaseintheheatcontent.Thisamountiscalledthespecificheatofthefluid,whichinwateris1caloriepergramperdegreeCelsius.Notethatheatincreaseisachange,notastate,asrequiredbythedefinitionofheat.

WecanvisualisetheCarnotcyclebythinkingofachambercontaininggas,withamovablepistonatoneend.Thecyclehasfoursteps:

1Heatthegassorapidlythatitstemperaturedoesn’tchange.Itexpands,performingworkonthepiston.

2Allowthegastoexpandfurther,reducingthepressure.Thegascools.

3Compressthegassorapidlythatitstemperaturedoesn’tchange.Thepistonnowperformsworkonthegas.

4Allowthegastoexpandfurther,increasingthepressure.Thegasreturnstoitsoriginaltemperature.

InaCarnotcycle,theheatintroducedinthefirststeptransferskineticenergytothepiston,allowingthepistontodowork.Thequantityofenergytransferredcanbecalculatedintermsoftheamountofheatintroducedandthetemperaturedifferencebetweenthegasanditssurroundings.Carnot’stheoremprovesthatinprincipleaCarnotcycleisthemostefficientwaytoconvertheatintowork.Thisplacesastringentlimitontheefficiencyofanyheatengine,andinparticularonasteamengine.

Inadiagramshowingthepressureandvolumeofthegas,aCarnotcyclelookslikeFigure47(left).TheGermanphysicistandmathematicianRudolfClausiusdiscoveredasimplerwaytovisualisethecycle,Figure47(right).Nowthetwoaxesaretemperatureandanewandfundamentalquantitycalledentropy.Inthesecoordinates,thecyclebecomesarectangle,andtheamountofworkperformedisjusttheareaoftherectangle.

Fig47Carnotcycle.Left:Intermsofpressureandvolume.Right:Intermsoftemperatureandentropy.

Entropyislikeheat:itisdefinedintermsofachangeofstate,notastateassuch.Supposethatafluidinsomeinitialstatechangestoanewstate.Thenthedifferenceinentropybetweenthetwostatesisthetotalchangeinthequantity‘heatdividedbytemperature’.Insymbols,forasmallstepalongthepathbetweenthetwostates,entropySisrelatedtoheatqandtemperatureTbythedifferentialequationdS=dq/T.Thechangeinentropyisthechangeinheatperunittemperature.Alargechangeofstatecanberepresentedasaseriesofsmallones,soweaddupallthesesmallchangesinentropytogettheoverallchangeofentropy.Calculustellsusthatthewaytodothisistouseanintegral.1

Havingdefinedentropy,thesecondlawofthermodynamicsisverysimple.Itstatesthatinanyphysicallyfeasiblethermodynamicprocess,theentropyofanisolatedsystemmustalwaysincrease.2Insymbols,dS≥0.Forexample,supposewedividearoomwithamovablepartition,putoxygenononesideofthepartitionandnitrogenontheother.Eachgashasaparticularentropy,relativetosomeinitialreferencestate.Nowremovethepartition,allowingthegasestomix.Thecombinedsystemalsohasaparticularentropy,relativetothesameinitialreferencestates.Andtheentropyofthecombinedsystemisalwaysgreaterthanthesumoftheentropiesofthetwoseparategases.

Classicalthermodynamicsisphenomenological:itdescribeswhatyoucanmeasure,butit’snotbasedonanycoherenttheoryoftheprocessesinvolved.Thatstepcamenextwiththekinetictheoryofgases,pioneeredbyDanielBernoulliin1738.Thistheoryprovidesaphysicalexplanationofpressure,temperature,thegaslaws,andthatmysteriousquantityentropy.Thebasicidea–highlycontroversialatthetime–isthatagasconsistsofalargenumberofidenticalmolecules,whichbouncearoundinspaceandoccasionallycollidewitheachother.Beingagasmeansthatthemoleculesarenottootightlypacked,soanygivenmoleculespendsalotofitstimetravellingthroughthevacuumofspaceataconstantspeedinastraightline.(Isay‘vacuum’eventhoughwe’rediscussingagas,becausethat’swhatthespacebetweenmoleculesconsistsof.)Sincemolecules,thoughtiny,havenonzerosize,occasionallytwoofthemwillcollide.Kinetictheorymakesthesimplifyingassumptionthattheybounceliketwocollidingbilliardballs,andthattheseballsareperfectlyelastic,sonoenergyislostinthecollision.Amongotherthings,thisimpliesthatthemoleculeskeepbouncingforever.

WhenBernoullifirstsuggestedthemodel,thelawofconservationofenergywasnotestablishedandperfectelasticityseemedunlikely.Thetheorygraduallywonsupportfromasmallnumberofscientists,whodevelopedtheirownversionsandaddedvariousnewideas,buttheirworkwasalmostuniversallyignored.TheGermanchemistandphysicistAugustKrönigwroteabookonthetopicin1856,simplifyingthephysicsbynotallowingthemoleculestorotate.Clausiusremovedthissimplificationayearlater.Heclaimedhehadarrivedathisresultsindependently,andisnowrankedasoneofthefirstsignificantfoundersofkinetictheory.Heproposedoneofthekeyconceptsofthetheory,themeanfreepathofamolecule:howfarittravels,onaverage,betweensuccessivecollisions.

BothKönigandClausiusdeducedtheidealgaslawfromkinetictheory.Thethreekeyvariablesarevolume,pressure,andtemperature.Volumeisdeterminedbythevesselthatcontainsthegas,itsets‘boundaryconditions’thataffecthowthegasbehaves,butisnotafeatureofthegasassuch.Pressureistheaverageforce(persquareunitofarea)exertedbythemoleculesofthegaswhentheycollidewiththewallsofthevessel.Thisdependsonhowmanymoleculesareinsidethevessel,andhowfasttheyaremoving.(Theydon’tallmoveatthesamespeed.)Mostinterestingistemperature.Thisalsodependsonhowfastthegasmoleculesaremoving,anditisproportionaltotheaveragekineticenergyofthemolecules.DeducingBoyle’slaw,thespecialcaseoftheidealgaslawforconstanttemperature,isespeciallystraightforward.Atafixedtemperature,thedistributionofvelocitiesdoesn’tchange,sopressureisdeterminedbyhowmanymoleculeshitthewall.Ifyoureducethevolume,thenumberofmoleculespercubicunitofspacegoesup,andthechanceofanymoleculehittingthewallgoesupaswell.Smallervolumemeansdensergasmeansmoremoleculeshittingthewall,andthisargumentcanbemadequantitative.Similarbutmorecomplicatedargumentsproducetheidealgaslawinallitsgloryaslongasthemoleculesaren’tsquashedtootightlytogether.SonowtherewasadeepertheoreticalbasisforBoyle’slaw,basedonthetheoryofmolecules.

MaxwellwasinspiredbyClausius’swork,andin1859heplacedkinetictheoryonmathematicalfoundationsbywritingdownaformulafortheprobabilitythatamoleculewilltravelwithagivenspeed.Itisbasedonthenormaldistributionorbellcurve(Chapter7).Maxwell’sformulaseemstohavebeenthefirstinstanceofaphysicallawbasedonprobabilities.HewasfollowedbytheAustrianphysicistLudwigBoltzmann,whodevelopedthesameformula,nowcalledtheMaxwell–Boltzmanndistribution.Boltzmannreinterpretedthermodynamicsintermsofthekinetictheoryofgases,foundingwhatisnow

calledstatisticalmechanics.Inparticular,hecameupwithanewinterpretationofentropy,relatingthethermodynamicconcepttoastatisticalfeatureofthemoleculesinthegas.

Thetraditionalthermodynamicquantities,suchastemperature,pressure,heat,andentropy,allrefertolarge-scaleaveragepropertiesofthegas.However,thefinestructureconsistsoflotsofmoleculeswhizzingaroundandbumpingintoeachother.Thesamelarge-scalestatecanarisefrominnumerabledifferentsmall-scalestates,becauseminordifferencesonthesmallscaleaverageout.Boltzmannthereforedistinguishedmacrostatesandmicrostatesofthesystem:large-scaleaveragesandtheactualstatesofthemolecules.Usingthis,heshowedthatentropy,amacrostate,canbeinterpretedasastatisticalfeatureofmicrostates.Heexpressedthisintheequation

S=klogW

HereSistheentropyofthesystem,Wisthenumberofdistinctmicrostatesthatcangiverisetotheoverallmacrostate,andkisaconstant.ItisnowcalledBoltzmann’sconstant,anditsvalueis1.38×10−23joulesperdegreekelvin.

Itisthisformulathatmotivatestheinterpretationofentropyasdisorder.Theideaisthatfewermicrostatescorrespondtoanorderedmacrostatethantoadisorderedone,andwecanunderstandwhybythinkingaboutapackofcards.Forsimplicity,supposethatwehavejustsixcards,marked2,3,4,J,Q,K.Putthemintwoseparatepiles,withthelow-valuecardsinonepileandthecourtcardsintheother.Thisisanorderedarrangement.Infact,itretainstracesoforderifyoushuffleeachpile,butkeepthepilesseparate,becausehoweveryoudothis,thelow-valuecardsareallinonepileandthecourtcardsareintheother.However,ifyoushufflebothpilestogether,thetwotypesofcardcanbecomemixed,witharrangementslike4QK2J3.Intuitively,thesemixed-uparrangementsaremoredisordered.

Let’sseehowthisrelatestoBoltzmann’sformula.Thereare36waystoarrangethecardsintheirtwopiles:sixforeachpile.Butthereare720ways(6!=1×2×3×4×5×6)toarrangeallsixcardsinorder.Thetypeoforderingofthecardsthatweallow–twopilesorone–isanalogoustothemacrostateofathermodynamicsystem.Theexactorderisthemicrostate.Themoreorderedmacrostatehas36microstates,thelessorderedonehas720.Sothemoremicrostatesthereare,thelessorderedthecorrespondingmacrostatebecomes.Sincelogarithmsgetbiggerwhenthenumbersdo,thegreaterthelogarithmofthenumberofmicrostates,themoredisorderedthemacrostatebecomes.Here

log36=3.58log720=6.58

Theseareeffectivelytheentropiesofthetwomacrostates.Boltzmann’sconstantjustscalesthevaluestofitthethermodynamicformalismwhenwe’redealingwithgases.

Thetwopilesofcardsareliketwonon-interactingthermodynamicstates,suchasaboxwithapartitionseparatingtwogases.Theirindividualentropiesareeachlog6,sothetotalentropyis2log6,whichequalslog36.Sothelogarithmmakesentropyadditivefornon-interactingsystems:togettheentropyofthecombined(butnotyetinteracting)system,addtheseparateentropies.Ifwenowletthesystemsinteract(removethepartition)theentropyincreasestolog720.

Themorecardsthereare,themorepronouncedthiseffectbecomes.Splitastandardpackof52playingcardsintotwopiles,withalltheredcardsinonepileandalltheblackcardsintheother.Thisarrangementcanoccurin(26!)2ways,whichisabout1.62×1053.Shufflingbothpilestogetherweget52!microstates,roughly8.07×1067.Thelogarithmsare122.52and156.36respectively,andagainthesecondislarger.

Boltzmann’sideaswerenotreceivedwithgreatacclaim.Atatechnicallevel,thermodynamicswasbesetwithdifficultconceptualissues.Onewastheprecisemeaningof‘microstate’.Thepositionandvelocityofamoleculearecontinuousvariables,abletotakeoninfinitelymanyvalues,butBoltzmannneededafinitenumberofmicrostatesinordertocounthowmanytherewereandthentakethelogarithm.Sothesevariableshadtobe‘coarse-grained’insomemanner,bysplittingthecontinuumofpossiblevaluesintofinitelymanyverysmallintervals.Anotherissue,morephilosophicalinnature,wasthearrowoftime–anapparentconflictbetweenthetime-reversibledynamicsofmicrostatesandtheone-waytimeofmacrostates,determinedbyentropyincrease.Thetwoissuesarerelated,aswewillshortlysee.

Thebiggestobstacletothetheory’sacceptance,however,wastheideathatmatterismadefromextremelytinyparticles,atoms.Thisconcept,andthewordatom,whichmeans‘indivisible’,goesbacktoancientGreece,butevenaround1900themajorityofphysicistsdidnotbelievethatmatterismadefromatoms.Sotheydidn’tbelieveinmolecules,either,andatheoryofgasesbasedonthemwasobviouslynonsense.Maxwell,Boltzmann,andotherpioneersofkinetictheorywereconvincedthatmoleculesandatomswerereal,buttotheskeptics,atomictheorywasjustaconvenientwaytopicturematter.Noatomshadever

beenobserved,sotherewasnoscientificevidencethattheyexisted.Molecules,specificcombinationsofatoms,weresimilarlycontroversial.Yes,atomictheoryfittedallsortsofexperimentaldatainchemistry,butthatwasnotproofthatatomsexisted.

OneofthethingsthatfinallyconvincedmostobjectorswastheuseofkinetictheorytomakepredictionsaboutBrownianmotion.ThiseffectwasdiscoveredbyaScottishbotanist,RobertBrown.3Hepioneeredtheuseofthemicroscope,discovering,amongotherthings,theexistenceofthenucleusofacell,nowknowntobetherepositoryofitsgeneticinformation.In1827Brownwaslookingthroughhismicroscopeatpollengrainsinafluid,andhespottedeventinierparticlesthathadbeenejectedbythepollen.Thesetinyparticlesjiggledaroundinarandommanner,andatfirstBrownwonderediftheyweresomediminutiveformoflife.However,hisexperimentsshowedthesameeffectinparticlesderivedfromnon-livingmatter,sowhatevercausedthejiggling,itdidn’thavetobealive.Atthetime,nooneknewwhatcausedthiseffect.Wenowknowthattheparticlesejectedbythepollenwereorganelles,tinysubsystemsofthecellwithspecificfunctions;inthiscase,tomanufacturestarchandfats.Andweinterprettheirrandomjigglesasevidenceforthetheorythatmatterismadefromatoms.

ThelinktoatomscomesfrommathematicalmodelsofBrownianmotion,whichfirstturnedupinstatisticalworkoftheDanishastronomerandactuaryThorvaldThielein1880.ThebigadvancewasmadebyEinsteinin1905andthePolishscientistMarianSmoluchowskiin1906.TheyindependentlyproposedaphysicalexplanationofBrownianmotion:atomsofthefluidinwhichtheparticleswerefloatingwererandomlybumpingintotheparticlesandgivingthemtinykicks.Onthisbasis,Einsteinusedamathematicalmodeltomakequantitativepredictionsaboutthestatisticsofthemotion,whichwereconfirmedbyJeanBaptistePerrinin1908–9.

Boltzmanncommittedsuicidein1906–justwhenthescientificworldwasstartingtoappreciatethatthebasisofhistheorywasreal.

InBoltzmann’sformulationofthermodynamics,moleculesinagasareanalogoustocardsinapack,andthenaturaldynamicsofthemoleculesisanalogoustoshuffling.Supposethatatsomemomentalltheoxygenmoleculesinaroomareconcentratedatoneend,andallthenitrogenmoleculesareattheother.Thisisanorderedthermodynamicstate,liketwoseparatepilesofcards.Afteraveryshortperiod,however,randomcollisionswillmixallthemoleculestogether,moreorlessuniformlythroughouttheroom,likeshufflingthecards.

We’vejustseenthatthisprocesstypicallycausesentropytoincrease.Thisistheorthodoxpictureoftherelentlessincreaseofentropy,anditisthestandardinterpretationofthesecondlaw:‘theamountofdisorderintheuniversesteadilyincreases’.I’mprettysurethatthischaracterisationofthesecondlawwouldhavesatisfiedSnowifanyonehadofferedit.Inthisform,onedramaticconsequenceofthesecondlawisthescenarioofthe‘heatdeathoftheuniverse’,inwhichtheentireuniversewilleventuallybecomealukewarmgaswithnointerestingstructurewhatsoever.

Entropy,andthemathematicalformalismthatgoeswithit,providesanexcellentmodelformanythings.Itexplainswhyheatenginescanonlyreachaparticularlevelofefficiency,whichpreventsengineerswastingvaluabletimeandmoneylookingforamare’snest.That’snotjusttrueofVictoriansteamengines,itappliestomoderncarenginesaswell.Enginedesignisoneofthepracticalareasthathasbenefitedfromknowingthelawsofthermodynamics.Refrigeratorsareanother.Theyusechemicalreactionstotransferheatoutofthefoodinthefridge.Ithastogosomewhere:youcanoftenfeeltheheatrisingfromtheoutsideofthefridge’smotorhousing.Thesamegoesforair-conditioning.Powergenerationisanotherapplication.Inacoal,gas,ornuclearpowerstation,whatitinitiallygeneratedisheat.Theheatcreatessteam,whichdrivesaturbine.Theturbine,followingprinciplesthatgobacktoFaraday,turnsmotionintoelectricity.

Thesecondlawofthermodynamicsalsogovernstheamountofenergywecanhopetoextractfromrenewableresources,suchaswindandwaves.Climatechangehasaddednewurgencytothisquestion,becauserenewableenergysourcesproducelesscarbondioxidethanconventionalones.Evennuclearpowerhasabigcarbonfootprint,becausethefuelhastobemade,transported,andstoredwhenitisnolongerusefulbutstillradioactive.AsIwritethereisasimmeringdebateaboutthemaximumamountofenergythatwecanextractfromtheoceanandtheatmospherewithoutcausingthekindsofchangethatwearehopingtoavoid.Itisbasedonthermodynamicestimatesoftheamountoffreeenergyinthosenaturalsystems.Thisisanimportantissue:ifrenewablesinprinciplecannotsupplytheenergyweneed,thenwehavetolookelsewhere.Solarpanels,whichextractenergydirectlyfromsunlight,arenotdirectlyaffectedbythethermodynamiclimits,buteventhoseinvolvemanufacturingprocessesandsoon.Atthemoment,thecasethatsuchlimitsareaseriousobstaclereliesonsomesweepingsimplifications,andeveniftheyarecorrect,thecalculationsdonotruleoutrenewablesasasourceformostoftheworld’spower.Butit’sworthrememberingthatsimilarlybroadcalculationsabout

carbondioxideproduction,performedinthe1950s,haveprovedsurprisinglyaccurateasapredictorofglobalwarming.

Thesecondlawworksbrilliantlyinitsoriginalcontext,thebehaviourofgases,butitseemstoconflictwiththerichcomplexitiesofourplanet,inparticular,life.Itseemstoruleoutthecomplexityandorganisationexhibitedbylivingsystems.SothesecondlawissometimesinvokedtoattackDarwinianevolution.However,thephysicsofsteamenginesisnotparticularlyappropriatetothestudyoflife.Inthekinetictheoryofgases,theforcesthatactbetweenthemoleculesareshort-range(activeonlywhenthemoleculescollide)andrepulsive(theybounce).Butmostoftheforcesofnaturearen’tlikethat.Forexample,gravityactsatenormousdistances,anditisattractive.TheexpansionoftheuniverseawayfromtheBigBanghasnotsmearedmatteroutintoauniformgas.Instead,thematterhasformedintoclumps–planets,stars,galaxies,supergalacticclusters…Theforcesthatholdmoleculestogetherarealsoattractive–exceptatveryshortdistanceswheretheybecomerepulsive,whichstopsthemoleculecollapsing–buttheireffectiverangeisfairlyshort.Forsystemssuchasthese,thethermodynamicmodelofindependentsubsystemswhoseinteractionsswitchonbutnotoffissimplyirrelevant.Thefeaturesofthermodynamicseitherdon’tapply,oraresolong-termthattheydon’tmodelanythinginteresting.

Thelawsofthermodynamics,then,underliemanythingsthatwetakeforgranted.Andtheinterpretationofentropyas‘disorder’helpsustounderstandthoselawsandgainanintuitivefeelingfortheirphysicalbasis.However,thereareoccasionswheninterpretingentropyasdisorderseemstoleadtoparadoxes.Thisisamorephilosophicalrealmofdiscourse–andit’sfascinating.

Oneofthedeepmysteriesofphysicsistime’sarrow.Timeseemstoflowinoneparticulardirection.However,itseemslogicallyandmathematicallypossiblefortimetoflowbackwardsinstead–apossibilityexploitedbybookssuchasMartinAmis’sTime’sArrow,themuchearliernovelCounter-ClockWorldbyPhilipK.Dick,andtheBBCtelevisionseriesRedDwarf,whoseprotagonistsmemorablydrankbeerandengagedinabarbrawlinreversetime.Sowhycan’ttimeflowtheotherway?Atfirstsight,thermodynamicsoffersasimpleexplanationforthearrowoftime:itisthedirectionofentropyincrease.Thermodynamicprocessesareirreversible:oxygenandnitrogenwillspontaneouslymix,butnotspontaneouslyunmix.

Thereisapuzzlehere,however,becauseanyclassicalmechanicalsystem,suchasthemoleculesinaroom,istime-reversible.Ifyoukeepshufflingapack

ofcardsatrandom,theneventuallyitwillgetbacktoitsoriginalorder.Inthemathematicalequations,ifatsomeinstantthevelocitiesofallparticlesaresimultaneouslyreversed,thenthesystemwillretraceitssteps,back-to-frontintime.Theentireuniversecanbounce,obeyingthesameequationsinbothdirections.Sowhydoweneverseeaneggunscrambling?

Theusualthermodynamicansweris:ascrambledeggismoredisorderedthananunscrambledone,entropyincreases,andthat’sthewaytimeflows.Butthere’sasubtlerreasonwhyeggsdon’tunscramble:theuniverseisvery,veryunlikelytobounceintherequiredmanner.Theprobabilityofthathappeningisridiculouslysmall.Sothediscrepancybetweenentropyincreaseandtime–reversibilitycomesfromtheinitialconditions,nottheequations.Theequationsformovingmoleculesaretime-reversible,buttheinitialconditionsarenot.Whenwereversetime,wemustuse‘initial’conditionsgivenbythefinalstateoftheforward-timemotion.

Themostimportantdistinctionhereisbetweensymmetryofequationsandsymmetryoftheirsolutions.Theequationsforbouncingmoleculeshavetime-reversalsymmetry,butindividualsolutionscanhaveadefinitearrowoftime.Themostyoucandeduceaboutasolution,fromtime-reversibilityoftheequation,isthattheremustalsoexistanothersolutionthatisthetime-reversalofthefirst.IfAlicethrowsaballtoBob,thetime-reversedsolutionhasBobthrowingaballtoAlice.Similarly,sincetheequationsofmechanicsallowavasetofalltothegroundandsmashintoathousandpieces,theymustalsoallowasolutioninwhichathousandshardsofglassmysteriouslymovetogether,assemblethemselvesintoanintactvase,andleapintotheair.

There’sclearlysomethingfunnygoingonhere,anditrepaysinvestigation.Wedon’thaveaproblemwithBobandAlicetossingaballeitherway.Weseesuchthingseveryday.Butwedon’tseeasmashedvaseputtingitselfbacktogether.Wedon’tseeaneggunscrambling.

Supposewesmashavaseandfilmtheresult.Westartwithasimple,orderedstate–anintactvase.Itfallstothefloor,wheretheimpactbreaksitintopiecesandpropelsthosepiecesalloverthefloor.Theyslowdownandcometoahalt.Italllooksentirelynormal.Nowplaythemoviebackwards.Bitsofglass,whichjusthappentobetherightshapetofittogether,arelyingonthefloor.Spontaneously,theystarttomove.Theymoveatjusttherightspeed,andinjusttherightdirection,tomeet.Theyassembleintoavase,whichheadsskywards.Thatdoesn’tseemright.

Infact,asdescribed,it’snotright.Severallawsofmechanicsappeartobe

violated,amongthemconservationofmomentumandconservationofenergy.Stationarymassescan’tsuddenlymove.Avasecan’tgainenergyfromnowhereandleapintotheair.

Ah,yes…butthat’sbecausewe’renotlookingcarefullyenough.Thevasedidn’tleapintotheairofitsownaccord.Thefloorstartedtovibrate,andthevibrationscametogethertogivethevaseasharpkickintotheair.Thebitsofglassweresimilarlyimpelledtomovebyincomingwavesofvibrationofthefloor.Ifwetracethosevibrationsback,theyspreadout,andseemtodiedown.Eventuallyfrictiondissipatesallmovement…Oh,yes,friction.Whathappenstokineticenergywhenthere’sfriction?Itturnsintoheat.Sowe’vemissedsomedetailsofthetime-reversedscenario.Momentumandenergydobalance,butthemissingamountscomefromthefloorlosingheat.

Inprinciple,wecouldsetupaforward-timesystemtomimicthetime-reversedvase.Wejusthavetoarrangeformoleculesinthefloortocollideinjusttherightwaytoreleasesomeoftheirheatasmotionofthefloor,kickthepiecesofglassinjusttherightway,thenhurlthevaseintotheair.Thepointisnotthatthisisimpossibleinprinciple:ifitwere,time-reversibilitywouldfail.Butit’simpossibleinpractice,becausethereisnowaytocontrolthatmanymoleculesthatprecisely.

This,too,isanissueaboutboundaryconditions–inthiscase,initialconditions.Theinitialconditionsforthevase-smashingexperimentareeasytoimplement,andtheapparatusiseasytoacquire.It’sallveryrobust,too:useanothervase,dropitfromadifferentheight…muchthesamewillhappen.Thevase-assemblingexperiment,incontrast,requiresextraordinarilyprecisecontrolofgazillionsofindividualmoleculesandexquisitelycarefullymadepiecesofglass.Withoutallthatcontrolequipmentdisturbingasinglemolecule.That’swhywecan’tactuallydoit.

However,noticehowwe’rethinkinghere:we’refocusingoninitialconditions.Thatsetsupanarrowoftime:therestoftheactioncomeslaterthanthestart.Ifwelookedatthevase-smashingexperiment’sfinalconditions,rightdowntothemolecularlevel,theywouldbesocomplexthatnooneintheirrightmindwouldevenconsidertryingtoreplicatethem.

Themathematicsofentropyfudgesouttheseverysmallscaleconsiderations.Itallowsvibrationstodieawaybutnottoincrease.Itallowsfrictiontoturnintoheatbutnotheattoturnintofriction.Thediscrepancybetweenthesecondlawofthermodynamicsandmicroscopicreversibilityarisesfromcoarse-graining,themodellingassumptionsmadewhenpassingfromadetailedmolecular

descriptiontoastatisticalone.Theseassumptionsimplicitlyspecifyanarrowoftime:large-scaledisturbancesareallowedtodiedownbelowtheperceptiblelevelastimepasses,butsmall-scaledisturbancesarenotallowedtofollowthetime-reversedscenario.Oncethedynamicspassesthroughthistemporaltrapdoor,it’snotallowedtocomeback.

Ifentropyalwaysincreases,howdidthechickenevercreatetheorderedeggtobeginwith?Acommonexplanation,advancedbytheAustrianphysicistErwinSchrödingerin1944inabriefandcharmingbookWhatisLife?,isthatlivingsystemssomehowborroworderfromtheirenvironment,andpayitbackbymakingtheenvironmentevenmoredisorderedthanitwouldotherwisehavebeen.Thisextraordercorrespondsto‘negativeentropy’,whichthechickencanusetomakeaneggwithoutviolatingthesecondlaw.InChapter15wewillseethatnegativeentropycan,inappropriatecircumstances,bethoughtofasinformation,anditisoftenclaimedthatthechickenaccessesinformation–providedbyitsDNA,forexample–toobtainthenecessarynegativeentropy.However,theidentificationofinformationwithnegativeentropymakessenseonlyinveryspecificcontexts,andtheactivitiesoflivingcreaturesarenotoneofthem.Organismscreateorderthroughtheprocessesthattheycarryout,butthoseprocessesarenotthermodynamic.Chickensdon’taccesssomestorehouseofordertomakethethermodynamicbooksbalance:theyuseprocessesforwhichathermodynamicmodelisinappropriate,andthrowthebooksawaybecausetheydon’tapply.

Thescenarioinwhichaneggiscreatedbyborrowingentropywouldbeappropriateiftheprocessthatthechickenusedwerethetime-reversalofaneggbreakingupintoitsconstituentmolecules.Atfirstsightthisisvaguelyplausible,becausethemoleculesthateventuallyformtheeggarescatteredthroughouttheenvironment;theycometogetherinthechicken,wherebiochemicalprocessesputthemtogetherinanorderedmannertoformtheegg.However,thereisadifferenceintheinitialconditions.Ifyouwentroundbeforehandlabellingmoleculesinthechicken’senvironment,tosay‘thisonewillendupintheeggatsuchandsuchalocation’,youwouldineffectbecreatinginitialconditionsascomplexandunlikelyasthoseforunscramblinganegg.Butthat’snothowthechickenoperates.Somemoleculeshappentoendupintheeggandareconceptuallylabelledaspartofitaftertheprocessiscomplete.Othermoleculescouldhavedonethesamejob–onemoleculeofcalciumcarbonateisjustasgoodformakingashellasanyother.Sothechickenisnotcreatingorderfromdisorder.Theorderisassignedtotheendresultoftheegg-makingprocess–like

shufflingapackofcardsintoarandomorderandthennumberingthem1,2,3,andsoonwithafelt-tippedpen.Amazing–they’reinnumericalorder!

Tobesure,theegglooksmoreorderedthanitsingredients,evenifwetakeaccountofthisdifferenceininitialconditions.Butthat’sbecausetheprocessthatmakesaneggisnotthermodynamic.Manyphysicalprocessesdo,ineffect,unscrambleeggs.Anexampleisthewaymineralsdissolvedinwatercancreatestalactitesandstalagmitesincaves.Ifwespecifiedtheexactformofstalactitewewanted,aheadoftime,we’dbeinthesamepositionassomeonetryingtounsmashavase.Butifwe’rewillingtosettleforanyoldstalactite,wegetone:orderfromdisorder.Thosetwotermsareoftenusedinasloppyway.Whatmattersarewhatkindoforderandwhatkindofdisorder.Thatsaid,Istilldon’texpecttoseeaneggunscrambling.Thereisnofeasiblewaytosetupthenecessaryinitialconditions.Thebestwecandoisturnthescrambledeggintochickenfeedandwaitforthebirdtolayanewone.

Infact,thereisareasonwhywewouldn’tseeaneggunscrambling,eveniftheworlddidrunbackwards.Becauseweandourmemoriesarepartofthesystemthatisbeingreversed,wewouldn’tbesurewhichwaytimewas‘really’running.Oursenseoftheflowoftimeisproducedbymemories,physico-chemicalpatternsinthebrain.Inconventionallanguage,thebrainstoresrecordsofthepastbutnotofthefuture.Imaginemakingaseriesofsnapshotsofthebrainwatchinganeggbeingscrambled,alongwithitsmemoriesoftheprocess.Atonestagethebrainremembersacold,unscrambledegg,andsomeofitshistorywhentakenfromthefridgeandputintothesaucepan.Atanotherstageitremembershavingwhiskedtheeggwithafork,andhavingmoveditfromthefridgetothesaucepan.

Ifwenowruntheentireuniverseinreverse,wereversetheorderinwhichthosememoriesoccur,in‘real’time.Butwedon’treversetheorderingofagivenmemoryinthebrain.Atthestart(inreversedtime)oftheprocessthatunscramblestheegg,thebraindoesnotrememberthe‘past’ofthategg–howitemergedfromamouthontoaspoon,wasunwhisked,graduallybuildingupacompleteegg…Instead,therecordinthebrainatthatmomentisoneinwhichitremembershavingcrackedopenanegg,alongwiththeprocessofmovingitfromthefridgetothesaucepanandscramblingit.Butthismemoryisexactlythesameasoneoftherecordsintheforward-timescenario.Thesamegoesforalltheothermemorysnapshots.Ourperceptionoftheworlddependsonwhatweobservenow,andwhatmemoriesourbrainholds,now.Inatime-reverseduniverse,wewouldineffectrememberthefuture,notthepast.

Theparadoxesoftime-reversibilityandentropyarenotproblemsabouttherealworld.Theyareproblemsabouttheassumptionswemakewhenwetrytomodelit.

13Onethingisabsolute

Relativity

Whatdoesitsay?

Mattercontainsenergyequaltoitsmassmultipliedbythesquareofthespeedoflight.

Whyisthatimportant?

Thespeedoflightishugeanditssquareisabsolutelyhumongous.Onekilogramofmatterwouldreleaseabout40%oftheenergyinthelargestnuclearweaponeverexploded.It’spartofapackageofequationsthatchangedourviewofspace,time,matter,andgravity.

Whatdiditleadto?

Radicalnewphysics,definitely.Nuclearweapons…well,justmaybe–thoughnotasdirectlyorconclusivelyastheurbanmythsclaim.Blackholes,theBigBang,GPSandsatnav.

JustasAlbertEinstein,withhisstartledmophairdo,isthearchetypalscientistinpopularculture,sohisequationE=mc2isthearchetypalequation.Itiswidelybelievedthattheequationledtotheinventionofnuclearweapons,thatitcomesfromEinstein’stheoryofrelativity,andthatthistheory(obviously)hassomethingtodowithvariousthingsbeingrelative.Infact,manysocialrelativistshappilychant‘everythingisrelative’,andthinkithassomethingtodowithEinstein.

Itdoesn’t.Einsteinnamedhistheory‘relativity’becauseitwasamodificationoftherulesforrelativemotionthathadtraditionallybeenusedinNewtonianmechanics,wheremotionisrelative,dependinginaverysimpleandintuitivewayontheframeofreferenceinwhichitisobserved.EinsteinhadtotweakNewtonianrelativitytomakesenseofabafflingexperimentaldiscovery:thatoneparticularphysicalphenomenonisnotrelativeatall,butabsolute.Fromthishederivedanewkindofphysicsinwhichobjectsshrinkwhentheymoveveryfast,timeslowstoacrawl,andmassincreaseswithoutlimit.Anextensionincorporatinggravityhasgivenusthebestunderstandingweyethaveoftheoriginsoftheuniverseandthestructureofthecosmos.Itisbasedontheideathatspaceandtimecanbecurved.

Relativityisreal.TheGlobalPositioningSystem(GPS,usedamongotherthingsforcarsatnav)worksonlywhencorrectionsaremadeforrelativisticeffects.ThesamegoesforparticleacceleratorssuchastheLargeHadronCollider,currentlysearchingfortheHiggsboson,thoughttobetheoriginofmass.Moderncommunicationshavebecomesofastthatmarkettradersarebeginningtorunupagainstarelativisticlimitation:thespeedoflight.Thisisthefastestthatanymessage,suchasanInternetinstructiontobuyorsellstock,cantravel.Someseethisasanopportunitytocutadealnanosecondsearlierthanthecompetition,butsofar,relativisticeffectshaven’thadaseriouseffectoninternationalfinance.However,peoplehavealreadyworkedoutthebestlocationsfornewstockmarketsordealerships.It’sonlyamatteroftime.

Atanyrate,notonlyisrelativitynotrelative:eventheiconicequationisnotwhatitseems.WhenEinsteinfirstderivedthephysicalideathatitrepresents,hedidn’twriteitinthefamiliarway.Itisnotamathematicalconsequenceofrelativity,thoughitbecomesoneifvariousphysicalassumptionsanddefinitionsareaccepted.Itisperhapstypicalofhumanculturethatourmosticonicequationisnot,andwasnot,whatitseemstobe,andneitheristhetheorythatgavebirth

toit.Eventheconnectionwithnuclearweaponsisnotclear-cut,anditshistoricalinfluenceonthefirstatomicbombwassmallcomparedwithEinstein’spoliticalcloutastheiconicscientist.

‘Relativity’coverstwodistinctbutrelatedtheories:specialrelativityandgeneralrelativity.I’lluseEinstein’scelebratedequationasanexcusetotalkaboutboth.Specialrelativityisaboutspace,time,andmatterintheabsenceofgravity;generalrelativitytakesgravityintoaccountaswell.Thetwotheoriesarepartofonebigpicture,butittookEinsteintenyearsofintensiveefforttodiscoverhowtomodifyspecialrelativitytoincorporategravity.BoththeorieswereinspiredbydifficultiesinreconcilingNewtonianphysicswithobservations,buttheiconicformulaaroseinspecialrelativity.

PhysicsseemedfairlystraightforwardandintuitiveinNewton’sday.Spacewasspace,timewastime,andneverthetwainshouldmeet.ThegeometryofspacewasthatofEuclid.Timewasindependentofspace,thesameforallobservers–providedtheyhadsynchronisedtheirclocks.Themassandsizeofabodydidnotchangewhenitmoved,andtimealwayspassedatthesamerateeverywhere.ButwhenEinsteinhadfinishedreformulatingphysics,allofthesestatements–sointuitivethatitisverydifficulttoimaginehowanyofthemcouldfailtorepresentreality–turnedouttobewrong.

Theywerenottotallywrong,ofcourse.Iftheyhadbeennonsense,thenNewton’sworkwouldneverhavegotofftheground.TheNewtonianpictureofthephysicaluniverseisanapproximation,notanexactdescription.Theapproximationisextremelyaccurateprovidedeverythinginvolvedismovingslowlyenough,andinmosteverydaycircumstancesthatisthecase.Evenajetfighter,travellingattwicethespeedofsound,ismovingslowlyforthispurpose.Butonethingthatdoesplayaroleineverydaylifemovesveryfastindeed,andsetstheyardstickforallotherspeeds:light.Newtonandhissuccessorshaddemonstratedthatlightwasawave,andMaxwell’sequationsconfirmedthis.Butthewavenatureoflightraisedanewissue.Oceanwavesarewavesinwater,soundwavesarewavesinair,earthquakesarewavesintheEarth.Solightwavesarewavesin…what?

Mathematicallytheyarewavesintheelectromagneticfield,whichisassumedtopervadethewholeofspace.Whentheelectromagneticfieldisexcited–persuadedtosupportelectricityandmagnetism–weobserveawave.Butwhathappenswhenit’snotexcited?Withoutwaves,anoceanwouldstillbeanocean,airwouldstillbeair,andtheEarthwouldstillbetheEarth.Analogously,theelectromagneticfieldwouldstillbe…theelectromagneticfield.Butyoucan’t

observetheelectromagneticfieldifthere’snoelectricityormagnetismgoingon.Ifyoucan’tobserveit,whatisit?Doesitexistatall?

Allwavesknowntophysics,excepttheelectromagneticfield,arewavesinsomethingtangible.Allthreetypesofwave–water,air,earthquake–arewavesofmovement.Themediummovesupanddownorfromsidetoside,butusuallyitdoesn’ttravelwiththewave.(Tiealongropetoawallandshakeoneend:awavetravelsalongtherope.Buttheropedoesn’ttravelalongtherope.)Thereareexceptions:whenairtravelsalongwiththewavewecallit‘wind’,andoceanwavesmovewaterupabeachwhentheyhitone.Buteventhoughwedescribeatsunamiasamovingwallofwater,itdoesn’trollacrossthetopoftheoceanlikeafootballrollingalongthepitch.Mostly,thewaterinanygivenlocationgoesupanddown.Itisthelocationofthe‘up’thatmoves.Untilthewatergetsclosetoshore;thenyougetsomethingmuchmorelikeamovingwall.

Light,andelectromagneticwavesingeneral,didn’tseemtobewavesinanythingtangible.InMaxwell’sday,andforfiftyyearsormoreafterwards,thatwasdisturbing.Newton’slawofgravityhadlongbeencriticisedbecauseitimpliesthatgravitysomehow‘actsatadistance’,asmiraculousinphilosophicalprincipleaskickingaballintothegoalwhenyou’resittinginthestands.Sayingthatitistransmittedby‘thegravitationalfield’doesn’treallyexplainwhat’shappening.Thesamegoesforelectromagnetism.Sophysicistscameroundtotheideathattherewassomemedium–nooneknewwhat,theycalleditthe‘luminiferousaether’orjustplain‘ether’–thatsupportedelectromagneticwaves.Vibrationstravelfasterthemorerigidthemedium,andlightwasveryfastindeed,sotheetherhadtobeextremelyrigid.Yetplanetscouldmovethroughitwithoutresistance.Tohaveavoidedeasydetection,theethermusthavenomass,noviscosity,beincompressible,andbetotallytransparenttoallformsofradiation.

Itwasadauntingcombinationofattributes,butalmostallphysicistsassumedtheetherexisted,becauselightclearlydidwhatlightdid.Somethinghadtocarrythewave.Moreover,theether’sexistencecouldinprinciplebedetected,becauseanotherfeatureoflightsuggestedawaytoobserveit.Inavacuum,lightmoveswithafixedspeedc.Newtonianmechanicshadtaughteveryphysicisttoask:speedrelativetowhat?Ifyoumeasureavelocityintwodifferentframesofreference,onemovingwithrespecttotheother,yougetdifferentanswers.Theconstancyofthespeedoflightsuggestedanobviousreply:relativetotheether.Butthiswasalittlefacile,becausetwoframesofreferencethataremovingwithrespecttoeachothercan’tbothbeatrestrelativetotheether.

AstheEarthploughsitswaythroughtheether,miraculouslyunresisted,itgoesroundandroundtheSun.Atoppositepointsonitsorbititismovinginoppositedirections.SobyNewtonianmechanics,thespeedoflightshouldvarybetweentwoextremes:cplusacontributionfromtheEarth’smotionrelativetotheether,andcminusthesamecontribution.Measurethespeed,measureitsixmonthslater,findthedifference:ifthereisone,youhaveproofthattheetherexists.Inthelate1800smanyexperimentsalongtheselineswerecarriedout,buttheresultswereinconclusive.Eithertherewasnodifference,ortherewasonebuttheexperimentalmethodwasn’taccurateenough.Worse,theEarthmightbedraggingtheetheralongwithit.ThiswouldsimultaneouslyexplainwhytheEarthcanmovethroughsucharigidmediumwithoutresistance,andimplythatyououghtnottoseeanydifferenceinthespeedoflightanyway.TheEarth’smotionrelativetotheetherwouldalwaysbezero.

In1887AlbertMichelsonandEdwardMorleycarriedoutoneofthemostfamousphysicsexperimentsofalltime.Theirapparatuswasdesignedtodetectextremelysmallvariationsinthespeedoflightintwodirections,atrightanglestoeachother.HowevertheEarthwasmovingrelativetotheether,itcouldn’tbemovingwiththesamerelativespeedintwodifferentdirections…unlessithappenedbycoincidencetobemovingalongthelinebisectingthosedirections,inwhichcaseyoujustrotatedtheapparatusalittleandtriedagain.

Theapparatus,Figure48,wassmallenoughtofitonalaboratorydesk.Itusedahalf-silveredmirrortosplitabeamoflightintotwoparts,onepassingthroughthemirrorandtheotherbeingreflectedatrightangles.Eachseparatebeamwasreflectedbackalongitspath,andthetwobeamscombinedagain,tohitadetector.Theapparatuswasadjustedtomakethepathsthesamelength.Theoriginalbeamwassetuptobecoherent,meaningthatitswaveswereinsynchronywitheachother–allhavingthesamephase,peakscoincidingwithpeaks.Anydifferencebetweenthespeedoflightinthedirectionsfollowedbythetwobeamswouldcausetheirphasestoshiftrelativetoeachother,sotheirpeakswouldbeindifferentplaces.Thiswouldcauseinterferencebetweenthetwowaves,resultinginastripedpatternof‘diffractionfringes’.MotionoftheEarthrelativetotheetherwouldcausethefringestomove.Theeffectwouldbetiny:givenwhatwasknownabouttheEarth’smotionrelativetotheSun,thediffractionfringeswouldshiftbyabout4%ofthewidthofonefringe.Byusingmultiplereflections,thiscouldbeincreasedto40%,bigenoughtobedetected.ToavoidthepossiblecoincidenceoftheEarthmovingexactlyalongthebisectorofthetwobeams,MichelsonandMorleyfloatedtheapparatusonabathofmercury,sothatitcouldbespunroundeasilyandrapidly.Itshouldthenbe

possibletowatchthefringesshiftingwithequalrapidity.

Fig48TheMichelson–Morleyexperiment.

Itwasacareful,accurateexperiment.Itsresultwasentirelynegative.Thefringesdidnotmoveby40%oftheirwidth.Asfarasanyonecouldtellwithcertainty,theydidn’tmoveatall.Laterexperiments,capableofdetectingashift0.07%aswideasafringe,alsogaveanegativeresult.Theetherdidnotexist.

Thisresultdidn’tjustdisposeoftheether:itthreatenedtodisposeofMaxwell’stheoryofelectromagnetism,too.ItimpliedthatlightdoesnotbehaveinaNewtonianmanner,relativetomovingframesofreference.ThisproblemcanbetracedrightbacktothemathematicalpropertiesofMaxwell’sequationsandhowtheytransformrelativetoamovingframe.TheIrishphysicistandchemistGeorgeFitzGeraldandtheDutchphysicistHendrikLorenzindependentlysuggested(in1892and1895respectively)anaudaciouswaytogetroundtheproblem.Ifamovingbodycontractsslightlyinitsdirectionofmotion,byjusttherightamount,thenthechangeinphasethattheMichelson–Morleyexperimentwashopingtodetectwouldbeexactlycancelledoutbythechangeinthelengthofthepaththatthelightwasfollowing.Lorenzshowedthatthis‘Lorenz–FitzGeraldcontraction’sortedoutthemathematicaldifficultieswiththeMaxwellequationsaswell.Thejointdiscoveryshowedthattheresultsofexperimentsonelectromagnetism,includinglight,donotdependontherelativemotionofthereferenceframe.Poincaré,whohadalsobeenworking

alongsimilarlines,addedhispersuasiveintellectualweighttotheidea.

ThestagewasnowsetforEinstein.In1905hedevelopedandextendedpreviousspeculationsaboutanewtheoryofrelativemotioninapaper‘Ontheelectrodynamicsofmovingbodies’.Hisworkwentbeyondthatofhispredecessorsintwoways.Heshowedthatthenecessarychangetothemathematicalformulationofrelativemotionwasmorethanjustatricktosortoutelectromagnetism.Itwasrequiredforallphysicallaws.Itfollowedthatthenewmathematicsmustbeagenuinedescriptionofreality,withthesamephilosophicalstatusthathadbeenaccordedtotheprevailingNewtoniandescription,butprovidingabetteragreementwithexperiments.Itwasrealphysics.

TheviewofrelativemotionemployedbyNewtonwentbackevenfurther,toGalileo.Inhis1632Dialogosopraiduemassimisistemidelmondo(‘DialogueConcerningtheTwoChiefWorldSystems’)Galileodiscussedashiptravellingatconstantvelocityonaperfectlysmoothsea,arguingthatnoexperimentinmechanicscarriedoutbelowdeckscouldrevealthattheshipwasmoving.ThisisGalileo’sprincipleofrelativity:inmechanics,thereisnodifferencebetweenobservationsmadeintwoframesthataremovingwithuniformvelocityrelativetoeachother.Inparticular,thereisnospecialframeofreferencethatis‘atrest’.Einstein’sstarting-pointwasthesameprinciple,butwithanextratwist:itmustapplynotjusttomechanics,buttoallphysicallaws.Amongthem,ofcourse,beingMaxwell’sequationsandtheconstancyofthespeedoflight.

ToEinstein,theMichelson–Morleyexperimentwasasmallpieceofextraevidence,butitwasn’tproofofthepudding.Theproofthathisnewtheorywascorrectlayinhisextendedprincipleofrelativity,andwhatitimpliedforthemathematicalstructureofthelawsofphysics.Ifyouacceptedtheprinciple,allelsefollowed.Thisiswhythetheorybecameknownas‘relativity’.Notbecause‘everythingisrelative’,butbecauseyouhavetotakeintoaccountthemannerinwhicheverythingisrelative.Andit’snotwhatyouexpect.

ThisversionofEinstein’stheoryisknownasspecialrelativitybecauseitappliesonlytoframesofreferencethataremovinguniformlywithrespecttoeachother.AmongitsconsequencesaretheLorenz–FitzGeraldcontraction,nowinterpretedasanecessaryfeatureofspace-time.Infact,therewerethreerelatedeffects.Ifoneframeofreferenceismovinguniformlyrelativetoanotherone,thenlengthsmeasuredinthatframecontractalongthedirectionofmotion,massesincrease,andtimerunsmoreslowly.Thesethreeeffectsaretiedtogetherbythebasicconservationlawsofenergyandmomentum;onceyouacceptone

ofthem,theothersarelogicalconsequences.Thetechnicalformulationoftheseeffectsisaformulathatdescribeshow

measurementsinoneframerelatetothoseintheother.Theexecutivesummaryis:ifabodycouldmoveclosetothespeedoflight,thenitslengthwouldbecomeverysmall,timewouldslowtoacrawl,anditsmasswouldbecomeverylarge.I’lljustgiveaflavourofthemathematics:thephysicaldescriptionshouldnotbetakentooliterallyanditwouldtaketoolongtosetitupinthecorrectlanguage.Itallcomesfrom…Pythagoras’stheorem.Oneoftheoldestequationsinscienceleadstooneofthenewest.

Supposethataspaceshipispassingoverheadwithvelocityv,andthecrewperformsanexperiment.Theysendapulseoflightfromthefloorofthecabintotheroof,andmeasurethetimetakentobeT.Meanwhileanobserveronthegroundwatchestheexperimentthroughatelescope(assumethespaceshipistransparent),measuringthetimetobet.

Fig49Left:Theexperimentinthecrew’sframeofreference.Right:Thesameexperimentinthegroundobserver’sframeofreference.Greyshowstheship’spositionasseenfromthegroundwhenthelightbeamstartsitsjourney;blackshowstheship’spositionwhenthelightcompletesitsjourney.

Figure49(left)showsthegeometryoftheexperimentfromthecrew’spointofview.Tothem,thelighthasgonestraightup.Sincelighttravelsatspeedc,thedistancetravellediscT,shownbythedottedarrow.Figure49(right)showsthegeometryoftheexperimentfromthegroundobserver’spointofview.Thespaceshiphasmovedadistancevt,sothelighthastravelleddiagonally.Sincelightalsotravelsatspeedcforthegroundobserver,thediagonalhaslengthct.Butthedottedlinehasthesamelengthasthedottedarrowinthefirstpicture,namelycT.ByPythagoras’stheorem,

(ct)2=(CT)2+(vt)2

WesolveforT,getting

whichissmallerthant.ToderivetheLorenz–FitzGeraldcontraction,wenowimaginethatthe

spaceshiptravelstoaplanetdistancexfromEarthatspeedv.Thentheelapsedtimeist=x/v.Butthepreviousformulashowsthattothecrew,thetimetakenisT,nott.Forthem,thedistanceXmustsatisfyT=X/v.Therefore

whichissmallerthanx.Thederivationofthemasschangeisslightlymoreinvolved,anditdepends

onaparticularinterpretationofmass,‘restmass’,soIwon’tgivedetails.Theformulais

whichislargerthanm.Theseequationstellusthatthereissomethingveryspecialaboutthespeedof

light(andindeedaboutlight).Animportantconsequenceofthisformalismisthatthespeedoflightisanimpenetrablebarrier.Ifabodystartsoutslowerthanlight,itcannotbeacceleratedtoaspeedgreaterthanthatoflight.InSeptember2011physicistsworkinginItalyannouncedthatsubatomicparticlescalledneutrinosappearedtobetravellingfasterthanlight.1Theirobservationiscontroversial,butifitisconfirmed,itwillleadtoimportantnewphysics.

Pythagorasturnsupinrelativityinotherways.Oneistheformulationofspecialrelativityintermsofthegeometryofspace-time,originallyintroducedbyHermannMinkowski.OrdinaryNewtonianspacecanbecapturedmathematicallybymakingitspointscorrespondtothreecoordinates(x,y,z),anddefiningthedistancedbetweensuchapointandanotherone(X,Y,Z)usingPythagoras’stheorem:

d2=(x−X)2+(y−Y)2+(z−Z)2

Nowtakethesquareroottogetd.Minkowskispace-timeissimilar,butnowtherearefourcoordinates(x,y,z,t),threeofspaceplusoneoftime,andapointiscalledanevent–alocationinspace,observedataspecifiedtime.The

distanceformulaisverysimilar:

d2=(x−X)2+(y−Y)2+(z−Z)2−c2(t−T)2

Thefactorc2isjustaconsequenceoftheunitsusedtomeasuretime,buttheminussigninfrontofitiscrucial.The‘distance’discalledtheinterval,andthesquarerootisrealonlywhentheright-handsideoftheequationispositive.Thatboilsdowntothespatialdistancebetweenthetwoeventsbeinglargerthanthetemporaldifference(incorrectunits:light-yearsandyears,forinstance).Thatinturnmeansthatinprincipleabodycouldtravelfromthefirstpointinspaceatthefirsttime,andarriveatthesecondpointinspaceatthesecondtime,withoutgoingfasterthanlight.

Inotherwords,theintervalisrealifandonlyifitisphysicallypossible,inprinciple,totravelbetweenthetwoevents.Theintervaliszeroifandonlyiflightcouldtravelbetweenthem.Thisphysicallyaccessibleregioniscalledthelightconeofanevent,anditcomesintwopieces:thepastandthefuture.Figure50showsthegeometrywhenspaceisreducedtoonedimension.

I’venowshownyouthreerelativisticequations,andsketchedhowtheyarose,butnoneofthemisEinstein’siconicequation.However,we’renowreadytounderstandhowhederivedit,onceweappreciateonemoreinnovationofearlytwentieth-centuryphysics.Aswe’veseen,physicistshadpreviouslyperformedexperimentstodemonstrateconclusivelythatlightisawave,andMaxwellhadshownthatitisawaveofelectromagnetism.However,by1905itwasbecomingclearthatdespitetheweightofevidenceforthewavenatureoflight,therearecircumstancesinwhichitbehaveslikeaparticle.InthatyearEinsteinusedthisideatoexplainsomefeaturesofthephotoelectriceffect,inwhichlightthathitsasuitablemetalgenerateselectricity.Hearguedthattheexperimentsmadesenseonlyiflightcomesindiscretepackages:ineffect,particles.Theyarenowcalledphotons.

Fig50Minkowskispace-time,withspaceshownasone-dimensional.

Thispuzzlingdiscoverywasoneofthekeystepstowardsquantummechanics,andI’llsaymoreaboutitinChapter14.Curiously,thisquintessentiallyquantum-mechanicalideawasvitaltoEinstein’sformulationofrelativity.Toderivehisequationrelatingmasstoenergy,Einsteinthoughtaboutwhathappenstoabodythatemitsapairofphotons.Tosimplifythecalculationsherestrictedattentiontoonedimensionofspace,sothatthebodymovedalongastraightline.Thissimplificationdoesnotaffecttheanswer.Thebasicideaistoconsiderthesystemintwodifferentframesofreference.2Onemoveswiththebody,sothatthebodyappearstobestationarywithinthatframe.Theotherframemoveswithasmall,nonzerovelocityrelativetothebody.Letmecallthesethestationaryandmovingframes.Theyarelikethespaceship(initsownframeofreferenceitisstationary)andmygroundobserver(towhomitappearstobemoving).

Einsteinassumedthatthetwophotonsareequallyenergetic,butemittedinoppositedirections.Theirvelocitiesareequalandopposite,sothevelocityofthebody(ineitherframe)doesnotchangewhenthephotonsareemitted.Thenhecalculatedtheenergyofthesystembeforethebodyemitsthepairofphotons,andafterwards.Byassumingthatenergymustbeconserved,heobtainedanexpressionthatrelatesthechangeinthebody’senergy,causedbyemittingthephotons,tothechangeinits(relativistic)mass.Theupshotwas:

(changeinenergy)=(changeinmass)×c2

Makingthereasonableassumptionthatabodyofzeromasshaszeroenergy,itthenfollowedthat

energy=mass×c2

This,ofcourse,isthefamousformula,inwhichEsymbolisesenergyandmmass.

Aswellasdoingthecalculations,Einsteinhadtointerprettheirmeaning.Inparticular,hearguedthatinaframeforwhichthebodyisatrest,theenergygivenbytheformulashouldbeconsideredtobeits‘internal’energy,whichitpossessesbecauseitismadefromsubatomicparticles,eachofwhichhasitsownenergy.Inamovingframe,thereisalsoacontributionfromkineticenergy.Thereareothermathematicalsubtletiestoo,suchastheuseofasmallvelocityandapproximationstotheexactformulas.

Einsteinisoftencredited,ifthat’stheword,withtherealisationthatanatomicbombwouldreleasestupendousquantitiesofenergy.CertainlyTimemagazinegavethatimpressioninJuly1946whenitputhisfaceonthecoverwithanatomicmushroomcloudinthebackgroundbearinghisiconicequation.Theconnectionbetweentheequationandahugeexplosionseemsclear:theequationtellsusthattheenergyinherentinanyobjectisitsmassmultipliedbythesquareofthespeedoflight.Sincethespeedoflightishuge,itssquareisevenbigger,whichequatestoalotofenergyinasmallamountofmatter.Theenergyinonegramofmatterturnsouttobe90terajoules,equivalenttoaboutoneday’soutputofelectricityfromanuclearpowerstation.

However,itdidn’thappenlikethat.Theenergyreleasedinanatomicbombisonlyatinyfractionoftherelativisticrestmass,andphysicistswerealreadyaware,onexperimentalgrounds,thatcertainnuclearreactionscouldreleasealotofenergy.Themaintechnicalproblemwastoholdalumpofsuitableradioactivematerialtogetherlongenoughtogetachainreaction,inwhichthedecayofoneradioactiveatomcausesittoemitradiationthattriggersthesameeffectinotheratoms,growingexponentially.Nevertheless,Einstein’sequationquicklybecameestablishedinthepublicmindastheprogenitoroftheatomicbomb.TheSmythreport,anAmericangovernmentdocumentreleasedtothepublictoexplaintheatomicbomb,placedtheequationonitssecondpage.IsuspectthatwhathappenediswhatJackCohenandIhavecalled‘liestochildren’–simplifiedstoriestoldforlegitimatepurposes,whichpavethewaytomoreaccurateenlightenment.3Thisishoweducationworks:thefullstoryisalwaystoocomplicatedforanyoneexcepttheexperts,andtheyknowsomuchthattheydon’tbelievemostofit.

However,Einstein’sequationcan’tjustbedismissedoutofhand.Itdidplaya

roleinthedevelopmentofnuclearweapons.Thenotionofnuclearfission,whichpowerstheatombomb,arosefromdiscussionsbetweenthephysicistsLiseMeitnerandOttoFrischinNaziGermanyin1938.Theyweretryingtounderstandtheforcesthatheldtheatomtogether,whichwereabitlikethesurfacetensionofadropofliquid.Theywereoutwalking,discussingphysics,andtheyappliedEinstein’sequationtoworkoutwhetherfissionwaspossibleonenergygrounds.Frischlaterwrote:4

Webothsatdownonatreetrunkandstartedtocalculateonscrapsofpaper…Whenthetwodropsseparatedtheywouldbedrivenapartbyelectricalrepulsion,about200MeVinall.FortunatelyLiseMeitnerrememberedhowtocomputethemassesofnuclei…andworkedoutthatthetwonucleiformed…wouldbelighterbyaboutone-fifththemassofaproton…accordingtoEinstein’sformulaE=mc2…themasswasjustequivalentto200MeV.Itallfitted!

AlthoughE=mc2wasnotdirectlyresponsiblefortheatombomb,itwasoneofthebigdiscoveriesinphysicsthatledtoaneffectivetheoreticalunderstandingofnuclearreactions.Einstein’smostimportantroleregardingtheatomicbombwaspolitical.UrgedbyLeoSzilard,EinsteinwrotetoPresidentRoosevelt,warningthattheNazismightbedevelopingatomicweaponsandexplainingtheirawesomepower.Hisreputationandinfluencewereenormous,andthepresidentheededthewarning.TheManhattanProject,HiroshimaandNagasaki,andtheensuingColdWarwerejustsomeoftheconsequences.

Einsteinwasn’tsatisfiedwithspecialrelativity.Itprovidedaunifiedtheoryofspace,time,matter,andelectromagnetism,butitmissedoutonevitalingredient.

Gravity.Einsteinbelievedthat‘allthelawsofphysics’mustsatisfyhisextended

versionofGalileo’sprincipleofrelativity.Thelawofgravitationsurelyoughttobeamongthem.Butthatwasn’tthecaseforthecurrentversionofrelativity.Newton’sinversesquarelawdidnottransformcorrectlybetweenframesofreference.SoEinsteindecidedhehadtochangeNewton’slaw.He’dalreadychangedvirtuallyeverythingelseintheNewtonianuniverse,sowhynot?

Ittookhimtenyears.Hisstarting-pointwastoworkouttheimplicationsoftheprincipleofrelativityforanobservermovingfreelyundertheinfluenceofgravity–inaliftthatisdroppingfreely,forexample.Eventuallyhehomedinonasuitableformulation.Inthishewasaidedbyaclosefriend,themathematician

MarcelGrossmann,whopointedhimtowardsarapidlygrowingfieldofmathematics:differentialgeometry.ThishaddevelopedfromRiemann’sconceptofamanifoldandhischaracterisationofcurvature,discussedinChapter1.ThereImentionedthatRiemann’smetriccanbewrittenasa3×3matrix,andthattechnicallythisisasymmetrictensor.AschoolofItalianmathematicians,notablyTullioLevi-CivitaandGregorioRicci-Curbastro,tookupRiemann’sideasanddevelopedthemintotensorcalculus.

From1912,Einsteinwasconvincedthatthekeytoarelativistictheoryofgravityrequiredhimtoreformulatehisideasusingtensorcalculus,butina4-dimensionalspace-timeratherthan3-dimensionalspace.ThemathematicianswerehappilyfollowingRiemannandallowinganynumberofdimensions,sotheyhadalreadysetthingsupinmorethanenoughgenerality.Tocutalongstoryshort,heeventuallyderivedwhatwenowcalltheEinsteinfieldequations,whichhewroteas:

HereR,g,andTaretensors–quantitiesthatdefinephysicalpropertiesandtransformaccordingtotherulesofdifferentialgeometry–andkisaconstant.Thesubscriptsμandνrunoverthefourcoordinatesofspace-time,soeachtensorisa4×4tableof16numbers.Botharesymmetric,meaningthattheydon’tchangewhenμandνareswapped,whichreducesthemtoalistof10distinctnumbers.Soreallytheformulapackagestogether10equations,whichiswhyweoftenrefertothemusingtheplural–compareMaxwell’sequations.RisRiemann’smetric:itdefinestheshapeofspace-time.gistheRiccicurvaturetensor,whichisamodificationofRiemann’snotionofcurvature.AndTistheenergy–momentumtensor,whichdescribeshowthesetwofundamentalquantitiesdependonthespace-timeeventconcerned.EinsteinpresentedhisequationstothePrussianAcademyofSciencein1915.Hecalledhisnewworkthegeneraltheoryofrelativity.

WecaninterpretEinstein’sequationsgeometrically,andwhenwedo,theyprovideanewapproachtogravity.Thebasicinnovationisthatgravityisnotrepresentedasaforce,butasthecurvatureofspace-time.Intheabsenceofgravity,space-timereducestoMinkowskispace.Theformulafortheintervaldeterminesthecorrespondingcurvaturetensor.Itsinterpretationis‘notcurved’,justasPythagoras’stheoremappliestoaflatplanebutnottoapositivelyornegativelycurvednon-Euclideanspace.Minkowskispace-timeisflat.Butwhengravityoccurs,space-timebends.

Theusualwaytopicturethisistoforgettime,dropthedimensionsofspacedowntotwo,andgetsomethinglikeFigure51(left).TheflatplaneofMinkowskispace(-time)isdistorted,shownherebyanactualbend,creatingadepression.Farfromthestar,matterorlighttravelsinastraightline(dotted).Butthecurvaturecausesthepathtobend.Infact,itlookssuperficiallyasthoughsomeforcecomingfromthestarattractsthemattertowardsit.Butthereisnoforce,justwarpedspace-time.However,thisimageofcurvaturedeformsspacealonganextradimension,whichisnotrequiredmathematically.Analternativeimageistodrawagridofgeodesics,shortestpaths,equallyspacedaccordingtothecurvedmetric.Theybunchtogetherwherethecurvatureisgreater,Figure51(right).

Fig51Left:Warpedspacenearastar,andhowitbendsthepathsofpassingmatterorlight.Right:alternativeimageusingagridofgeodesics,whichbunchtogetherinregionsofhighercurvature.

Ifthecurvatureofspace-timeissmall,thatis,ifwhat(intheoldpicture)wethinkofasgravitationalforcesarenottoolarge,thenthisformulationleadstoNewton’slawofgravity.Comparingthetwotheories,Einstein’sconstantkturnsouttobe8πG/c4,whereGisNewton’sgravitationalconstant.Thislinksthenewtheorytotheoldone,andprovesthatinmostcasesthenewonewillagreewiththeold.Theinterestingnewphysicsoccurswhenthisisnolongertrue:whengravityislarge.WhenEinsteincameupwithhistheory,anytestofrelativityhadtotakeplaceoutsidethelaboratory,onagrandscale.Whichmeantastronomy.

Einsteinthereforewentlookingforunexplainedpeculiaritiesinthemotionoftheplanets,effectsthatdidn’tsquarewithNewton.Hefoundonethatmightbesuitable:anobscurefeatureoftheorbitofMercury,theplanetclosesttotheSun,subjectedtothegreatestgravitationalforces–andso,ifEinsteinwasright,insidearegionofhighcurvature.

Likeallplanets,Mercuryfollowsapaththatisveryclosetoanellipse,sosomepointsinitsorbitareclosertotheSunthanothers.Theclosestofallis

calleditsperihelion(‘nearSun’inGreek).Theexactlocationofthisperihelionhadbeenobservedformanyyears,andtherewassomethingfunnyaboutit.TheperihelionslowlyrotatedabouttheSun,aneffectcalledprecession;ineffect,thelongaxisoftheorbitalellipsewasslowlychangingdirection.Thatwasallright;Newton’slawspredictedit,becauseMercuryisnottheonlyplanetintheSolarSystemandotherplanetswereslowlychangingitsorbit.TheproblemwasthatNewtoniancalculationsgavethewrongrateofprecession.Theaxiswasrotatingtooquickly.

Thathadbeenknownsince1840whenFrançoisArago,directoroftheParisObservatory,askedUrbainLeVerriertocalculatetheorbitofMercuryusingNewton’slawsofmotionandgravitation.ButwhentheresultsweretestedbyobservingtheexacttimingofatransitofMercury–apassageacrossthefaceoftheSun,asviewedfromEarth–theywerewrong.LeVerrierdecidedtotryagain,eliminatingpotentialsourcesoferror,andin1859hepublishedhisnewresults.OntheNewtonianmodel,therateofprecessionwasaccuratetoabout0.7%.Thedifferencecomparedwithobservationswastiny:38secondsofarceverycentury(laterrevisedto43arc-seconds).That’snotmuch,lessthanonetenthousandthofadegreeperyear,butitwasenoughtointerestLeVerrier.In1846hehadmadehisreputationbyanalysingirregularitiesintheorbitofUranusandpredictingtheexistence,andlocation,ofathenundiscoveredplanet:Neptune.Nowhewashopingtorepeatthefeat.HeinterpretedtheunexpectedperihelionmovementasevidencethatsomeunknownworldwasperturbingMercury’sorbit.HedidthesumsandpredictedtheexistenceofasmallplanetwithanorbitclosertotheSunthanthatofMercury.Heevenhadanameforit:Vulcan,theRomangodoffire.

ObservingVulcan,ifitexisted,wouldbedifficult.TheglareoftheSunwasanobstacle,sothebestbetwastocatchVulcanintransit,whereitwouldbeatinydarkdotagainstthebrightdiscoftheSun.ShortlyafterLeVerrier’sprediction,anamateurastronomernamedEdmondLescarbaultinformedthedistinguishedastronomerthathehadseenjustthat.Hehadinitiallyassumedthatthedotmustbeasunspot,butitmovedatthewrongspeed.In1860LeVerrierannouncedthediscoveryofVulcantotheParisAcademyofScience,andthegovernmentawardedLescarbaulttheprestigiousLégiond’Honneur.

Amidtheclamour,someastronomersremainedunimpressed.OnewasEmmanuelLiais,whohadbeenstudyingtheSunwithmuchbetterequipmentthanLescarbault.Hisreputationwasontheline:hehadbeenobservingtheSunfortheBraziliangovernment,anditwouldhavebeendisgracefultohavemissedsomethingofsuchimportance.Heflatlydeniedthatatransithadtakenplace.

Foratime,everythinggotveryconfused.AmateursrepeatedlyclaimedtheyhadseenVulcan,sometimesyearsbeforeLeVerrierannouncedhisprediction.In1878JamesWatson,aprofessional,andLewisSwift,anamateur,saidtheyhadseenaplanetlikeVulcanduringasolareclipse.LeVerrierhaddiedayearearlier,stillconvincedhehaddiscoveredanewplanetneartheSun,butwithouthisenthusiasticnewcalculationsoforbitsandpredictionsoftransits–noneofwhichhappened–interestinVulcanquicklydiedaway.Astronomersbecameskeptical.

In1915,Einsteinadministeredthecoupdegrâce.Hereanalysedthemotionusinggeneralrelativity,withoutassuminganynewplanet,andasimpleandtransparentcalculationledhimtoavalueof43secondsofarcfortheprecession–theexactfigureobtainedbyupdatingLeVerrier’soriginalcalculations.AmodernNewtoniancalculationpredictsaprecessionof5560arcsecondspercentury,butobservationsgive5600.Thedifferenceis40secondsofarc,soabout3arc-secondspercenturyremainsunaccountedfor.Einstein’sannouncementdidtwothings:itwasseenasavindicationofrelativity,andasfarasmostastronomerswereconcerned,itrelegatedVulcantothescrapheap.5

AnotherfamousastronomicalverificationofgeneralrelativityisEinstein’spredictionthattheSunbendslight.Newtoniangravitationalsopredictsthis,butgeneralrelativitypredictsanamountofbendingthatistwiceaslarge.Thetotalsolareclipseof1919providedanopportunitytodistinguishthetwo,andSirArthurEddingtonmountedanexpedition,eventuallyannouncingthatEinsteinprevailed.Thiswasacceptedwithenthusiasmatthetime,butlateritbecameclearthatthedatawerepoor,andtheresultwasquestioned.Furtherindependentobservationsfrom1922seemedtoagreewiththerelativisticprediction,asdidalaterreanalysisofEddington’sdata.Bythe1960sitbecamepossibletomaketheobservationsforradio-frequencyradiation,andonlythenwasitcertainthatthedatadidindeedshowadeviationtwicethatpredictedbyNewtonandequaltothatpredictedbyEinstein.

Themostdramaticpredictionsfromgeneralrelativityariseonamuchgranderscale:blackholes,whicharebornwhenamassivestarcollapsesunderitsowngravitation,andtheexpandinguniverse,currentlyexplainedbytheBigBang.

SolutionstoEinstein’sequationsarespace-timegeometries.Thesemightrepresenttheuniverseasawhole,orsomepartofit,assumedtobegravitationallyisolatedsothattherestoftheuniversehasnoimportanteffect.ThisisanalogoustoearlyNewtonianassumptionsthatonlytwobodiesareinteracting,forexample.SinceEinstein’sfieldequationsinvolvetenvariables,

explicitsolutionsintermsofmathematicalformulasarerare.Todaywecansolvetheequationsnumerically,butthatwasapipedreambeforethe1960sbecausecomputerseitherdidn’texistorweretoolimitedtobeuseful.Thestandardwaytosimplifyequationsistoinvokesymmetry.Supposethattheinitialconditionsforaspace-timearesphericallysymmetric,thatis,allphysicalquantitiesdependonlyonthedistancefromthecentre.Thenthenumberofvariablesinanymodelisgreatlyreduced.In1916theGermanastrophysicistKarlSchwarzschildmadethisassumptionforEinstein’sequations,andmanagedtosolvetheresultingequationswithanexactformula,knownastheSchwarzschildmetric.Hisformulahadacuriousfeature:asingularity.Thesolutionbecameinfiniteataparticulardistancefromthecentre,calledtheSchwarzschildradius.Atfirstitwasassumedthatthissingularitywassomekindofmathematicalartefact,anditsphysicalmeaningwasthesubjectofconsiderabledispute.Wenowinterpretitastheeventhorizonofablackhole.

Imagineastarsomassivethatitsradiationcannotcounteritsgravitationalfield.Thestarwillbegintocontract,suckedtogetherbyitsownmass.Thedenseritgets,thestrongerthiseffectbecomes,sothecontractionhappenseverfaster.Thestar’sescapevelocity,thespeedwithwhichanobjectmustmovetoescapethegravitationalfield,alsoincreases.TheSchwarzschildmetrictellsusthatatsomestage,theescapevelocitybecomesequaltothatoflight.Nownothingcanescape,becausenothingcantravelfasterthanlight.Thestarhasbecomeablackhole,andtheSchwarzschildradiustellsustheregionfromwhichnothingcanescape,boundedbytheblackhole’seventhorizon.

Blackholephysicsiscomplex,andthereisn’tspacetodoitjusticehere.Sufficeittosaythatmostcosmologistsarenowsatisfiedthatthepredictionisvalid,thattheuniversecontainsinnumerableblackholes,andindeedthatatleastonelurksattheheartofourGalaxy.Indeed,ofmostgalaxies.

In1917Einsteinappliedhisequationstotheentireuniverse,assuminganotherkindofsymmetry:homogeneity.Theuniverseshouldlookthesame(onlargeenoughscale)atallpointsinspaceandtime.Bythenhehadmodifiedtheequationstoincludea‘cosmologicalconstant’Λ,andsortedoutthemeaningoftheconstantκ.Theequationswerenowwrittenlikethis:

Thesolutionshadasurprisingimplication:theuniverseshouldshrinkastimepasses.ThisforcedEinsteintoaddtheterminvolvingthecosmologicalconstant:

hewasseekinganunchanging,stableuniverse,andbyadjustingtheconstanttotherightvaluehecouldstophismodeluniversecontractingtoapoint.In1922AlexanderFriedmannfoundanotherequation,whichpredictedtheuniverseshouldexpandanddidnotrequirethecosmologicalconstant.Italsopredictedtherateofexpansion.Einsteinstillwasn’thappy:hewantedtheuniversetobestableandunchanging.

ForonceEinstein’simaginationfailedhim.In1929AmericanastronomersEdwinHubbleandMiltonHumasonfoundevidencethattheuniverseisexpanding.Distantgalaxiesaremovingawayfromus,asshownbyshiftsinthefrequencyofthelighttheyemit–thefamousDopplereffect,inwhichthesoundofaspeedingambulancedropsasitpassesby,becausethesoundwavesareaffectedbytherelativespeedofemitterandreceiver.Nowthewavesareelectromagneticandthephysicsisrelativistic,butthereisstillaDopplereffect.Notonlydodistantgalaxiesmoveawayfromus:themoredistanttheyare,thefastertheyrecede.

Runningtheexpansionbackwardsintime,itturnsoutthatatsomepointinthepast,theentireuniversewasessentiallyjustapoint.Beforethat,itdidn’texistatall.Atthatprimevalpoint,spaceandtimebothcameintoexistenceinthefamousBigBang,atheoryproposedbyFrenchmathematicianGeorgesLemaîtrein1927,andalmostuniversallyignored.Whenradiotelescopesobservedthecosmologicalmicrowavebackgroundradiationin1964,atatemperaturethatfittedtheBigBangmodel,cosmologistsdecidedLemaîtrehadbeenrightafterall.Again,thetopicdeservesabookofisown,andmanyhavebeenwritten.SufficeittosaythatourcurrentmostwidelyacceptedtheoryofcosmologyisanelaborationoftheBigBangscenario.

Scientificknowledge,however,isalwaysprovisional.Newdiscoveriescanchangeit.TheBigBanghasbeentheacceptedcosmologicalparadigmforthelast30years,butitisbeginningtoshowsomecracks.Severaldiscoverieseithercastseriousdoubtonthetheory,orrequirenewphysicalparticlesandforcesthathavebeeninferredbutnotobserved.Therearethreemainsourcesofdifficulty.I’llsummarisethemfirst,andthendiscusseachinmoredetail.Thefirstisgalacticrotationcurves,whichsuggestthatmostofthematterintheuniverseismissing.Thecurrentproposalisthatthisisasignofanewkindofmatter,darkmatter,whichconstitutesabout90%ofthematterintheuniverse,andisdifferentfromanymatteryetobserveddirectlyonEarth.Thesecondisanaccelerationintheexpansionoftheuniverse,whichrequiresanewforce,darkenergy,ofunknownoriginbutmodelledusingEinstein’scosmologicalconstant.

Thethirdisacollectionoftheoreticalissuesrelatedtothepopulartheoryofinflation,whichexplainswhytheobservableuniverseissouniform.Thetheoryfitsobservations,butitsinternallogicislookingshaky.

Darkmatterfirst.In1938theDopplereffectwasusedtomeasurethespeedsofgalaxiesinclusters,andtheresultswereinconsistentwithNewtoniangravitation.Becausegalaxiesareseparatedbylargedistances,space-timeisalmostflatandNewtoniangravityisagoodmodel.FritzZwickysuggestedthattheremustbesomeunobservedmattertoaccountforthediscrepancy,anditwasnameddarkmatterbecauseitcouldnotbeseeninphotographs.In1959,usingtheDopplereffecttomeasurethespeedofrotationofstarsinthegalaxyM33,LouiseVoldersdiscoveredthattheobservedrotationcurve–aplotofspeedagainstdistancefromthecentre–wasalsoinconsistentwithNewtoniangravitation,whichagainisagoodmodel.Insteadofthespeedfallingoffatgreaterdistances,itremainedalmostconstant,Figure52.Thesameproblemarisesformanyothergalaxies.

Ifitexists,darkmattermustbedifferentfromordinary‘baryonic’matter,theparticlesobservedinexperimentsonEarth.Itsexistenceisacceptedbymostcosmologists,whoarguethatdarkmatterexplainsseveraldifferentanomaliesinobservations,notjustrotationcurves.Candidateparticleshavebeensuggested,suchasWIMPs(weaklyinteractingmassiveparticles),butsofartheseparticleshavenotbeendetectedinexperiments.Thedistributionofdarkmatteraroundgalaxieshasbeenplottedbyassumingdarkmatterexistsandworkingoutwhereithastobetomaketherotationcurvesflat.Itgenerallyseemstoformtwoglobesofgalacticproportions,oneabovetheplaneofthegalaxyandtheotherbelowit,likeagiantdumb-bell.ThisisabitlikepredictingtheexistenceofNeptunefromdiscrepanciesintheorbitofUranus,butsuchpredictionsrequireconfirmation:Neptunehadtobefound.

Fig52GalacticrotationcurvesforM33:theoryandobservations.

Darkenergyissimilarlyproposedtoexplaintheresultsofthe1998High-zSupernovaSearchTeam,whichexpectedtofindevidencethattheexpansionoftheuniverseisslowingdownastheinitialimpulsefromtheBigBangdiesaway.Instead,theobservationsindicatedthattheexpansionoftheuniverseisspeedingup,afindingconfirmedbytheSupernovaCosmologyProjectin1999.Itisasthoughsomeantigravityforcepervadesspace,pushinggalaxiesapartatanever-increasingrate.Thisforceisnotanyofthefourbasicforcesofphysics:gravity,electromagnetism,strongnuclearforce,weaknuclearforce.Itwasnameddarkenergy.Again,itsexistenceseemedtosolvesomeothercosmologicalproblems.

InflationwasproposedbytheAmericanphysicistAlanGuthin1980toexplainwhytheuniverseisextremelyuniforminitsphysicalpropertiesonverylargescales.TheoryshowedthattheBigBangoughttohaveproducedauniversethatwasfarmorecurved.Guthsuggestedthatan‘inflatonfield’(that’sright,nosecondi:it’sthoughttobeascalarquantumfieldcorrespondingtoahypotheticalparticle,theinflaton)causedtheearlyuniversetoexpandwithextremerapidity.Between10−36and10−32secondsaftertheBigBang,thevolumeoftheuniversegrewbyamindbogglingfactorof1078.Theinflatonfieldhasnotbeenobserved(thiswouldrequireunfeasiblyhighenergies)butinflationexplainssomanyfeaturesoftheuniverse,andfitsobservationssoclosely,thatmostcosmologistsareconvincedithappened.It’snotsurprisingthatdarkmatter,darkenergy,andinflationwerepopularamongcosmologists,becausetheyletthemcontinuetousetheirfavouritephysicalmodels,andtheresultsagreedwithobservations.Butthingsarestartingtofallapart.

Thedistributionsofdarkmatterdon’tprovideasatisfactoryexplanationofrotationcurves.Enormousamountsofdarkmatterareneededtokeeptherotationcurveflatouttothelargedistancesobserved.Thedarkmatterhastohaveunrealisticallylargeangularmomentum,whichisinconsistentwiththeusualtheoriesofgalaxyformation.Thesameratherspecialinitialdistributionofdarkmatterisrequiredineverygalaxy,whichseemsunlikely.Thedumb-bellshapeisunstablebecauseitplacestheadditionalmassontheoutsideofthegalaxy.

Darkenergyfaresbetter,anditisthoughttobesomekindofquantum-mechanicalvacuumenergy,arisingfromfluctuationsinthevacuum.However,currentcalculationsofthesizeofthevacuumenergyaretoobigbyafactorof10122,whichisbadnewsevenbythestandardsofcosmology.6

Themainproblemsaffectinginflationarenotobservations–itfitsthoseamazinglywell–butitslogicalfoundations.Mostinflationaryscenarioswouldleadtoauniversethatdiffersconsiderablyfromours;whatcountsistheinitialconditionsatthetimeoftheBigBang.Inordertomatchobservations,inflationrequirestheearlystateoftheuniversetobeveryspecial.However,therearealsoveryspecialinitialconditionsthatproduceauniversejustlikeourswithoutinvokinginflation.Althoughbothsetsofconditionsareextremelyrare,calculationsperformedbyRogerPenrose7showthattheinitialconditionsthatdonotrequireinflationoutnumberthosethatproduceinflationbyafactorofonegoogolplex–tentothepowertentothepower100.Soexplainingthecurrentstateoftheuniversewithoutinflationwouldbemuchmoreconvincingthanexplainingitwithinflation.

Penrose’scalculationreliesonthermodynamics,whichmightnotbeanappropriatemodel,butanalternativeapproach,carriedoutbyGaryGibbonsandNeilTurok,leadstothesameconclusion.Thisisto‘unwind’theuniversebacktoitsinitialstate.Itturnsoutthatalmostallofthepotentialinitialstatesdonotinvolveaperiodofinflation,andthosethatdorequireitareanexceedinglysmallproportion.Butthebiggestproblemofallisthatwheninflationisweddedtoquantummechanics,itpredictsthatquantumfluctuationswilloccasionallytriggerinflationinasmallregionofanapparentlysettleduniverse.Althoughsuchfluctuationsarerare,inflationissorapidandsogiganticthatthenetresultistinyislandsofnormalspace-timesurroundedbyever-growingregionsofrunawayinflation.Inthoseregions,thefundamentalconstantsofphysicscanbedifferentfromtheirvaluesinouruniverse.Ineffect,anythingispossible.Canatheorythatpredictsanythingbetestablescientifically?

Therearealternatives,anditisstartingtolookasthoughtheyneedtobetakenseriously.DarkmattermightnotbeanotherNeptune,butanotherVulcan–anattempttoexplainagravitationalanomalybyinvokingnewmatter,whenwhatreallyneedschangingisthelawofgravitation.

Themainwell-developedproposalisMOND,modifiedNewtoniandynamics,proposedbyIsraeliphysicistMordehaiMilgromin1983.Thismodifiesnotthelawofgravity,infact,butNewton’ssecondlawofmotion.Itassumesthataccelerationisnotproportionaltoforcewhentheaccelerationisverysmall.ThereisatendencyamongcosmologiststoassumethattheonlyviablealternativetheoriesaredarkmatterorMOND–soifMONDdisagreeswithobservations,thatleavesonlydarkmatter.However,therearemanypotentialwaystomodifythelawofgravity,andweareunlikelytofindtherightone

straightaway.ThedemiseofMONDhasbeenproclaimedseveraltimes,butonfurtherinvestigationnodecisiveflawhasyetbeenfound.ThemainproblemwithMOND,tomymind,isthatitputsintoitsequationswhatithopestogetout;it’slikeEinsteinmodifyingNewton’slawtochangetheformulanearalargemass.Instead,hefoundaradicallynewwaytothinkofgravity,thecurvatureofspace-time.

EvenifweretaingeneralrelativityanditsNewtonianapproximation,theremaybenoneedfordarkenergy.In2009,usingthemathematicsofshockwaves,AmericanmathematiciansJoelSmollerandBlakeTempleshowedthattherearesolutionsofEinstein’sfieldequationsinwhichthemetricexpandsatanacceleratingrate.8ThesesolutionsshowthatsmallchangestotheStandardModelcouldaccountfortheobservedaccelerationofgalaxieswithoutinvokingdarkenergy.

Generalrelativitymodelsoftheuniverseassumethatitformsamanifold;thatis,onverylargescalesthestructuresmoothesout.However,theobserveddistributionofmatterintheuniverseisclumpyonverybigscales,suchastheSloanGreatWall,afilamentcomposedofgalaxies1.37billionlightyearslong,Figure53.Cosmologistsbelievethatonevenlargerscalesthesmoothnesswillbecomeapparent–buttodate,everytimetherangeofobservationshasbeenextended,theclumpinesshaspersisted.

Fig53Theclumpinessoftheuniverse.

RobertMacKayandColinRourke,twoBritishmathematicians,havearguedthataclumpyuniverseinwhichtherearemanylocalsourcesoflargecurvaturecouldexplainallofthecosmologicalpuzzles.9Suchastructureisclosertowhatisobservedthansomelarge-scalesmoothing,andisconsistentwiththegeneralprinciplethattheuniverseoughttobemuchthesameeverywhere.Insucha

universethereneedbenoBigBang;infact,thewholethingcouldbeinasteadystate,andbefar,farolderthanthecurrentfigureof13.8billionyears.Individualgalaxieswouldgothroughalifecycle,survivingrelativelyunchangedforaround1016years.Theywouldhaveaverymassivecentralblackhole.Galacticrotationcurveswouldbeflatbecauseofinertialdrag,aconsequenceofgeneralrelativityinwhicharotatingmassivebodydragsspace-timeroundwithitinitsvicinity.Theredshiftobservedinquasarswouldbecausedbyalargegravitationalfield,notbytheDopplereffect,andwouldnotbeindicativeofanexpandinguniverse–thistheoryhaslongbeenadvancedbyAmericanastronomerHaltonArp,andneversatisfactorilydisproved.Thealternativemodelevenindicatesatemperatureof5°Kforthecosmologicalmicrowavebackground,themainevidence(asidefromredshiftinterpretedasexpansion)fortheBigBang.

MacKayandRourkesaythattheirproposal‘overturnsvirtuallyeverytenetofcurrentcosmology.Itdoesnot,however,contradictanyobservationalevidence.’Itmaywellbewrong,butthefascinatingpointisthatyoucanretainEinstein’sfieldequationsunchanged,dispensewithdarkmatter,darkenergy,andinflation,andstillgetbehaviourreasonablylikeallofthosepuzzlingobservations.Sowhateverthetheory’sfate,itsuggeststhatcosmologistsshouldconsidermoreimaginativemathematicalmodelsbeforeresortingtonewandotherwiseunsupportedphysics.Darkmatter,darkenergy,inflation,eachrequiringradicallynewphysicsthatnoonehasobserved…Inscience,evenonedeusexmachinaraiseseyebrows.Threewouldbeconsideredintolerableinanythingotherthancosmology.Tobefair,it’sdifficulttoexperimentontheentireuniverse,sospeculativelyfittingtheoriestoobservationsisaboutallthatcanbedone.Butimaginewhatwouldhappenifabiologistexplainedlifebysomeunobservable‘lifefield’,letalonesuggestingthatanewkindof‘vitalmatter’andanewkindof‘vitalenergy’werealsonecessary–whileprovidingnoevidencethatanyofthemexisted.

Leavingasidetheperplexingrealmofcosmology,therearenowmorehomelywaystoverifyrelativity,bothspecialandgeneral,onahumanscale.Specialrelativitycanbetestedinthelaboratory,andmodernmeasuringtechniquesprovideexquisiteaccuracy.ParticleacceleratorssuchastheLargeHadronCollidersimplywouldnotworkunlessthedesignerstookspecialrelativityintoaccount,becausetheparticlesthatwhirlroundthesemachinesdosoatspeedsverycloseindeedtothatoflight.Mosttestsofgeneralrelativityarestillastronomical,rangingfromgravitationallensingtopulsardynamics,andthe

levelofaccuracyishigh.ArecentNASAexperimentinlow-Earthorbit,usinghigh-precisiongyroscopes,confirmedtheoccurrenceofinertialframe-dragging,butfailedtoreachtheintendedprecisionbecauseofunexpectedelectrostaticeffects.Bythetimethedatawerecorrectedforthisproblem,otherexperimentshadalreadyachievedthesameresults.

However,oneinstanceofrelativisticdynamics,bothspecialandgeneral,isclosertohome:carsatellitenavigation.Thesatnavsystemsusedbymotoristscalculatethecar’spositionusingsignalsfromanetworkof24orbitingsatellites,theGlobalPositioningSystem.GPSisastonishinglyaccurate,anditworksbecausemodernelectronicscanreliablyhandleandmeasureverytinyinstantsoftime.Itisbasedonveryprecisetimingsignals,pulsesemittedbythesatellitesandpickedupontheground.Comparingthesignalsfromseveralsatellitestriangulatesthelocationofthereceivertowithinafewmetres.Thislevelofaccuracyrequiresknowingthetimingtowithinabout25nanoseconds(billionthsofasecond).Newtoniandynamicsdoesn’tgivecorrectlocations,becausetwoeffectsthatarenotaccountedforinNewton’sequationsaltertheflowoftime:thesatellite’smotionandEarth’sgravitationalfield.

Specialrelativitydealswiththemotion,anditpredictsthattheatomicclocksonthesatellitesshouldlose7microseconds(millionthsofasecond)perdaycomparedwithclocksontheground,thankstorelativistictimedilation.Generalrelativitypredictsagainof45microsecondsperdaycausedbytheEarth’sgravity.Thenetresultisthattheclocksonthesatellitesgain38microsecondsperdayforrelativisticreasons.Smallasthismayseem,itseffectonGPSsignalsisbynomeansnegligible.Anerrorof38microsecondsis38,000nanoseconds,about1500timestheerrorthatGPScantolerate.Ifthesoftwarecalculatedyourcar’slocationusingNewtoniandynamics,yoursatnavwouldquicklybecomeuseless,becausetheerrorwouldgrowatarateof10kilometresperday.TenminutesfromnowNewtonianGPSwouldplaceyouonthewrongstreet;bytomorrowitwouldplaceyouinthewrongtown.Withinaweekyou’dbeinthewrongcounty;withinamonth,thewrongcountry.Withinayear,you’dbeonthewrongplanet.Ifyoudisbelieverelativity,butusesatnavtoplanyourjourneys,youhavesomeexplainingtodo.

14Quantumweirdness

Schrödinger’sEquation

Whatdoesitsay?

Theequationmodelsmatternotasaparticle,butasawave,anddescribeshowsuchawavepropagates.

Whyisthatimportant?

Schrödinger’sequationisfundamentaltoquantummechanics,whichtogetherwithgeneralrelativityconstitutetoday’smosteffectivetheoriesofthephysicaluniverse.

Whatdiditleadto?

Aradicalrevisionofthephysicsoftheworldatverysmallscales,inwhicheveryobjecthasa‘wavefunction’thatdescribesaprobabilitycloudofpossiblestates.Atthisleveltheworldisinherentlyuncertain.Attemptstorelatethemicroscopicquantumworldtoourmacroscopicclassicalworldledtophilosophicalissuesthatstillreverberate.Butexperimentally,quantumtheoryworksbeautifully,andtoday’scomputerchipsandlaserswouldn’tworkwithoutit.

In1900thegreatphysicistLordKelvindeclaredthatthethencurrenttheoryofheatandlight,consideredtobeanalmostcompletedescriptionofnature,was‘obscuredbytwoclouds.Thefirstinvolvesthequestion:HowcouldtheEarthmovethroughanelasticsolid,suchasisessentiallytheluminiferousether?ThesecondistheMaxwell–Boltzmanndoctrineregardingthepartitionofenergy.’Kelvin’snoseforanimportantproblemwasspoton.InChapter13wesawhowthefirstquestionledto,andwasresolvedby,relativity.Nowwewillseehowthesecondledtotheothergreatpillarofpresent-dayphysics,quantumtheory.

Thequantumworldisnotoriouslyweird.Manyphysicistsfeelthatifyoudon’tappreciatejusthowweirditis,youdon’tappreciateitatall.Thereisalottobesaidforthatopinion,becausethequantumworldissodifferentfromourcomfortablehuman-scaleonethateventhesimplestconceptschangeoutofallrecognition.Itis,forexample,aworldinwhichlightisbothaparticleandawave.Itisaworldwhereacatinaboxcanbebothaliveanddeadatthesametime…untilyouopenthebox,thatis,whensuddenlytheunfortunateanimal’swavefunction‘collapses’toonestateortheother.Inthequantummultiverse,thereexistsonecopyofouruniverseinwhichHitlerlostWorldWarII,andanotherinwhichhewonit.Wejusthappentolivein–thatis,existasquantumwavefunctionsin–thefirstone.Otherversionsofus,justasrealbutinaccessibletooursenses,liveintheotherone.

Quantummechanicsisdefinitelyweird.Whetheritisquitethatweird,though,isanothermatteraltogether.

Itallbeganwithlightbulbs.Thiswasappropriate,becausethosewereoneofthemostspectacularapplicationstoemergefromtheburgeoningsubjectsofelectricityandmagnetism,whichMaxwellsobrilliantlyunified.In1894aGermanphysicistnamedMaxPlanckwashiredbyanelectricalcompanytodesignthemostefficientlightbulbpossible,onegivingthemostlightwhileconsumingtheleastelectricalenergy.Hesawthatthekeytothisquestionwasafundamentalissueinphysics,raisedin1859byanotherGermanphysicist,GustavKirchhoff.Itconcernedatheoreticalconstructknownasablackbody,whichabsorbsallelectromagneticradiationthatfallsonit.Thebigquestionwas:howdoessuchabodyemitradiation?Itcan’tstoreitall;somehastocomebackoutagain.Inparticular,howdoestheintensityoftheemittedradiationdependonitsfrequencyandthebody’stemperature?

Therewasalreadyananswerfromthermodynamics,inwhichablackbodycanbemodelledasaboxwhosewallsareperfectmirrors.Electromagneticradiationbouncestoandfro,reflectedbythemirrors.Howistheenergyintheboxdistributedamongthevariousfrequencieswhenthesystemhassettledtoanequilibriumstate?In1876Boltzmannprovedthe‘equipartitiontheorem’:theenergyisapportionedequallytoeachindependentcomponentofthemotion.Thesecomponentsarejustlikethebasicwavesinaviolinstring:normalmodes.

Therewasonlyoneproblemwiththisanswer:itcouldn’tpossiblybecorrect.Itimpliedthatthetotalpowerradiatedoverallfrequenciesmustbeinfinite.Thisparadoxicalconclusionbecameknownastheultravioletcatastrophe:ultravioletbecausethatwasthebeginningofthehigh-frequencyrange,andcatastrophebecauseitwas.Norealbodycanemitaninfiniteamountofpower.

AlthoughPlanckwasawareofthisproblem,itdidn’tbotherhim,becausehedidn’tbelievetheequipartitiontheoremanyway.Ironically,hisworkresolvedtheparadoxanddidawaywiththeultravioletcatastrophe,buthenoticedthisonlylater.Heusedexperimentalobservationsofhowenergydependedonfrequency,andfittedamathematicalformulatothedata.Hisformula,derivedearlyin1900,didnotinitiallyhaveanyphysicalbasis.Itwasjustaformulathatworked.Butlaterthesameyearhetriedtoreconcilehisformulawiththeclassicalthermodynamicone,anddecidedthattheenergylevelsoftheblackbody’svibrationalmodescouldnotformacontinuum,asthermodynamicsassumed.Instead,theselevelshadtobediscrete–separatedbytinygaps.Infact,foranygivenfrequency,theenergyhadtobeanintegermultipleofthatfrequency,multipliedbyaverytinyconstant.WenowcallthisnumberPlanck’sconstantanddenoteitbyh.Itsvalue,inunitsofjouleseconds,is6.62606957(29)×10−34,wherethefiguresinbracketsmaybeinaccurate.ThisvalueisdeducedfromtheoreticalrelationshipsbetweenPlanck’sconstantandotherquantitiesthatareeasiertomeasure.ThefirstsuchmeasurementwasmadebyRobertMillikanusingthephotoelectriceffect,describedbelow.Thetinypacketsofenergyarenowcalledquanta(pluralofquantum),fromtheLatinquantus,‘howmuch.’

Planck’sconstantmaybetiny,butifthesetofenergylevelsforagivenfrequencyisdiscrete,thetotalenergyturnsouttobefinite.Sotheultravioletcatastrophewasasignthatacontinuummodelfailedtoreflectnature.Andthatimpliedthatnature,onverysmallscales,mustbediscrete.Initiallythisdidn’toccurtoPlanck:hethoughtofhisdiscreteenergylevelsasamathematicaltricktogetasensibleformula.Infact,Boltzmannhadentertainedasimilarideain1877,butdidn’tgetanywherewithit.EverythingchangedwhenEinstein

broughthisfertileimaginationtobear,andphysicsenteredanewrealm.In1905,thesameyearashisworkonspecialrelativity,heinvestigatedthephotoelectriceffect,inwhichlighthittingasuitablemetalcausesittoemitelectrons.ThreeyearsearlierPhilippLenardhadnoticedthatwhenthelighthasahigherfrequency,theelectronshavehigherenergies.Butthewavetheoryoflight,amplyconfirmedbyMaxwell,impliesthattheenergyoftheelectronsshoulddependontheintensityofthelight,notonitsfrequency.EinsteinrealisedthatPlanck’squantawouldexplainthediscrepancy.Hesuggestedthatlight,ratherthanbeingawave,wascomposedoftinyparticles,nowcalledphotons.Theenergyinasinglephoton,ofagivenfrequency,shouldbethefrequencytimesPlanck’sconstant–justlikeoneofPlanck’squanta.Aphotonwasaquantumoflight.

There’sanobviousproblemwithEinstein’stheoryofthephotoelectriceffect:itassumeslightisaparticle.Buttherewasabundantevidencethatlightwasawave.Ontheotherhand,thephotoelectriceffectwasincompatiblewithlightbeingawave.Sowaslightawave,oraparticle?

Yes.Itwas–orhadaspectsthatmanifestedthemselvesas–either.Insome

experiments,lightseemedtobehavelikeawave.Inothers,itbehavedlikeaparticle.Asphysicistscametogripswithverysmallscalesoftheuniverse,theydecidedthatlightwasn’ttheonlythingtohavethisstrangedualnature,sometimesparticle,sometimeswave.Allmatterdid.Theycalleditwave–particleduality.ThefirstpersontograspthisdualnatureofmatterwasLouis-VictordeBroglie,in1924.HerephrasedPlanck’slawintermsnotofenergy,butofmomentum,andsuggestedthatthemomentumoftheparticleaspectandthefrequencyofthewaveaspectshouldberelated:multiplythemtogetherandyougetPlanck’sconstant.Threeyearslaterhewasprovedright,atleastforelectrons.Onetheonehand,electronsareparticles,andcanbeobservedbehavingthatway.Ontheotherhand,theydiffractlikewaves.In1988atomsofsodiumwerealsospottedbehavinglikeawave.

Matterwasneitherparticlenorwave,butabitofboth–awavicle.Severalmoreorlessintuitiveimagesofthisdualnatureofmatterwere

devised.Inone,aparticleisalocalisedclumpofwaves,knownasawavepacket,Figure54.Thepacketasawholecanbehavelikeaparticle,butsomeexperimentscanprobeitsinternalwavelikestructure.Attentionshiftedfromprovidingimagesforwaviclestosortingouthowtheybehaved.Thequest

quicklyattaineditsgoal,andthecentralequationofquantumtheoryemerged.

Fig54Wavepacket.

TheequationbearsthenameofErwinSchrödinger.In1927,buildingontheworkofseveralotherphysicists,notablyWernerHeisenberg,hewrotedownadifferentialequationforanyquantumwavefunction.Itlookedlikethis:

HereΨ(Greekcapitalpsi)istheformofthewave,tistime(so∂/∂tappliedtoΨgivesitsrateofchangewithrespecttotime),ĤisanexpressioncalledtheHamiltonianoperator,andħish/2π,wherehisPlanck’sconstant.Andi?Thatwastheweirdestfeatureofall.It’sthesquarerootofminusone(Chapter5).Schrödinger’sequationappliestowavesdefinedoverthecomplexnumbers,notjusttherealnumbersasinthefamiliarwaveequation.

Wavesinwhat?Theclassicalwaveequation(Chapter8)defineswavesinspace,anditssolutionisanumericalfunctionofspaceandtime.ThesamegoesforSchrödinger’sequation,butnowthewavefunctionΨtakescomplexvalues,notjustrealones.It’sabitlikeanoceanwavewhoseheightis2+3i.Theappearanceofiisinmanywaysthemostmysteriousandprofoundfeatureofquantummechanics.Previouslyihadturnedupinsolutionsofequations,andinmethodsforfindingthosesolutions,buthereitwaspartoftheequation,anexplicitfeatureofthephysicallaw.

Onewaytointerpretthisisthatquantumwavesarelinkedpairsofrealwaves,asifmycomplexoceanwavewerereallytwowaves,oneofheight2andtheotherofheight3,withthetwodirectionsofheightatrightanglestoeachother.Butit’snotquitethatstraightforward,becausethetwowavesdon’thaveafixedshape.Astimepasses,theycyclethroughawholeseriesofshapes,andeachismysteriouslylinkedtotheother.It’sabitliketheelectricandmagnetic

componentsofalightwave,butnowelectricitycananddoes‘rotate’intomagnetism,andconversely.Thetwowavesaretwofacetsofasingleshape,whichspinssteadilyaroundtheunitcircleinthecomplexplane.Boththerealandtheimaginarypartsofthisrotatingshapechangeinaveryspecificway:theyarecombinedinsinusoidallyvaryingamounts.Mathematicallythisleadstotheideathataquantumwavefunctionhasaspecialkindofphase.Thephysicalinterpretationofthatphaseissimilarto,butdifferentfrom,theroleofphaseintheclassicalwaveequation.

RememberhowFourier’stricksolvesboththeheatequationandthewaveequation?Somespecialsolutions,Fourier’ssinesandcosines,haveespeciallypleasantmathematicalproperties.Allothersolutions,howevercomplicated,aresuperpositionsofthesenormalmodes.WecansolveSchrödinger’sequationusingasimilaridea,butnowthebasicpatternsaremorecomplicatedthansinesandcosines.Theyarecalledeigenfunctions,andtheycanbedistinguishedfromallothersolutions.Insteadofbeingsomegeneralfunctionofbothspaceandtime,aneigenfunctionisafunctiondefinedonlyonspace,multipliedbyonedependingonlyontime.Thespaceandtimevariables,inthejargon,areseparable.TheeigenfunctionsdependontheHamiltonianoperator,whichisamathematicaldescriptionofthephysicalsystemconcerned.Differentsystems–anelectroninapotentialwell,apairofcollidingphotons,whatever–havedifferentHamiltonianoperators,hencedifferenteigenfunctions.

Forsimplicity,considerastandingwavefortheclassicalwaveequation–avibratingviolinstring,whoseendsarepinneddown.Atallinstantsoftime,theshapeofthestringisalmostthesame,buttheamplitudeismodulated:multipliedbyafactorthatvariessinusoidallywithtime,asinFigure35(page138).Thecomplexphaseofaquantumwavefunctionissimilar,buthardertovisualise.Foranyindividualeigenfunction,theeffectofthequantumphaseisjustashiftofthetimecoordinate.Forasuperpositionofseveraleigenfunctions,yousplitthewavefunctionintothesecomponents,factoreachintoapurelyspatialparttimesapurelytemporalone,spinthetemporalpartroundtheunitcircleinthecomplexplaneattheappropriatespeed,andaddthepiecesbacktogether.Eachseparateeigenfunctionhasacomplexamplitude,andthismodulatesatitsownparticularfrequency.

Itmaysoundcomplicated,butitwouldbecompletelybafflingifyoudidn’tsplitthewavefunctionintoeigenfunctions.Atleastthenyou’vegotachance.

Despitethesecomplexities,quantummechanicswouldbejustafancyversionoftheclassicalwaveequation,resultingintwowavesratherthanone,wereitnot

forapuzzlingtwist.Youcanobserveclassicalwaves,andseewhatshapetheyare,eveniftheyaresuperpositionsofseveralFouriermodes.Butinquantummechanics,youcanneverobservetheentirewavefunction.Allyoucanobserveonanygivenoccasionisasinglecomponenteigenfunction.Roughlyspeaking,ifyouattempttomeasuretwoofthesecomponentsatthesametime,themeasurementprocessononeofthemdisturbstheotherone.

Thisimmediatelyraisesadifficultphilosophicalissue.Ifyoucan’tobservetheentirewavefunction,doesitactuallyexist?Isitagenuinephysicalobject,orjustaconvenientmathematicalfiction?Isanunobservablequantityscientificallymeaningful?ItisherethatSchrödinger’scelebratedfelineentersthestory.Itarisesbecauseofastandardwaytointerpretwhataquantummeasurementis,calledtheCopenhageninterpretation.1

Imagineaquantumsysteminsomesuperposedstate:say,anelectronwhosestateisamixtureofspin-upandspin-down,whicharepurestatesdefinedbyeigenfunctions.(Itdoesn’tmatterwhatspin-upandspin-downmean.)Whenyouobservethestate,however,youeithergetspin-up,oryougetspin-down.Youcan’tobserveasuperposition.Moreover,onceyou’veobservedoneofthese–sayspin-up–thatbecomestheactualstateoftheelectron.Somehowyourmeasurementseemstohaveforcedthesuperpositiontochangeintoaspecificcomponenteigenfunction.ThisCopenhageninterpretationtakesthisstatementliterally:yourmeasurementprocesshascollapsedtheoriginalwavefunctionintoasinglepureeigenfunction.

Ifyouobservealotofelectrons,sometimesyougetspin-up,sometimesspin-down.Youcaninfertheprobabilitythattheelectronisinoneofthosestates.Sothewavefunctionitselfcanbeinterpretedasakindofprobabilitycloud.Itdoesn’tshowtheactualstateoftheelectron:itshowshowprobableitisthatwhenyoumeasureit,yougetaparticularresult.Butthatmakesitastatisticalpattern,notarealthing.ItnomoreprovesthewavefunctionisrealthanQuetelet’smeasurementsofhumanheightprovethatadevelopingembryopossessessomesortofbellcurve.

TheCopenhageninterpretationisstraightforward,reflectswhathappensinexperiments,andmakesnodetailedassumptionsaboutwhathappenswhenyouobserveaquantumsystem.Forthesereasons,mostworkingphysicistsareveryhappytouseit.Butsomewerenot,intheearlydayswhentheytheorywasstillbeingthrashedout,andsomestillarenot.AndoneofthedissenterswasSchrödingerhimself.

In1935,SchrödingerwasworryingabouttheCopenhageninterpretation.Hecouldseethatitworked,onapragmaticlevel,forquantumsystemslikeelectronsandphotons.Buttheworldaroundhim,eventhoughdeepdowninsideitwasjustaseethingmassofquantumparticles,seemeddifferent.Seekingawaytomakethedifferenceasglaringashecould,Schrödingercameupwithathoughtexperimentinwhichaquantumparticlehadadramaticandobviouseffectonacat.

Imagineabox,whichwhenshutisimpervioustoallquantuminteractions.Insideit,placeanatomofradioactivematter,aradiationdetector,aflaskofpoison,andalivecat.Nowshutthebox,andwait.Atsomepointtheradioactiveatomwilldecay,andemitaparticleofradiation.Thedetectorwillspotit,andisriggedsothatwhenitdoesso,itcausestheflasktobreakandreleasethepoisoninside.Thiskillsthecat.

Inquantummechanics,thedecayofaradioactiveatomisarandomevent.Fromoutside,noobservercantellwhethertheatomhasdecayedornot.Ifithas,thecatisdead;ifnot,it’salive.AccordingtotheCopenhageninterpretation,untilsomeoneobservestheatom,itisinasuperpositionoftwoquantumstates:decayedandnotdecayed.Thesamegoesforthestatesofthedetector,theflask,andthecat.Sothecatisinasuperpositionoftwostates:deadandalive.

Sincetheboxisimpervioustoallquantuminteractions,theonlywaytofindoutwhethertheatomhasdecayedandkilledthecatistoopenthebox.TheCopenhageninterpretationtellsusthattheinstantwedothis,thewavefunctionscollapseandthecatsuddenlyswitchestoapurestate:eitherdead,oralive.However,theinsideoftheboxisnodifferentfromtheexternalworld,whereweneverobserveacatthatisinasuperposedalive/deadstate.Sobeforeweopentheboxandobserveitscontents,theremusteitherbeadeadcatinside,oraliveone.

SchrödingerintendedthisthoughtexperimentasacriticismoftheCopenhageninterpretation.Microscopicquantumsystemsobeythesuperpositionprincipleandcanexistinmixedstates;macroscopiconescan’t.Bylinkingamicroscopicsystem,theatom,toamacroscopicone,thecat,SchrödingerwaspointingoutwhathebelievedtobeaflawintheCopenhageninterpretation:itgavenonsensewhenappliedtoacat.Hemusthavebeenstartledwhenthemajorityofphysicistsresponded,ineffect:‘Yes,Erwin,you’reabsolutelyright:untilsomeoneopensthebox,thecatreallyissimultaneouslydeadandalive.’Especiallywhenitdawnedonhimthathecouldn’tfindoutwhowasright,evenifheopenedthebox.Hewouldobserveeitheralivecatora

deadone.Hemightinferthatthecathadbeeninthatstatebeforeheopenedthebox,buthecouldn’tbesure.TheobservableresultwasconsistentwiththeCopenhageninterpretation.

Verywell:addafilmcameratothecontentsofthebox,andfilmwhatactuallyhappens.Thatwilldecidethematter.‘Ah,no,’thephysicistsreplied.‘Youcanonlyseewhatthecamerahasfilmedafteryouopenthebox.Beforethat,thefilmisinasuperposedstate:containingamovieofalivecat,andcontainingamovieofadeadone.’

TheCopenhageninterpretationfreedupphysiciststodotheircalculationsandsortoutwhatquantummechanicspredicted,withoutfacinguptothedifficult,ifnotimpossible,issueofhowtheclassicalworldemergedfromaquantumsubstrate–howamacroscopicdevice,unimaginablycomplexonaquantumscale,somehowmadeameasurementofaquantumstate.SincetheCopenhageninterpretationdidthejob,theyweren’treallyinterestedinphilosophicalquestions.SogenerationsofphysicistsweretaughtthatSchrödingerhadinventedhiscattoshowthatquantumsuperpositionextendedintothemacroscopicworldtoo:theexactoppositeofwhatSchrödingerhadbeentryingtotellthem.

It’snotreallyagreatsurprisethatmatterbehavesstrangelyonthelevelofelectronsandatoms.Wemayinitiallyrebelatthethought,outofunfamiliarity,butifanelectronisreallyatinyclumpofwavesratherthanatinyclumpofstuff,wecanlearntolivewithit.Ifthatmeansthatthestateoftheelectronisitselfabitweird,spinningnotjustaboutanupaxisoradownaxisbutabitofboth,wecanlivewiththattoo.Andifthelimitationsofourmeasuringdevicesimplythatwecannevercatchtheelectrondoingthatkindofthing–thatanymeasurementwemakenecessarilysettlesforsomepurestate,upordown–thenthat’showitis.Ifthesameappliestoaradioactiveatom,andthestatesare‘decayed’or‘notdecayed’,becauseitscomponentparticleshavestatesaselusiveasthoseoftheelectron,wecanevenacceptthattheatomitself,initsentirety,maybeinasuperpositionofthosestatesuntilwemakeameasurement.Butacatisacat,anditseemstobeaverybigstretchoftheimaginationtoimaginethattheanimalcanbebothaliveanddeadatthesametime,onlytomiraculouslycollapseintooneortheotherwhenweopentheboxthatcontainsit.Ifquantumrealityrequiresasuperposedalive/deadcat,whyisitsoshythatitwon’tletusobservesuchastate?

Therearesoundreasonsintheformalismofquantumtheorythat(untilveryrecently)requireanymeasurement,any‘observable’,tobeaneigenfunction.

Thereareevensounderreasonswhythestateofaquantumsystemshouldbeawave,obeyingSchrödinger’sequation.Howcanyougetfromonetotheother?TheCopenhageninterpretationdeclaresthatsomehow(don’taskhow)themeasurementprocesscollapsesthecomplex,superposedwavefunctiondowntoasinglecomponenteigenfunction.Havingbeenprovidedwiththisformofwords,yourtaskasaphysicististogetonwithmakingmeasurementsandcalculatingeigenfunctionsandsoon,andstopaskingawkwardquestions.Itworksamazinglywell,ifyoumeasuresuccessbygettinganswersthatagreewithexperiment.AndeverythingwouldhavebeenfineifSchrödinger’sequationpermittedthewavefunctiontobehaveinthismanner,butitdoesn’t.InTheHiddenRealityBrianGreeneputsitthisway:‘Evenpoliteproddingrevealsanuncomfortablefeature…Theinstantaneouscollapseofawave…can’tpossibleemergefromSchrödinger’smath.’Instead,theCopenhageninterpretationwasapragmaticbolt-ontothetheory,awaytohandlemeasurementswithoutunderstandingorfacinguptowhattheyreallywere.

Thisisallverywell,butit’snotwhatSchrödingerwastryingtopointout.Heintroducedacat,ratherthananelectronoranatom,becauseitputwhatheconsideredtobethemainissueinsharprelief.Acatbelongstothemacroscopicworldinwhichwelive,inwhichmatterdoesnotbehavethewayquantummechanicsdemands.Wedonotseesuperposedcats.2Schrödingerwasaskingwhyourfamiliar‘classical’universefailstoresembletheunderlyingquantumreality.Ifeverythingfromwhichtheworldisbuiltcanexistinsuperposedstates,whydoestheuniverselookclassical?ManyphysicistshaveperformedwonderfulexperimentsshowingthatelectronsandatomsreallydobehavethewayquantumandCopenhagensaytheyshould.Butthismissesthepoint:youhavetodoitwithacat.Theoristswonderedwhetherthecatcouldobserveitsownstate,orwhethersomeoneelsecouldsecretlyopentheboxandwritedownwhatwasinside.Theyconcluded,followingthesamelogicasSchrödinger,thatifthecatobserveditsstatethentheboxcontainedasuperpositionofadeadcatthathadcommittedsuicidebyobservingitself,andalivecatthathadobserveditselftobealive,untilthelegitimateobserver(aphysicist)openedthebox.Thenthewholeshebangcollapsedtooneortheother.Similarlythefriendbecameasuperpositionoftwofriends:oneofwhomhadseenadeadcatwhiletheotherhadseenaliveone,untilaphysicistopenedthebox,causingthefriend’sstatetocollapse.Youcouldproceedinthiswayuntilthestateoftheentireuniversewasasuperpositionofauniversewithadeadcatandauniversewithaliveone,andthenthestateoftheuniversecollapsedwhenaphysicistopenedthebox.

Itwasallabitembarrassing.Physicistscouldgetonwiththeirworkwithoutsortingitout,theycouldevendenytherewasanythingtobesortedout,butsomethingwasmissing.Forexample,whathappenstousifanalienphysicistontheplanetApellobetneesIIIopensabox?DowesuddenlydiscoverweactuallyblewourselvesupinanuclearwarwhentheCubanmissilecrisisof1962escalated,andhavebeenlivingonborrowedtimeeversince?

Themeasurementprocessisnottheneat,tidymathematicaloperationthattheCopenhageninterpretationassumes.Whenaskedtodescribehowtheapparatuscomestoitsdecision,theCopenhageninterpretationreplies‘itjustdoes’.Theimageofthewavefunctioncollapsingtoasingleeigenfunctiondescribestheinputandtheoutputofthemeasurementprocess,butnothowtogetfromonetotheother.Butwhenyoumakearealmeasurementyoudon’tjustwaveamagicwandandcausethewavefunctiontodisobeySchrödinger’sequationandcollapse.Instead,youdosomethingsoenormouslycomplicated,fromaquantumviewpoint,thatitisobviouslyhopelesstomodelitrealistically.Tomeasureanelectron’sspin,forexample,youmakeitinteractwithasuitablepieceofapparatus,whichhasapointerthateithermovestothe‘up’positionorthe‘down’one.Oranumericaldisplay,orasignalsenttoacomputer…Thisdeviceyieldsonestate,andonestateonly.Youdon’tseethepointerinasuperpositionofupanddown.

Weareusedtothis,becausethat’showtheclassicalworldworks.Butunderneathit’ssupposedtobeaquantumworld.Replacethecatwiththespinapparatus,anditshouldindeedexistinasuperposedstate.Theapparatus,viewedasaquantumsystem,isextraordinarilycomplicated.Itcontainsgazillionsofparticles–between1025and1030,ataroughestimate.Themeasurementemergessomehowfromtheinteractionofthatsingleelectronwiththesegazillionparticles.Admirationfortheexpertiseofthecompanythatmanufacturestheinstrumentmustbeboundless;toextractanythingsensiblefromsomethingsomessyisalmostbeyondbelief.It’sliketryingtoworkoutsomeone’sshoesizebymakingthempassthroughacity.Butifyou’reclever(arrangeforthemtoencounterashoeshop)youcangetasensibleresult,andacleverinstrumentdesignercanproducemeaningfulmeasurementsofelectronspin.Butthere’snorealisticprospectofmodellingindetailhowsuchadeviceworksasabonafidequantumsystem.There’stoomuchdetail,thebiggestcomputerintheworldwouldflounder.ThatmakesitdifficulttoanalysearealmeasurementprocessusingSchrödinger’sequation.

Evenso,wedohavesomeunderstandingofhowourclassicalworldemergesfromanunderlyingquantumone.Let’sstartwithasimpleversion,arayoflight

hittingamirror.Theclassicalanswer,Snell’slaw,statesthatthereflectedraybouncesoffatthesameangleastheonethathit.InhisbookQEDonquantumelectrodynamics,thephysicistRichardFeynmanexplainedthatthisisnotwhathappensinthequantumworld.Therayisreallyastreamofphotons,andeachphotoncanbounceallovertheplace.However,ifyousuperposeallthepossiblethingsthephotoncoulddo,thenyougetSnell’slaw.Theoverwhelmingproportionofphotonsbouncebackatanglesveryclosetotheoneatwhichtheyhit.Feynmanevenmanagedtoshowwhywithoutusinganycomplicatedmathematics,butbehindthiscalculationisageneralmathematicalidea:theprincipleofstationaryphase.Ifyousuperposeallquantumstatesforanopticalsystem,yougettheclassicaloutcomeinwhichlightraysfollowtheshortestpath,measuredbytimetaken.Youcanevenaddbellsandwhistlestodecoratetheraypathswithclassicalwave-opticaldiffractionfringes.

Thisexampleshows,veryexplicitly,thatthesuperpositionofallpossibleworlds–inthisopticalframework–yieldstheclassicalworld.Themostimportantfeatureisnotsomuchthedetailedgeometryofthelightray,butthefactthatityieldsonlyoneworldattheclassicallevel.Downinthequantumdetailsofindividualphotons,youcanobservealltheparaphernaliaofsuperposition,eigenfunctions,andsoon.Butupatthehumanscale,allthatcancelsout–well,addstogether–toproduceaclean,classicalworld.

Theotherpartoftheexplanationiscalleddecoherence.We’veseenthatquantumwaveshaveaphaseaswellasanamplitude.It’saveryfunnyphase,acomplexnumber,butit’saphasenonetheless.Thephaseisabsolutelycrucialtoanysuperposition.Ifyoutaketwosuperposedstates,changethephaseofone,andaddthembacktogether,whatyougetisnothingliketheoriginal.Ifyoudothesamewithalotofcomponents,thereassembledwavecanbealmostanything.LossofphaseinformationwrecksanySchrödinger’scat-likesuperposition.Youdon’tjustlosetrackofwhetherit’saliveordead:youcan’ttellitwasacat.Whenquantumwavesceasetohavenicephaserelations,theydecohere–theystarttobehavemorelikeclassicalphysics,andsuperpositionsloseanymeaning.Whatcausesthemtodecohereisinteractionswithsurroundingparticles.Thisispresumablyhowapparatuscanmeasureelectronspinandgetaspecific,uniqueresult.

Bothoftheseapproachesleadtothesameconclusion:classicalphysicsiswhatyouobserveifyoutakeahuman-scaleviewofaverycomplicatedquantumsystemwithgazillionsofparticles.Specialexperimentalmethods,specialdevices,mightpreservesomeofthequantumeffects,makingthempokeupintoourcomfortableclassicalexistence,butgenericquantumsystemsquicklycease

toappearquantumaswemovetolargerscalesofbehaviour.

That’sonewaytoresolvethefateofthepoorcat.Onlyiftheboxistotallyimpervioustoquantumdecoherencecantheexperimentproducethesuperposedcat,andnosuchboxexists.Whatwouldyoumakeitfrom?

Butthere’sanotherway,onethatgoestotheoppositeextreme.EarlierIsaidthat‘Youcouldproceedinthiswayuntilthestateoftheentireuniversewasasuperposition.’In1957HughEverettJr.pointedoutthatinasense,youhaveto.Theonlywaytoprovideanaccuratequantummodelofasystemistoconsideritswavefunction.Everyonewashappytodosowhenthesystemwasanelectron,oranatom,or(morecontroversially)acat.Everetttookthesystemtobetheentireuniverse.

Hearguedthatyouhadnochoiceifthat’swhatyouwantedtomodel.Nothinglessthantheuniversecanbetrulyisolated.Everythinginteractswitheverythingelse.Andhediscoveredthatifyoutookthatstep,thentheproblemofthecat,andtheparadoxicalrelationbetweenquantumandclassicalreality,iseasilyresolved.Thequantumwavefunctionoftheuniverseisnotapureeigenmode,butasuperpositionofallpossibleeigenmodes.Althoughwecan’tcalculatesuchthings(wecan’tforacat,andauniverseisatadmorecomplicated)wecanreasonaboutthem.Ineffect,wearerepresentingtheuniverse,quantum-mechanically,asacombinationofallthepossiblethingsthatauniversecando.

Theupshotwasthatthewavefunctionofthecatdoesnothavetocollapsetogiveasingleclassicalobservation.Itcanremaincompletelyunchanged,withnoviolationofSchrödinger’sequation.Instead,therearetwocoexistinguniverses.Inone,thecatdied;intheother,itdidn’t.Whenyouopenthebox,therearecorrespondinglytwoyousandtwoboxes.Oneofthemispartofthewavefunctionofauniversewithadeadcat;theotherispartofadifferentwavefunctionwithalivecat.Inplaceofauniqueclassicalworldthatsomehowemergesfromthesuperpositionofquantumpossibilities,wehaveavastrangeofclassicalworlds,eachcorrespondingtoonequantumpossibility.

Everett’soriginalversion,whichhecalledtherelativestateformulation,cametopopularattentioninthe1970sthroughBryceDeWitt,whogaveitamorecatchyname:themany-worldsinterpretationofquantummechanics.Itisoftendramatisedinhistoricalterms:forexample,thatthereisauniverseinwhichAdolfHitlerwonWorldWarII,andanotheroneinwhichhedidn’t.TheoneinwhichIamwritingthisbookisthelatter,butsomewherealongsideitin

thequantumrealmanotherIanStewartiswritingabookverysimilartothisone,butinGerman,remindinghisreadersthattheyareintheuniversewhereHitlerwon.Mathematically,Everett’sinterpretationcanbeviewedasalogicalequivalentofconventionalquantummechanics,anditleads–inmorelimitedinterpretations–toefficientwaystosolvephysicsproblems.Hisformalismwillthereforesurviveanyexperimentaltestthatconventionalquantummechanicssurvives.Sodoesthatimplythattheseparalleluniverses,‘alternateworlds’intransatlanticparlance,reallyexist?IsanothermetypingawayhappilyonacomputerkeyboardinaworldwhereHitlerwon?Oristheset-upaconvenientmathematicalfiction?

Thereisanobviousproblem:howcanwebesurethatinaworlddominatedbyHitler’sdream,theThousandYearReich,computersliketheoneI’musingwouldexist?Clearlytheremustbemanymoreuniversesthantwo,andeventsinthemmustfollowsensibleclassicalpatterns.SomaybeStewart-2doesn’texistbutHitler-2does.Acommondescriptionoftheformationandevolutionofparalleluniversesinvolvesthem‘splittingoff’wheneverthereisachoiceofquantumstate.Greenepointsoutthatthisimageiswrong:nothingsplits.Theuniverse’swavefunctionhasbeen,andalwayswillbe,split.Itscomponenteigenfunctionsarethere:weimagineasplitwhenweselectoneofthem,butthewholepointofEverett’sexplanationisthatnothinginthewavefunctionactuallychanges.

Withthatasacaveat,asurprisingnumberofquantumphysicistsacceptthemany-worldsinterpretation.Schrödinger’scatreallyisaliveanddead.Hitlerreallydidwinandlose.Oneversionofuslivesinoneofthoseuniverses,othersdonot.That’swhatthemathematicssays.It’snotaninterpretation,aconvenientwaytoarrangethecalculations.It’sasrealasyouandI.ItisyouandI.

I’mnotconvinced.It’snotthesuperpositionthatbothersme,though.Idon’tfindtheexistenceofaparallelNaziworldunthinkable,orimpossible.3ButIdoobject,strenuously,totheideathatyoucanseparateaquantumwavefunctionaccordingtohuman-scalehistoricalnarratives.Themathematicalseparationoccursatthelevelofquantumstatesofconstituentparticles.Mostcombinationsofparticlestatesmakenosensewhatsoeverasahumannarrative.Asimplealternativetoadeadcatisnotalivecat.Itisadeadcatwithoneelectroninadifferentstate.Complexalternativesarefarmorenumerousthanalivecat.Theyincludeacatthatsuddenlyexplodesfornoapparentreason,onethatturnsintoaflowervase,onethatgetselectedpresidentoftheUnitedStates,andonethatsurvivedeventhoughtheradioactiveatomreleasedthepoison.Thosealternativecatsarerhetoricallyusefulbutunrepresentative.Mostalternativesarenotcatsat

all;infact,theyareindescribableinclassicalterms.Ifso,mostofthealternativeStewartsaren’trecognisableaspeople–indeedasanything–andalmostallofthosethatexistdosowithinaworldthatmakesabsolutelynosenseinhumanterms.Sothechancethatanotherversionoflittleoldmehappenstoliveinanotherworldthatmakesnarrativesensetoahumanbeingisnegligible.

Theuniversemaywellbeanincrediblycomplexsuperpositionofalternativestates.Ifyouthinkquantummechanicsisbasicallyright,ithastobe.In1983thephysicistStephenHawkingsaidthatthemany-worldsinterpretationwas‘self-evidentlycorrect’inthissense.Butitdoesn’tfollowthatthereexistsasuperpositionofuniversesinwhichacatisaliveordead,andHitlerdidordidnotwin.Thereisnoreasontosupposethatthemathematicalcomponentscanbeseparatedintosetsthatfittogethertocreatehumannarratives.Hawkingdismissednarrativeinterpretationsofthemany-worldsformalism,saying‘Allthatonedoes,really,istocalculateconditionalprobabilities–inotherwords,theprobabilityofAhappening,givenB.Ithinkthatthat’sallthemany-worldsinterpretationis.Somepeopleoverlayitwithalotofmysticismaboutthewavefunctionsplittingintodifferentparts.Butallthatyou’recalculatingisconditionalprobabilities.’

It’sworthcomparingtheHitlertalewithFeynman’sstoryofthelightray.InthestyleofalternativeHitlers,Feynmanwouldbetellingusthatthereisoneclassicalworldwherethelightraybouncesoffthemirroratthesameangleatwhichithit,anotherclassicalworldinwhichitbouncesatananglethat’sonedegreewrong,anotherwhereit’stwodegreeswrong,andsoon.Buthedidn’t.Hetoldusthatthereisoneclassicalworld,emergingfromthesuperpositionofthequantumalternatives.Theremaybeinnumerableparallelworldsatthequantumlevel,butthesedonotcorrespondinanymeaningfulwaytoparallelworldsthataredescribableattheclassicallevel.Snell’slawisvalidinanyclassicalworld.Ifitweren’t,theworldcouldn’tbeclassical.AsFeynmanexplainedforlightrays,theclassicalworldemergeswhenyousuperposeallofthequantumalternatives.Thereisonlyonesuchsuperposition,sothereisonlyoneclassicaluniverse.Ours.

Quantummechanicsisn’tconfinedtothelaboratory.Thewholeofmodernelectronicsdependsonit.Semiconductortechnology,thebasisofallintegratedcircuits–siliconchips–isquantum-mechanical.Withoutthephysicsofthequantum,noonewouldhavedreamedthatsuchdevicescouldwork.Computers,mobilephones,CDplayers,gamesconsoles,cars,refrigerators,ovens,virtuallyallmodernhouseholdgadgets,containmemorychips,tocontaintheinstructions

thatmakethesedevicesdowhatwewant.Manycontainmorecomplexcircuitry,suchasmicroprocessors,anentirecomputeronachip.Mostmemorychipsarevariationsonthefirsttruesemiconductordevice:thetransistor.

Inthe1930s,AmericanphysicistsEugeneWignerandFrederickSeitzanalysedhowelectronsmovethoughacrystal,aproblemthatrequiresquantummechanics.Theydiscoveredsomeofthebasicfeaturesofsemiconductors.Somematerialsareconductorsofelectricity:electronscanflowthroughthemwithease.Metalsaregoodconductors,andineverydayusecopperwireiscommonplaceforthispurpose.Insulatorsdonotpermitelectronstoflow,sotheystoptheflowofelectricity:theplasticsthatsheatheelectricalwires,topreventuselectrocutingourselvesontheTVpowerlead,areinsulators.Semiconductorsareabitofboth,dependingoncircumstances.Siliconisthebestknown,andcurrentlythemostwidelyused,butseveralotherelementssuchasantimony,arsenic,boron,carbon,germanium,andseleniumarealsosemiconductors.Becausesemiconductorscanbeswitchedfromonestatetotheother,theycanbeusedtomanipulateelectricalcurrents,andthisisthebasisofallelectroniccircuits.

WignerandSeitzdiscoveredthatthepropertiesofsemiconductorsdependontheenergylevelsoftheelectronswithinthem,andtheselevelscanbecontrolledby‘doping’thebasicsemiconductormaterialbyaddingsmallquantitiesofspecificimpurities.Twoimportanttypesarep-typesemiconductors,whichcarrycurrentasaflowofelectrons,andn-typesemiconductors,inwhichcurrentflowsintheoppositedirectiontotheelectrons,carriedby‘holes’–placeswheretherearefewerelectronsthannormal.In1947JohnBardeenandWalterBrattainatBellLaboratoriesdiscoveredthatacrystalofgermaniumcouldactasanamplifier.Ifanelectricalcurrentwasfedintoit,theoutputcurrentwashigher.WilliamShockley,leaderoftheSolidStatePhysicsGroup,realisedhowimportantthiscouldbe,andinitiatedaprojecttoinvestigatesemiconductors.Outofthiscamethetransistor–shortfor‘transferresistor’.Thereweresomeearlierpatentsbutnoworkingdevicesorpublishedpapers.TechnicallytheBellLabs’devicewasaJFET(junctiongatefield-effecttransistor,Figure55).Sincethisinitialbreakthrough,manyotherkindsoftransistorhavebeeninvented.TexasInstrumentsmanufacturedthefirstsilicontransistorin1954.Thesameyearsawatransistor-basedcomputer,TRIDAC,builtbytheUSmilitary.Itwasthreecubicfeetinsizeanditspowerrequirementwasthesameasonelightbulb.ThiswasanearlystepinahugeAmericanmilitaryprogrammetodevelopalternativestovacuumtubeelectronics,whichwastoocumbersome,fragile,andunreliableformilitaryuse.

Fig55StructureofaJFET.Thesourceanddrainareattheends,inap-typelayer,whilethegateisann-typelayerthatcontrolstheflow.Ifyouthinkoftheflowofelectronsfromsourcetodrainasahose,thegateineffectsqueezesthehose,increasingthepressure(voltage)atthedrain.

Becausesemiconductortechnologyisbasedondopingsiliconorsimilarsubstanceswithimpurities,itlentitselftominiaturisation.Circuitscanbebuiltupinlayersonasiliconsubstrate,bybombardingthesurfacewiththedesiredimpurity,andetchingawayunwantedregionswithacid.Theareasaffectedaredeterminedbyphotographicallyproducedmasks,andthesecanbeshrunktoverysmallsizeusingopticallenses.Outofallthisemergedtoday’selectronics,includingmemorychipsthatcanholdbillionsofbytesofinformationandveryfastmicroprocessorsthatorchestratetheactivityofcomputers.

Anotherubiquitousapplicationofquantummechanicsisthelaser.Thisisadevicethatemitsastrongbeamofcoherentlight:oneinwhichthelightwavesareallinphasewitheachother.Itconsistsofanopticalcavitywithmirrorsateachend,filledwithsomethingthatreactstolightofaspecificwavelengthbyproducingmorelightofthesamewavelength–alightamplifier.Pumpinenergytostarttheprocessrolling,letthelightbouncetoandfroalongthecavity,amplifyingallthetime,andwhenitreachesasufficientlyhighintensity,letitout.Thegainmediumcanbeafluid,agas,acrystal,orasemiconductor.Differentmaterialsworkatdifferentwavelengths.Theamplificationprocessdependsonthequantummechanicsofatoms.Theelectronsintheatomscanexistindifferentenergystates,andtheycanbeswitchedbetweenthembyabsorbingoremittingphotons.

LASERmeanslightamplificationbystimulatedemissionofradiation.Whenthefirstlaserwasinvented,itwaswidelyderidedasananswerlookingforaproblem.Thiswasunimaginative:awholehostofsuitableproblemsquicklyappeared,oncetherewasasolution.Producingacoherentbeamoflightisbasictechnology,anditwasalwaysboundtohaveuses,justasanimprovedhammerwouldautomaticallyfindmanyuses.Wheninventinggenerictechnology,youdon’thavetohaveaspecificapplicationinmind.Todayweuselasersforsomanypurposesthatit’simpossibletolistthemall.ThereareprosaicuseslikelaserpointersforlecturesandlaserbeamsforDIY.CDplayers,DVDplayers,andBlu-rayalluselaserstoreadinformationfromtinypitsormarksondiscs.Surveyorsuselaserstomeasuredistancesandangles.AstronomersuselaserstomeasurethedistancefromtheEarthtotheMoon.Surgeonsuselasersforfinecuttingofdelicatetissues.Lasertreatmentofeyesisroutine,forrepairingdetachedretinasandremouldingthesurfaceofthecorneatocorrectvisioninsteadofusingglassesorcontactlenses.The‘StarWars’antimissilesystemwasintendedtousepowerfullaserstoshootdownenemymissiles,andalthoughitwasneverbuilt,someofthelaserswere.Militaryusesoflasers,akintothepulpscience-fictionray-gun,arebeinginvestigatedrightnow.AnditmayevenbepossibletolaunchspacevehiclesfromEarthbymakingthemrideapowerfullaserbeam.

Newusesofquantummechanicsarrivealmostbytheweek.Oneofthelatestisquantumdots,tinypiecesofsemiconductorwhoseelectronicproperties,includingthelightthattheyemit,varyaccordingtotheirsizeandshape.Theycanthereforebetailoredtohavemanydesirablefeatures.Theyalreadyhaveavarietyofapplications,includingbiologicalimaging,wheretheycanreplacetraditional(andoftentoxic)dyes.Theyalsoperformmuchbetter,emittingbrighterlight.

Furtherdowntheline,someengineersandphysicistsareworkingonthebasiccomponentsofaquantumcomputer.Insuchadevice,thebinarystatesof0and1canbesuperposedinanycombination,ineffectallowingcomputationstoassumebothvaluesatthesametime.Thiswouldallowmanydifferentcalculationstobeperformedinparallel,speedingthemupenormously.Theoreticalalgorithmshavebeendevised,carryingoutsuchtasksassplittinganumberintoitsprimefactors.Conventionalcomputersrunintotroublewhenthenumbershavemorethanahundreddigitsorso,butaquantumcomputershouldbeabletofactorisemuchbiggernumberswithease.Themainobstacletoquantumcomputingisdecoherence,whichdestroyssuperposedstates.Schrödinger’scatisexactingrevengeforitsinhumanetreatment.

15Codes,communications,andcomputers

InformationTheory

Whatdoesitsay?

Itdefineshowmuchinformationamessagecontains,intermsoftheprobabilitieswithwhichthesymbolsthatmakeituparelikelytooccur.

Whyisthatimportant?

Itistheequationthatusheredintheinformationage.Itestablishedlimitsontheefficiencyofcommunications,allowingengineerstostoplookingforcodesthatweretooeffectivetoexist.Itisbasictotoday’sdigitalcommunications–phones,CDs,DVDs,theinternet.

Whatdiditleadto?

Efficienterror-detectinganderror-correctingcodes,usedineverythingfromCDstospaceprobes.Applicationsincludestatistics,artificialintelligence,cryptography,andextractingmeaningfromDNAsequences.

In1977NASAlaunchedtwospaceprobes,Voyager1and2.TheplanetsoftheSolarSystemhadarrangedthemselvesinunusuallyfavourablepositions,makingitpossibletofindreasonablyefficientorbitsthatwouldlettheprobesvisitseveralplanets.TheinitialaimwastoexamineJupiterandSaturn,butiftheprobesheldout,theirtrajectorieswouldtakethemonpastUranusandNeptune.Voyager1couldhavegonetoPluto(atthattimeconsideredaplanet,andequallyinteresting–indeedtotallyunchanged–nowthatit’snot)butanalternative,Saturn’sintriguingmoonTitan,tookprecedence.Bothprobeswerespectacularlysuccessful,andVoyager1isnowthemostdistanthuman-madeobjectfromEarth,morethan10billionmilesawayandstillsendingbackdata.

Signalstrengthfallsoffwiththesquareofthedistance,sothesignalreceivedonEarthis10–20timesthestrengththatitwouldbeifreceivedfromadistanceofonemile.Thatis,onehundredquintilliontimesweaker.Voyager1musthaveareallypowerfultransmitter…No,it’satinyspaceprobe.Itispoweredbyaradioactiveisotope,plutonium-238,butevensothetotalpoweravailableisnowaboutoneeighththatofatypicalelectrickettle.Therearetworeasonswhywecanstillobtainusefulinformationfromtheprobe:powerfulreceiversonEarth,andspecialcodesusedtoprotectthedatafromerrorscausedbyextraneousfactorssuchasinterference.Voyager1cansenddatausingtwodifferentsystems.One,thelow-rate

channel,cansend40binarydigits,0sor1s,everysecond,butitdoesnotallowcodingtodealwithpotentialerrors.Theother,thehigh-ratechannel,cantransmitupto120,000binarydigitseverysecond,andtheseareencodedsothaterrorscanbespottedandputrightprovidedthey’renottoofrequent.Thepricepaidforthisabilityisthatthemessagesaretwiceaslongastheywouldotherwisebe,sotheycarryonlyhalfasmuchdataastheycould.Sinceerrorscouldruinthedata,thisisapriceworthpaying.

Codesofthiskindarewidelyusedinallmoderncommunications:spacemissions,landlinephones,mobilephones,theInternet,CDsandDVDs,Blu-ray,andsoon.Withoutthem,allcommunicationswouldbeliabletoerrors;thiswouldnotbeacceptableif,forinstance,youwereusingtheInternettopayabill.Ifyourinstructiontopay£20wasreceivedas£200,youwouldn’tbepleased.ACDplayerusesatinylens,whichfocusesalaserbeamontoverythintracksimpressedinthematerialofthedisc.Thelenshoversaverytinydistanceabovethespinningdisc.YetyoucanlistentoaCDwhiledrivingalongabumpyroad,

becausethesignalisencodedinawaythatallowserrorstobefoundandputrightbytheelectronicswhilethediscisbeingplayed.Thereareothertricks,too,butthisoneisfundamental.

Ourinformationagereliesondigitalsignals–longstringsof0sand1s,pulsesandnon-pulsesofelectricityorradio.Theequipmentthatsends,receives,andstoresthesignalsreliesonverysmall,verypreciseelectroniccircuitsontinysliversofsilicon–‘chips’.Butforalltheclevernessofthecircuitdesignandmanufacture,noneofitwouldworkwithouterror-detectinganderror-correctingcodes.Anditwasinthiscontextthattheterm‘information’ceasedtobeaninformalwordfor‘know-how’,andbecameameasurablenumericalquantity.Andthatprovidedfundamentallimitationsontheefficiencywithwhichcodescanmodifymessagestoprotectthemagainsterrors.Knowingtheselimitationssavedengineerslotsofwastedtime,tryingtoinventcodesthatwouldbesoefficientthey’dbeimpossible.Itprovidedthebasisfortoday’sinformationculture.

I’moldenoughtorememberwhentheonlywaytotelephonesomeoneinanothercountry(shockhorror)wastomakeabookingaheadoftimewiththephonecompany–intheUKtherewasonlyone,PostOfficeTelephones–foraspecifictimeandlength.Saytenminutesat3.45p.m.on11January.Anditcostafortune.AfewweeksagoafriendandIdidanhour-longinterviewforasciencefictionconventioninAustralia,fromtheUnitedKingdom,usingSkypeTM.Itwasfree,anditsentvideoaswellassound.Alothaschangedinfiftyyears.Nowadays,weexchangeinformationonlinewithfriends,bothrealonesandthephoniesthatlargenumbersofpeoplecollectlikebutterfliesusingsocialnetworkingsites.WenolongerbuymusicCDsormovieDVDs:webuytheinformationthattheycontain,downloadedovertheInternet.Booksareheadingthesameway.Marketresearchcompaniesamasshugequantitiesofinformationaboutourpurchasinghabitsandtrytouseittoinfluencewhatwebuy.Eveninmedicine,thereisagrowingemphasisontheinformationthatiscontainedinourDNA.Oftentheattitudeseemstobethatifyouhavetheinformationrequiredtodosomething,thenthatalonesuffices;youdon’tneedactuallytodoit,orevenknowhowtodoit.

Thereislittledoubtthattheinformationrevolutionhastransformedourlives,andagoodcasecanbemadethatinbroadtermsthebenefitsoutweighthedisadvantages–eventhoughthelatterincludelossofprivacy,potentialfraudulentaccesstoourbankaccountsfromanywhereinthewordattheclickofamouse,andcomputervirusesthatcandisableabankoranuclearpowerstation.

Whatisinformation?Whydoesithavesuchpower?Andisitreallywhatitisclaimedtobe?

TheconceptofinformationasameasurablequantityemergedfromtheresearchlaboratoriesoftheBellTelephoneCompany,themainprovideroftelephoneservicesintheUnitedStatesfrom1877toitsbreak-upin1984onanti-trust(monopoly)grounds.AmongitsengineerswasClaudeShannon,adistantcousinofthefamousinventorEdison.Shannon’sbestsubjectatschoolwasmathematics,andhehadanaptitudeforbuildingmechanicaldevices.BythetimehewasworkingforBellLabshewasamathematicianandcryptographer,aswellasanelectronicengineer.Hewasoneofthefirsttoapplymathematicallogic–so-calledBooleanalgebra–tocomputercircuits.Heusedthistechniquetosimplifythedesignofswitchingcircuitsusedbythetelephonesystem,andthenextendedittootherproblemsincircuitdesign.

DuringWorldWarIIheworkedonsecretcodesandcommunications,anddevelopedsomefundamentalideasthatwerereportedinaclassifiedmemorandumforBellin1945underthetitle‘Amathematicaltheoryofcryptography’.In1948hepublishedsomeofhisworkintheopenliterature,andthe1945article,declassified,waspublishedsoonafter.WithadditionalmaterialbyWarrenWeaver,itappearedin1949asTheMathematicalTheoryofCommunication.

Shannonwantedtoknowhowtotransmitmessageseffectivelywhenthetransmissionchannelwassubjecttorandomerrors,‘noise’intheengineeringjargon.Allpracticalcommunicationssufferfromnoise,beitfromfaultyequipment,cosmicrays,orunavoidablevariabilityincircuitcomponents.Onesolutionistoreducethenoisebybuildingbetterequipment,ifpossible.Analternativeistoencodethesignalsusingmathematicalproceduresthatcandetecterrors,andevenputthemright.

Thesimplesterror-detectingcodeistosendthesamemessagetwice.Ifyoureceive

thesamemassagetwicethesamemessagetwice

thenthereisclearlyanerrorinthethirdword,butwithoutunderstandingEnglishitisnotobviouswhichversioniscorrect.Athirdrepetitionwoulddecidethematterbymajorityvoteandbecomeanerror-correctingcode.Howeffectiveoraccuratesuchcodesaredependsonthelikelihood,andnature,ofthe

errors.Ifthecommunicationchannelisverynoisy,forinstance,thenallthreeversionsofthemessagemightbesobadlygarbledthatitwouldbeimpossibletoreconstructit.

Inpracticemererepetitionistoosimple-minded:therearemoreefficientwaystoencodemessagestorevealorcorrecterrors.Shannon’sstartingpointwastopinpointthemeaningofefficiency.Allsuchcodesreplacetheoriginalmessagebyalongerone.Thetwocodesabovedoubleortreblethelength.Longermessagestakemoretimetosend,costmore,occupymorememory,andclogthecommunicationchannel.Sotheefficiency,foragivenrateoferrordetectionorcorrection,canbequantifiedastheratioofthelengthofthecodedmessagetothatoftheoriginal.

Themainissue,forShannon,wastodeterminetheinherentlimitationsofsuchcodes.Supposeanengineerhaddevisedanewcode.Wastheresomewaytodecidewhetheritwasaboutasgoodastheyget,ormightsomeimprovementbepossible?Shannonbeganbyquantifyinghowmuchinformationamessagecontains.Bysodoing,heturned‘information’fromavaguemetaphorintoascientificconcept.

Therearetwodistinctwaystorepresentanumber.Itcanbedefinedbyasequenceofsymbols,forexampleitsdecimaldigits,oritcancorrespondtosomephysicalquantity,suchasthelengthofastickorthevoltageinawire.Representationsofthefirstkindaredigital,thesecondareanalogue.Inthe1930s,scientificandengineeringcalculationswereoftenperformedusinganaloguecomputers,becauseatthetimethesewereeasiertodesignandbuild.Simpleelectroniccircuitscanaddormultiplytwovoltages,forexample.However,machinesofthistypelackedprecision,anddigitalcomputersbegantoappear.Itveryquicklybecameclearthatthemostconvenientrepresentationofnumberswasnotdecimal,base10,butbinary,base2.Indecimalnotationtherearetensymbolsfordigits,0–9,andeverydigitmultipliestentimesinvalueforeverystepitmovestotheleft.So157represents

1×102+5×101+7×100

Binarynotationemploysthesamebasicprinciple,butnowthereareonlytwodigits,0and1.Abinarynumbersuchas10011101encodes,insymbolicform,thenumber

1×27+0×26+0×25+1×24+1×23+1×22

+0×21+1×20

sothateachdigitdoublesinvalueforeverystepitmovestotheleft.Indecimal,thisnumberequals157–sowehavewrittenthesamenumberintwodifferentforms,usingtwodifferenttypesofnotation.

Binarynotationisidealforelectronicsystemsbecauseitismucheasiertodistinguishbetweentwopossiblevaluesofacurrent,oravoltage,oramagneticfield,thanitistodistinguishbetweenmorethantwo.Incrudeterms,0canmean‘noelectriccurrent’and1canmean‘someelectriccurrent’,0canmean‘nomagneticfield’and1canmean‘somemagneticfield’,andsoon.Inpracticeengineerssetathresholdvalue,andthen0means‘belowthreshold’and1means‘abovethreshold’.Bykeepingtheactualvaluesusedfor0and1farenoughapart,andsettingthethresholdinbetween,thereisverylittledangerofconfusing0with1.Sodevicesbasedonbinarynotationarerobust.That’swhatmakesthemdigital.

Withearlycomputers,theengineershadtostruggletokeepthecircuitvariableswithinreasonablebounds,andbinarymadetheirlivesmucheasier.Moderncircuitsonsiliconchipsarepreciseenoughtopermitotherchoices,suchasbase3,butthedesignofdigitalcomputershasbeenbasedonbinarynotationforsolongnowthatitgenerallymakessensetosticktobinary,evenifalternativeswouldwork.Moderncircuitsarealsoverysmallandveryquick.Withoutsomesuchtechnologicalbreakthroughincircuitmanufacture,theworldwouldhaveafewthousandcomputers,ratherthanbillions.ThomasJ.Watson,whofoundedIBM,oncesaidthathedidn’tthinktherewouldbeamarketformorethanaboutfivecomputersworldwide.Atthetimeheseemedtobetalkingsense,becauseinthosedaysthemostpowerfulcomputerswereaboutthesizeofahouse,consumedasmuchelectricityasasmallvillage,andcosttensofmillionsofdollars.Onlybiggovernmentorganisations,suchastheUnitedStatesArmy,couldaffordone,ormakeenoughuseofit.Todayabasic,out-of-datemobilephonecontainsmorecomputingpowerthananythingthatwasavailablewhenWatsonmadehisremark.

Thechoiceofbinaryrepresentationfordigitalcomputers,hencealsofordigitalmessagestransmittedbetweencomputers–andlaterbetweenalmostanytwoelectronicgadgetsontheplanet–ledtothebasicunitofinformation:thebit.Thenameisshortfor‘binarydigit’,andonebitofinformationisone0orone1.Itisreasonabletodefinetheinformation‘containedin’asequenceofbinarydigitstobethetotalnumberofdigitsinthesequence.Sothe8-digitsequence10011101contains8bitsofinformation.

Shannonrealisedthatsimple-mindedbit-countingmakessenseasameasureofinformationonlyif0sand1sarelikeheadsandtailswithafaircoin,thatis,areequallylikelytooccur.Supposeweknowthatinsomespecificcircumstances0occursninetimesoutoften,and1onlyonce.Aswereadalongthestringofdigits,weexpectmostdigitstobe0.Ifthatexpectationisconfirmed,wehaven’treceivedmuchinformation,becausethisiswhatweexpectedanyway.However,ifwesee1,thatconveysalotmoreinformation,becausewedidn’texpectthatatall.

Wecantakeadvantageofthisbyencodingthesameinformationmoreefficiently.If0occurswithprobability9/10and1withprobability1/10,wecandefineanewcodelikethis:

000→00(usewheneverpossible)

00→01(ifno000remains)

0→10(ifno00remains)

1→11(always)

WhatImeanhereisthatamessagesuchas

00000000100010000010000001000000000

isfirstbrokenupfromlefttorightintoblocksthatread000,00,0,or1.Withstringsofconsecutive0s,weuse000wheneverwecan.Ifnot,what’sleftiseither00or0,followedbya1.Soherethemessagebreaksupas

000-000-00-1-000-1-000-00-1-000-000-1-000-000-000

andthecodedversionbecomes

00-00-01-11-00-11-00-01-11-00-00-11-11-00-00-00

Theoriginalmessagehas35digits,buttheencodedversionhasonly32.Theamountofinformationseemstohavedecreased.

Sometimesthecodedversionmightbelonger:forinstance,111turnsinto111111.Butthat’srarebecause1occursonlyonetimeintenonaverage.Therewillbequitealotof000s,whichdropto00.Anyspare00changesto01,thesamelength;aspare0increasesthelengthbychangingto00.Theupshotisthatinthelongrun,forrandomlychosenmessageswiththegivenprobabilitiesof0

and1,thecodedversionisshorter.Mycodehereisverysimple-minded,andaclevererchoicecanshortenthe

messageevenmore.OneofthemainquestionsthatShannonwantedtoanswerwas:howefficientcancodesofthisgeneraltypebe?Ifyouknowthelistofsymbolsthatisbeingusedtocreateamessage,andyoualsoknowhowlikelyeachsymbolis,howmuchcanyoushortenthemessagebyusingasuitablecode?Hissolutionwasanequation,definingtheamountofinformationintermsoftheseprobabilities.

Supposeforsimplicitythatthemessagesuseonlytwosymbols0and1,butnowthesearelikeflipsofabiasedcoin,sothat0hasprobabilitypofoccurring,and1hasprobabilityq=1–p.Shannon’sanalysisledhimtoaformulafortheinformationcontent:itshouldbedefinedas

H=−plogp−qlogq

wherelogisthelogarithmtobase2.Atfirstsightthisdoesn’tseemterriblyintuitive.I’llexplainhowShannon

deriveditinamoment,butthemainthingtoappreciateatthisstageishowHbehavesaspvariesfrom0to1,whichisshowninFigure56.ThevalueofHincreasessmoothlyfrom0to1asprisesfrom0to ,andthendropsbacksymmetricallyto0aspgoesfrom to1.

Fig56HowShannoninformationHdependsonp.Hrunsverticallyandprunshorizontally.

Shannonpointedoutseveral‘interestingproperties’ofH,sodefined:Ifp=0,inwhichcaseonlythesymbol1willoccur,theinformationHis

zero.Thatis,ifwearecertainwhichsymbolisgoingtobetransmittedtous,receivingitconveysnoinformationwhatsoever.Thesameholdswhenp=1.Onlythesymbol0willoccur,andagainwereceivenoinformation.Theamountofinformationislargestwhenp=q= ,correspondingtothetossofafaircoin.Inthiscase,

bearinginmindthatthelogarithmsaretobase2.Thatis,onetossofafaircoinconveysonebitofinformation,aswewereoriginallyassumingbeforewestartedworryingaboutcodingmessagestocompressthem,andbiasedcoins.Inallothercases,receivingonesymbolconveyslessinformationthanonebit.Themorebiasedthecoinbecomes,thelessinformationtheresultofonetossconveys.Theformulatreatsthetwosymbolsinexactlythesameway.Ifweexchangepandq,thenHstaysthesame.

Allofthesepropertiescorrespondtoourintuitivesenseofhowmuchinformationwereceivewhenwearetoldtheresultofacointoss.Thatmakestheformulaareasonableworkingdefinition.Shannonthenprovidedasolidfoundationforhisdefinitionbylistingseveralbasicprinciplesthatanymeasureofinformationcontentoughttoobeyandderivingauniqueformulathatsatisfiedthem.Hisset-upwasverygeneral:themessagecouldchoosefromanumberofdifferentsymbols,occurringwithprobabilitiesp1,p2,…,pnwherenisthenumberofsymbols.TheinformationHconveyedbyachoiceofoneofthesesymbolsshouldsatisfy:

Hisacontinuousfunctionofp1,p2,…,pn.Thatis,smallchangesintheprobabilitiesshouldleadtosmallchangesintheamountofinformation.Ifalloftheprobabilitiesareequal,whichimpliestheyareall1/n,thenHshouldincreaseifngetslarger.Thatis,ifyouarechoosingbetween3symbols,allequallylikely,thentheinformationyoureceiveshouldbemorethanifthechoicewerebetweenjust2equallylikelysymbols;achoicebetween4symbolsshouldconveymoreinformationthanachoicebetween3symbols,andsoon.

Ifthereisanaturalwaytobreakachoicedownintotwosuccessivechoices,thentheoriginalHshouldbeasimplecombinationofthenewHs.

Thisfinalconditionismosteasilyunderstoodusinganexample,andI’veputoneintheNotes.1ShannonprovedthattheonlyfunctionHthatobeyshisthreeprinciplesis

oraconstantmultipleofthisexpression,whichbasicallyjustchangestheunitofinformation,likechangingfromfeettometres.

Thereisagoodreasontotaketheconstanttobe1,andI’llillustrateitinonesimplecase.Thinkofthefourbinarystrings00,01,10,11assymbolsintheirownright.If0and1areequallylikely,eachstringhasthesameprobability,

namely .Theamountofinformationconveyedbyonechoiceasuchastringistherefore

Thatis,2bits.Whichisasensiblenumberfortheinformationinalength-2binarystringwhenthechoices0and1areequallylikely.Inthesameway,ifthesymbolsarealllength-nbinarystrings,andwesettheconstantto1,thentheinformationcontentisnbits.Noticethatwhenn=2weobtaintheformulapicturedinFigure56.TheproofofShannon’stheoremistoocomplicatedtogivehere,butitshowsthatifyouacceptShannon’sthreeconditionsthenthereisasinglenaturalwaytoquantifyinformation.2Theequationitselfismerelyadefinition:whatcountsishowitperformsinpractice.

Shannonusedhisequationtoprovethatthereisafundamentallimitonhowmuchinformationacommunicationchannelcanconvey.Supposethatyouaretransmittingadigitalsignalalongaphoneline,whosecapacitytocarryamessageisatmostCbitspersecond.Thiscapacityisdeterminedbythenumberofbinarydigitsthatthephonelinecantransmit,anditisnotrelatedtotheprobabilitiesofvarioussignals.SupposethatthemessageisbeinggeneratedfromsymbolswithinformationcontentH,alsomeasuredinbitspersecond.Shannon’stheoremanswersthequestion:ifthechannelisnoisy,canthesignalbeencodedsothattheproportionoferrorsisassmallaswewish?Theansweristhatthisisalwayspossible,nomatterwhatthenoiselevelis,ifHislessthanorequaltoC.ItisnotpossibleifHisgreaterthanC.Infact,theproportionof

errorscannotbereducedbelowthedifferenceH—C,nomatterwhichcodeisemployed,butthereexistcodesthatgetascloseasyouwishtothaterrorrate.

Shannon’sproofofhistheoremdemonstratesthatcodesoftherequiredkindexist,ineachofhistwocases,buttheproofdoesn’ttelluswhatthosecodesare.Anentirebranchofinformationscience,amixtureofmathematics,computing,andelectronicengineering,isdevotedtofindingefficientcodesforspecificpurposes.Itiscalledcodingtheory.Themethodsforcomingupwiththesecodesareverydiverse,drawingonmanyareasofmathematics.Itisthesemethodsthatareincorporatedintoourelectronicgadgetry,beitasmartphoneorVoyager1’stransmitter.Peoplecarrysignificantquantitiesofsophisticatedabstractalgebraaroundintheirpockets,intheformofsoftwarethatimplementserror-correctingcodesformobilephones.

I’lltrytoconveytheflavourofcodingtheorywithoutgettingtootangledinthecomplexities.Oneofthemostinfluentialconceptsinthetheoryrelatescodestomultidimensionalgeometry.ItwaspublishedbyRichardHammingin1950inafamouspaper,‘Errordetectinganderrorcorrectingcodes’.Initssimplestform,itprovidesacomparisonbetweenstringsofbinarydigits.Considertwosuchstrings,say10011101and10110101.Comparecorrespondingbits,andcounthowmanytimestheyaredifferent,likethis:

1001110110110101

whereI’vemarkedthedifferencesinboldtype.Heretherearetwolocationsatwhichthebit-stringsdiffer.WecallthisnumbertheHammingdistancebetweenthetwostrings.Itcanbethoughtofasthesmallestnumberofone-biterrorsthatcanconvertonestringintotheother.Soitiscloselyrelatedtothelikelyeffectoferrors,iftheseoccurataknownaveragerate.Thatsuggestsitmightprovidesomeinsightintohowtodetectsucherrors,andmaybeevenhowtoputthemright.

Multidimensionalgeometrycomesintoplaybecausethestringsofafixedlengthcanbeassociatedwiththeverticesofamultidimensional‘hypercube’.Riemanntaughtushowtothinkofsuchspacesbythinkingoflistsofnumbers.Forexample,aspaceoffourdimensionsconsistsofallpossiblelistsoffournumbers:(x1,x2,x3,x4).Eachsuchlistisconsideredtorepresentapointinthespace,andallpossiblelistscaninprincipleoccur.Theseparatexsarethecoordinatesofthepoint.Ifthespacehas157dimensions,youhavetouselistsof

157numbers:(x1,x2,…,x157).Itisoftenusefultospecifyhowfaraparttwosuchlistsare.In‘flat’EuclideangeometrythisisdoneusingasimplegeneralisationofPythagoras’stheorem.Supposewehaveasecondpoint(y1,y2,…,y157)inour157-dimensionalspace.Thenthedistancebetweenthetwopointsisthesquarerootofthesumofthesquaresofthedifferencesbetweencorrespondingcoordinates.Thatis,

Ifthespaceiscurved,Riemann’sideaofametriccanbeusedinstead.Hamming’sideaistodosomethingverysimilar,butthevaluesofthe

coordinatesarerestrictedtojust0and1.Then(x1–y1)2is0ifx1andy1arethesame,but1ifnot,andthesamegoesfor(x2–y2)2andsoon.Healsoomittedthesquareroot,whichchangestheanswer,butincompensationtheresultisalwaysawholenumber,equaltotheHammingdistance.Thisnotionhasallthepropertiesthatmake‘distance’useful,suchasbeingzeroonlywhenthetwostringsareidentical,andensuringthatthelengthofanysideofa‘triangle’(asetofthreestrings)islessthanorequaltothesumofthelengthsoftheothertwosides.

Wecandrawpicturesofallbitstringsoflengths2,3,and4(andwithmoreeffortandlessclarity,5,6,andpossiblyeven10,thoughnoonewouldfindthatuseful).TheresultingdiagramsareshowninFigure57.

Fig57Spacesofallbit-stringsoflengths2,3,and4.

Thefirsttwoarerecognisableasasquareandacube(projectedontoaplanebecauseithastobeprintedonasheetofpaper).Thethirdisahypercube,the4-dimensionalanalogue,andagainthishastobeprojectedontoaplane.ThestraightlinesjoiningthedotshaveHamminglength1–thetwostringsateitherenddifferinpreciselyonelocation,onecoordinate.TheHammingdistancebetweenanytwostringsisthenumberofsuchlinesintheshortestpaththat

connectsthem.Supposewearethinkingof3-bitstrings,livingonthecornersofacube.Pick

oneofthestrings,say101.Supposetherateoferrorsisatmostonebitineverythree.Thenthisstringmayeitherbetransmittedunchanged,oritcouldendupasanyof001,111,or100.Eachofthesediffersfromtheoriginalstringinjustonelocation,soitsHammingdistancefromtheoriginalstringis1.Inaloosegeometricalimage,theerroneousstringslieona‘sphere’centredatthecorrectstring,ofradius1.Thesphereconsistsofjustthreepoints,andifwewereworkingin157-dimensionalspacewitharadiusof5,say,itwouldn’tevenlookterriblyspherical.Butitplaysasimilarroletoanordinarysphere:ithasafairlycompactshape,anditcontainsexactlythepointswhosedistancefromthecentreislessthanorequaltotheradius.

Supposeweusethespherestoconstructacode,sothateachspherecorrespondstoanewsymbol,andthatsymbolisencodedwiththecoordinatesofthecentreofthesphere.Supposemoreoverthatthesespheresdon’toverlap.Forinstance,Imightintroduceasymbolaforthespherecentredat101.Thisspherecontainsfourstrings:101,001,111,and100.IfIreceiveanyofthesefourstrings,Iknowthatthesymbolwasoriginallya.Atleast,that’strueprovidedmyothersymbolscorrespondinasimilarwaytospheresthatdonothaveanypointsincommonwiththisone.

Nowthegeometrystartstomakeitselfuseful.Inthecube,thereareeightpoints(strings)andeachspherecontainsfourofthem.IfItrytofitspheresintothecube,withoutthemoverlapping,thebestIcanmanageistwoofthem,because8/4=2.Icanactuallyfindanotherone,namelythespherecentredon010.Thiscontains010,110,000,011,noneofwhichareinthefirstsphere.SoIcanintroduceasecondsymbolbassociatedwiththissphere.Myerror-correctingcodeformessageswrittenwithaandbsymbolsnowreplaceseveryaby101,andeverybby010.IfIreceive,say,

101-010-100-101-000

thenIcandecodetheoriginalmessageas

a-b-a-a-b

despitetheerrorsinthethirdandfifthstring.Ijustseewhichofmytwospherestheerroneousstringbelongsto.

Allverywell,butthismultipliesthelengthofthemessageby3,andwe

alreadyknowaneasierwaytoachievethesameresult:repeatthemessagethreetimes.Butthesameideatakesonanewsignificanceifweworkinhigher-dimensionalspaces.Withstringsoflength4,thehypercube,thereare16strings,andeachspherecontains5points.Soitmightbepossibletofitthreespheresinwithoutthemoverlapping.Ifyoutrythat,it’snotactuallypossible–twofitinbuttheremaininggapisthewrongshape.Butthenumbersincreasinglyworkinourfavour.Thespaceofstringsoflength5contains32strings,andeachsphereusesjust6ofthem–possiblyroomfor5,andifnot,abetterchanceoffittingin4.Length6givesus64points,andspheresthatuse7,soupto9spheresmightfitin.

Fromthispointonalotoffiddlydetailisneededtoworkoutjustwhatispossible,andithelpstodevelopmoresophisticatedmethods.Butwhatwearelookingatistheanalogue,inthespaceofstrings,ofthemostefficientwaystopackspherestogether.Andthisisalong-standingareaofmathematics,aboutwhichquitealotisknown.SomeofthattechniquecanbetransferredfromEuclideangeometrytoHammingdistances,andwhenthatdoesn’tworkwecaninventnewmethodsmoresuitedtothegeometryofstrings.Asanexample,Hamminginventedanewcode,moreefficientthananyknownatthetime,whichencodes4-bitstringsbyconvertingtheminto7-bitstrings.Itcandetectandcorrectanysingle-biterror.Modifiedtoan8-bitcode,itcandetect,butnotcorrect,any2-biterror.

ThiscodeiscalledtheHammingcode.Iwon’tdescribeit,butlet’sdothesumstoseeifitmightbepossible.Thereare16stringsoflength4,and128oflength7.Spheresofradius1inthe7-dimensionalhypercubecontain8points.And128/8=16.Sowithenoughcunning,itmightjustbepossibletosqueezetherequired16spheresintothe7-dimensionalhypercube.Theywouldhavetofitexactly,becausethere’snospareroomleftover.Asithappens,suchanarrangementexists,andHammingfoundit.Withoutthemultidimensionalgeometrytohelp,itwouldbedifficulttoguessthatitexisted,letalonefindit.Possible,buthard.Evenwiththegeometry,it’snotobvious.

Shannon’sconceptofinformationprovideslimitsonhowefficientcodescanbe.Codingtheorydoestheotherhalfofthejob:findingcodesthatareasefficientaspossible.Themostimportanttoolsherecomefromabstractalgebra.Thisisthestudyofmathematicalstructuresthatsharethebasicarithmeticalfeaturesofintegersorrealnumbers,butdifferfromtheminsignificantways.Inarithmetic,wecanaddnumbers,subtractthem,andmultiplythem,togetnumbersofthesamekind.Fortherealnumberswecanalsodividebyanythingotherthanzero

togetarealnumber.Thisisnotpossiblefortheintegers,because,forexample,isnotaninteger.However,itispossibleifwepasstothelargersystemofrationalnumbers,fractions.Inthefamiliarnumbersystems,variousalgebraiclawshold,forexamplethecommutativelawofaddition,whichstatesthat2+3=3+2andthesamegoesforanytwonumbers.

Thefamiliarsystemssharethesealgebraicpropertieswithlessfamiliarones.Thesimplestexampleusesjusttwonumbers,0and1.Sumsandproductsaredefinedjustasforintegers,withoneexception:weinsistthat1+1=0,not2.Despitethismodification,alloftheusuallawsofalgebrasurvive.Thissystemhasonlytwo‘elements’,twonumber-likeobjects.Thereisexactlyonesuchsystemwheneverthenumberofelementsisapowerofanyprimenumber:2,3,4,5,7,8,9,11,13,16,andsoon.SuchsystemsarecalledGaloisfieldsaftertheFrenchmathematicianÉvaristeGalois,whoclassifiedthemaround1830.Becausetheyhavefinitelymanyelements,theyaresuitedtodigitalcommunications,andpowersof2areespeciallyconvenientbecauseofbinarynotation.

GaloisfieldsleadtocodingsystemscalledReed–Solomoncodes,afterIrvingReedandGustaveSolomonwhoinventedthemin1960.Theyareusedinconsumerelectronics,especiallyCDsandDVDs.Theyareerror-correctingcodesbasedonalgebraicpropertiesofpolynomials,whosecoefficientsaretakenfromaGaloisfield.Thesignalbeingencoded–audioorvideo–isusedtoconstructapolynomial.Ifthepolynomialhasdegreen,thatis,thehighestpoweroccurringisxn,thenthepolynomialcanbereconstructedfromitsvaluesatanynpoints.Ifwespecifythevaluesatmorethannpoints,wecanloseormodifysomeofthevalueswithoutlosingtrackofwhichpolynomialitis.Ifthenumberoferrorsisnottoolarge,itisstillpossibletoworkoutwhichpolynomialitis,anddecodetogettheoriginaldata.

Inpracticethesignalisrepresentedasaseriesofblocksofbinarydigits.Apopularchoiceuses255bytes(8-bitstrings)perblock.Ofthese,223bytesencodethesignal,whiletheremaining32bytesare‘paritysymbols’,tellinguswhethervariouscombinationsofdigitsintheuncorrupteddataareoddoreven.ThisparticularReed–Solomoncodecancorrectupto16errorsperblock,anerrorratejustlessthan1%.

WheneveryoudrivealongabumpyroadwithaCDonthecarstereo,youareusingabstractalgebra,intheformofaReed–Solomoncode,toensurethatthemusiccomesovercrispandclear,insteadofbeingjerkyandcrackly,perhapswithsomepartsmissingaltogether.Informationtheoryiswidelyusedin

cryptographyandcryptanalysis–secretcodesandmethodsforbreakingthem.Shannonhimselfusedittoestimatetheamountofcodedmessagesthatmustbeinterceptedtostandachanceofbreakingthecode.Keepinginformationsecretturnsouttobemoredifficultthanmightbeexpected,andinformationtheoryshedslightonthisproblem,bothfromthepointofviewofthepeoplewhowantitkeptsecretandthosewhowanttofindoutwhatitis.Thisissueisimportantnotjusttothemilitary,buttoeveryonewhousestheInternettobuygoodsorengagesintelephonebanking.

Informationtheorynowplaysasignificantroleinbiology,particularlyintheanalysisofDNAsequencedata.TheDNAmoleculeisadouble-helix,formedbytwostrandsthatwindroundeachother.Eachstrandisasequenceofbases,specialmoleculesthatcomeinfourtypes–adenine,guanine,thymine,andcytosine.SoDNAislikeacodemessagewrittenusingfourpossiblesymbols:A,G,T,andC.Thehumangenome,forexample,is3billionbaseslong.BiologistscannowfindtheDNAsequencesofinnumerableorganismsatarapidlygrowingrate,leadingtoanewareaofcomputerscience:bioinformatics.Thiscentresonmethodsforhandlingbiologicaldataefficientlyandeffectively,andoneofitsbasictoolsisinformationtheory.

Amoredifficultissueisthequalityofinformation,ratherthanthequantity.Themessages‘twoplustwomakefour’and‘twoplustwomakefive’containexactlythesameamountofinformation,butoneistrueandtheotherisfalse.PaeansofpraisefortheinformationageignoretheuncomfortabletruththatmuchoftheinformationrattlingaroundtheInternetismisinformation.Therearewebsitesrunbycriminalswhowanttostealyourmoney,ordenialistswhowanttoreplacesolidsciencebywhicheverbeehappenstobebuzzingaroundinsidetheirownbonnet.

Thevitalconcepthereisnotinformationassuch,butmeaning.ThreebillionDNAbasesofhumanDNAinformationare,literally,meaninglessunlesswecanfindouthowtheyaffectourbodiesandbehaviour.OnthetenthanniversaryofthecompletionoftheHumanGenomeProject,severalleadingscientificjournalssurveyedmedicalprogressresultingsofarfromlistinghumanDNAbases.Theoveralltonewasmuted:afewnewcuresfordiseaseshavebeenfoundsofar,butnotinthequantityoriginallypredicted.ExtractingmeaningfromDNAinformationisprovingharderthanmostbiologistshadhoped.TheHumanGenomeProjectwasanecessaryfirststep,butithasrevealedjusthowdifficultsuchproblemsare,ratherthansolvingthem.

Thenotionofinformationhasescapedfromelectronicengineeringand

invadedmanyotherareasofscience,bothasametaphorandasatechnicalconcept.TheformulaforinformationlooksverylikethatforentropyinBoltzmann’sapproachtothermodynamics;themaindifferencesarelogarithmstobase2insteadofnaturallogarithms,andachangeinsign.Thissimilaritycanbeformalised,andentropycanbeinterpretedas‘missinginformation’.Sotheentropyofagasincreasesbecausewelosetrackofexactlywhereitsmoleculesare,andhowfastthey’removing.Therelationbetweenentropyandinformationhastobesetuprathercarefully:althoughtheformulasareverysimilar,thecontextinwhichtheyapplyisdifferent.Thermodynamicentropyisalarge-scalepropertyofthestateofagas,butinformationisapropertyofasignal-producingsource,notofasignalassuch.In1957theAmericanphysicistEdwinJaynes,anexpertinstatisticalmechanics,summeduptherelationship:thermodynamicentropycanbeviewedasanapplicationofShannoninformation,butentropyitselfshouldnotbeidentifiedwithmissinginformationwithoutspecifyingtherightcontext.Ifthisdistinctionisborneinmind,therearevalidcontextsinwhichentropycanbeviewedasalossofinformation.Justasentropyincreaseplacesconstraintsontheefficiencyofsteamengines,theentropicinterpretationofinformationplacesconstraintsontheefficiencyofcomputations.Forexample,itmusttakeatleast5.8×10-23joulesofenergytoflipabitfrom0to1orviceversaatthetemperatureofliquidhelium,whatevermethodyouuse.

Problemsarisewhenthewords‘information’and‘entropy’areusedinamoremetaphoricalsense.BiologistsoftensaythatDNAdetermines‘theinformation’requiredtomakeanorganism.Thereisasenseinwhichthisisalmostcorrect:delete‘the’.However,themetaphoricalinterpretationofinformationsuggeststhatonceyouknowtheDNA,thenyouknoweverythingthereistoknowabouttheorganism.Afterall,you’vegottheinformation,right?Andforatimemanybiologiststhoughtthatthisstatementwasclosetothetruth.However,wenowknowthatitisoveroptimistic.EveniftheinformationinDNAreallydidspecifytheorganismuniquely,youwouldstillneedtoworkouthowitgrowsandwhattheDNAactuallydoes.However,ittakesalotmorethanalistofDNAcodestocreateanorganism:so-calledepigeneticfactorsmustalsobetakenintoaccount.Theseincludechemical‘switches’thatmakeasegmentofDNAcodeactiveorinactive,butalsoentirelydifferentfactorsthataretransmittedfromparenttooffspring.Forhumanbeings,thosefactorsincludethecultureinwhichwegrowup.Soitpaysnottobetoocasualwhenyouusetechnicaltermslike‘information’.

16Theimbalanceofnature

ChaosTheory

Whatdoesitsay?

Itmodelshowapopulationoflivingcreatureschangesfromonegenerationtothenext,whentherearelimitstotheavailableresources.

Whyisthatimportant?

Itisoneofthesimplestequationsthatcangeneratedeterministicchaos–apparentlyrandombehaviourwithnorandomcause.

Whatdiditleadto?

Therealisationthatsimplenonlinearequationscancreateverycomplexdynamics,andthatapparentrandomnessmayconcealhiddenorder.Popularlyknownaschaostheory,thisdiscoveryhasinnumerableapplicationsthroughoutthesciencesincludingthemotionoftheplanetsintheSolarSystem,weatherforecasting,populationdynamicsinecology,variablestars,earthquakemodelling,andefficienttrajectoriesforspaceprobes.

Themetaphorofthebalanceofnaturetripsreadilyoffthetongueasadescriptionofwhattheworldwoulddoifnastyhumansdidn’tkeepinterfering.Nature,lefttoitsowndevices,wouldsettledowntoastateofperfectharmony.Coralreefswouldalwaysharbourthesamespeciesofcolourfulfishinsimilarnumbers,rabbitsandfoxeswouldlearntosharethefieldsandwoodlandssothatthefoxeswouldbewellfed,mostrabbitswouldsurvive,andneitherpopulationwouldexplodeorcrash.Theworldwouldsettledowntoafixedstateandstaythere.Untilthenextbigmeteorite,orasupervolcano,upsetthebalance.

It’sacommonmetaphor,perilouslyclosetobeingacliché.It’salsohighlymisleading.Nature’sbalanceisdistinctlywobbly.

We’vebeenherebefore.WhenPoincaréwasworkingonKingOscar’sprize,theconventionalwisdomheldthatastableSolarSystemisoneinwhichtheplanetsfollowmuchthesameorbitsforever,giveortakeaharmlessbitofjiggling.Technicallythisisnotasteadystate,butoneinwhicheachplanetrepeatssimilarmotionsoverandoveragain,subjecttominordisturbancescausedbyalltheothers,butnotdeviatinghugelyfromwhatitwouldhavedonewithoutthem.Thedynamicsis‘quasiperiodic’–combiningseveralseparateperiodicmotionswhoseperiodsarenotallmultiplesofthesametimeinterval.Intherealmofplanets,that’sascloseto‘steady’asanyonecanhopefor.

Butthedynamicswasn’tlikethat,asPoincarébelatedly,andtohiscost,foundout.Itcould,intherightcircumstances,bechaotic.Theequationshadnoexplicitrandomterms,sothatinprinciplethepresentstatecompletelydeterminedthefuturestate,yetparadoxicallytheactualmotioncouldappeartoberandom.Infact,ifyouaskedcoarse-grainedquestionslike‘whichsideoftheSunwillitbeon?’,theanswercouldbeagenuinelyrandomseriesofobservations.Onlyifyoucouldlookinfinitelycloselywouldyoubeabletoseethatthemotionreallywascompletelydetermined.

Thiswasthefirstintimationofwhatwenowcall‘chaos’,whichisshortfor‘deterministicchaos’,andquitedifferentfrom‘random’–eventhoughthat’swhatitcanlooklike.Chaoticdynamicshashiddenpatterns,butthey’resubtle;theydifferfromwhatwemightnaturallythinkofmeasuring.Onlybyunderstandingthecausesofchaoscanweextractthosepatternsfromanirregularmishmashofdata.

Asalwaysinscience,therewereafewisolatedprecursors,generallyviewed

asminorcuriositiesunworthyofseriousattention.Onlyinthe1960sdidmathematicians,physicists,andengineersbegintorealisejusthownaturalchaosisindynamics,andhowradicallyitdiffersfromanythingenvisagedinclassicalscience.Wearestilllearningtoappreciatewhatthattellsus,andwhattodoaboutit.Butalreadychaoticdynamics,‘chaostheory’inpopularparlance,pervadesmostareasofscience.Itmayevenhavethingstotellusabouteconomicsandthesocialsciences.It’snottheanswertoeverything:onlycriticseverclaimeditwas,andthatwastomakeiteasiertoshootitdown.Chaoshassurvivedallsuchattacks,andforagoodreason:itisabsolutelyfundamentaltoallbehaviourgovernedbydifferentialequations,andthosearethebasicstuffofphysicallaw.

Thereischaosinbiology,too.OneofthefirsttoappreciatethatthismightbethecasewastheAustralianecologistRobertMay,nowLordMayofOxfordandaformerpresidentoftheRoyalSociety.Hesoughttounderstandhowthepopulationsofvariousspecieschangeovertimeinnaturalsystemssuchascoralreefsandwoodlands.In1975MaywroteashortarticleforthejournalNature,pointingoutthattheequationstypicallyusedtomodelchangestoanimalandplantpopulationscouldproducechaos.Maydidn’tclaimthatthemodelshewasdiscussingwereaccuraterepresentationsofwhatrealpopulationsdid.Hispointwasmoregeneral:chaoswasnaturalinmodelsofthatkind,andthishadtobeborneinmind.

Themostimportantconsequenceofchaosisthatirregularbehaviourneednothaveirregularcauses.Previously,ifecologistsnoticedthatsomepopulationofanimalswasfluctuatingwildly,theywouldlookforsomeexternalcause–alsopresumedtobefluctuatingwildly,andgenerallylabelled‘random’.Theweather,perhaps,orasuddeninfluxofpredatorsfromelsewhere.May’sexamplesshowedthattheinternalworkingsoftheanimalpopulationscouldgenerateirregularitywithoutoutsidehelp.

Hismainexamplewastheequationthatdecoratestheopeningofthischapter.Itiscalledthelogisticequation,anditisasimplemodelofapopulationofanimalsinwhichthesizeofeachgenerationisdeterminedbythepreviousone.‘Discrete’meansthattheflowoftimeiscountedingenerations,andisthusaninteger.Sothemodelissimilartoadifferentialequation,inwhichtimeisacontinuousvariable,butconceptuallyandcomputationallysimpler.Thepopulationismeasuredasafractionofsomeoveralllargevalue,andcanthereforeberepresentedbyarealnumberthatliesbetween0(extinction)and1(thetheoreticalmaximumthatthesystemcansustain).Lettingtimettickin

integersteps,correspondingtogenerations,thisnumberisxtingenerationt.Thelogisticequationstatesthat

xt+1=kxt(1−xt)

wherekisaconstant.Wecaninterpretkasthegrowthrateofthepopulationwhendiminishingresourcesdonotslowitdown.1

Westartthemodelattime0withaninitialpopulationx0.Thenweusetheequationwitht=0tocalculatex1,thenwesett=1andcomputex2,andsoon.Withoutevendoingthesumswecanseestraightawaythat,foranyfixedgrowthratek,thepopulationsizeofgenerationzerocompletelydeterminesthesizesofallsucceedinggenerations.Sothemodelisdeterministic:knowledgeofthepresentdeterminesthefutureuniquelyandexactly.

Sowhatisthefuture?The‘balanceofnature’metaphorsuggeststhatthepopulationshouldsettletoasteadystate.Wecanevencalculatewhatthatsteadystateshouldbe:justsetthepopulationattimet+1tobethesameasthatattimet.Thisleadstotwosteadystates:populations0and1-1/k.Apopulationofsize0isextinct,sotheothervalueshouldapplytoanexistingpopulation.Unfortunately,althoughthisisasteadystate,itcanbeunstable.Ifitis,theninpracticeyou’llneverseeit:it’sliketryingtobalanceapencilverticallyonitssharpenedpoint.Theslightestdisturbancewillcauseittotopple.Thecalculationsshowthatthesteadystateisunstablewhenkisbiggerthan3.

What,then,doweseeinpractice?Figure58showsatypical‘timeseries’forthepopulationwhenk=4.It’snotsteady:it’sallovertheplace.However,ifyoulookcloselytherearehintsthatthedynamicsisnotcompletelyrandom.Wheneverthepopulationgetsreallybig,itimmediatelycrashestoaverylowvalue,andthengrowsinaregularmanner(roughlyexponentially)forthenexttwoorthreegenerations:seetheshortarrowsinFigure58.Andsomethinginterestinghappenswheneverthepopulationgetscloseto0.75orthereabouts:itoscillatesalternatelyaboveandbelowthatvalue,andtheoscillationsgrowgivingacharacteristiczigzagshape,gettingwidertowardstheright:seethelongerarrowsinthefigure.

Fig58Chaoticoscillationsinamodelanimalpopulation.Shortarrowsshowcrashesfollowedbyshort-termexponentialgrowth.Longerarrowsshowunstableoscillations.

Despitethesepatterns,thereisasenseinwhichthebehaviouristrulyrandom–butonlywhenyouthrowawaysomeofthedetail.SupposeweassignthesymbolH(heads)wheneverthepopulationisbiggerthan0.5,andT(tails)whenit’slessthan0.5.ThisparticularsetofdatabeginswiththesequenceTHTHTHHTHHTTHHandcontinuesunpredictably,justlikearandomsequenceofcointosses.Thiswayofcoarseningthedata,bylookingatspecificrangesofvaluesandnotingonlywhichrangethepopulationbelongsto,iscalledsymbolicdynamics.Inthiscase,itispossibletoprovethat,foralmostallinitialpopulationvaluesx0,thesequenceofheadsandtailsisinallrespectslikeatypicalsequenceofrandomtossesofafaircoin.Onlywhenwelookattheexactvaluesdowestarttoseesomepatterns.

It’saremarkablediscovery.Adynamicalsystemcanbecompletelydeterministic,withvisiblepatternsindetaileddata,yetacoarse-grainedviewofthesamedatacanberandom–inaprovable,rigoroussense.Determinismandrandomnessarenotopposites.Insomecircumstances,theycanbeentirelycompatible.

Maydidn’tinventthelogisticequation,andhedidn’tdiscoveritsastonishingproperties.Hedidn’tclaimtohavedoneeitherofthosethings.Hisaimwastoalertworkersinthelifesciences,especiallyecologists,totheremarkablediscoveriesemerginginthephysicalsciencesandmathematics:discoveriesthatfundamentallychangethewayscientistsshouldthinkaboutobservationaldata.Wehumansmayhavetroublesolvingequationsbasedonsimplerules,butnaturedoesn’thavetosolvetheequationsthewaywedo.Itjustobeystherules.

Soitcandothingsthatstrikeusasbeingcomplicated,forsimplereasons.Chaosemergedfromatopologicalapproachtodynamics,orchestratedin

particularbytheAmericanmathematicianStephenSmaleandtheRussianmathematicianVladimirArnoldinthe1960s.Bothweretryingtofindoutwhattypesofbehaviourweretypicalindifferentialequations.SmalewasmotivatedbyPoincaré’sstrangeresultsonthethree-bodyproblem(Chapter4),andArnoldwasinspiredbyrelateddiscoveriesofhisformerresearchsupervisorAndreiKolmogorov.Bothquicklyrealisedwhychaosiscommon:itisanaturalconsequenceofthegeometryofdifferentialequations,aswe’llseeinamoment.

Asinterestinchaosspread,exampleswerespottedlurkingunnoticedinearlierscientificpapers.Previouslyconsideredtobejustisolatedweirdeffects,theseexamplesnowslottedintoabroadertheory.Inthe1940stheEnglishmathematiciansJohnLittlewoodandMaryCartwrighthadseentracesofchaosinelectronicoscillators.In1958TsunejiRikitakeofTokyo’sAssociationfortheDevelopmentofEarthquakePredictionhadfoundchaoticbehaviourinadynamomodeloftheEarth’smagneticfield.Andin1963theAmericanmeteorologistEdwardLorenzhadpinneddownthenatureofchaoticdynamicsinconsiderabledetail,inasimplemodelofatmosphericconvectionmotivatedbyweather-forecasting.Theseandotherpioneershadpointedtheway;nowalloftheirdisparatediscoverieswerestartingtofittogether.

Inparticular,thecircumstancesthatledtochaos,ratherthansomethingsimpler,turnedouttobegeometricratherthanalgebraic.Inthelogisticmodelwithk=4,bothextremesofthepopulation,0and1,moveto0inthenextgeneration,whilethemidpoint, ,movesto1.Soateachtime-steptheintervalfrom0to1isstretchedtotwiceitslength,foldedinhalf,andslappeddowninitsoriginallocation.Thisiswhatacookdoestodoughwhenmakingbread,andbythinkingaboutdoughbeingkneaded,wegainahandleonchaos.Imagineatinyspeckinthelogisticdough–araisin,say.Supposethatithappenstolieonaperiodiccycle,sothatafteracertainnumberofstretch-and-foldoperationsitreturnstowhereitstarted.Nowwecanseewhythispointisunstable.Imagineanotherraisin,initiallyveryclosetothefirstone.Eachstretchmovesitfurtheraway.Foratime,though,itdoesn’tmovefarenoughawaytostoptrackingthefirstraisin.Whenthedoughisfolded,bothraisinsendupinthesamelayer.Sonexttime,thesecondraisinhasmovedevenfurtherawayfromthefirst.Thisiswhytheperiodicstateisunstable:stretchingmovesallnearbypointsawayfromit,nottowardsit.Eventuallytheexpansionbecomessogreatthatthetworaisinsendupindifferentlayerswhenthedoughisfolded.Afterthat,theirfatesare

prettymuchindependentofeachother.Whydoesacookkneaddough?Tomixuptheingredients(includingtrappedair).Ifyoumixstuffup,theindividualparticleshavetomoveinaveryirregularway.Particlesthatstartclosetogetherendupfarapart;pointsfarapartmaybefoldedbacktobeclosetogether.Inshort,chaosisthenaturalresultofmixing.

Isaidatthestartofthischapterthatyoudon’thaveanythingchaoticinyourkitchen,exceptperhapsthatdishwasher.Ilied.Youprobablyhaveseveralchaoticgadgets:afoodprocessor,anegg-beater.Thebladeofthefoodprocessorfollowsaverysimplerule:goroundandround,fast.Thefoodinteractswiththeblade:itoughttodosomethingsimpletoo.Butitdoesn’tgoroundandround:itgetsmixedup.Asthebladecutsthroughthefood,somebitsgoonesideofit,somegotheotherside:locally,thefoodgetspulledapart.Butitdoesn’tescapefromthemixingbowl,soitallgetsfoldedbackinonitself.

SmaleandArnoldrealisedthatallchaoticdynamicsislikethis.Theydidn’tphrasetheirresultsinquitethatlanguage,mindyou:‘pulledapart’was‘positiveLiapunovexponent’and‘foldedback’was‘thesystemhasacompactdomain’.Butinfancylanguage,theyweresayingthatchaosislikemixingdough.

Thisalsoexplainssomethingelse,noticedespeciallybyLorenzin1963.Chaoticdynamicsissensitivetoinitialconditions.Howeverclosethetworaisinsaretobeginwith,theyeventuallygetpulledsofarapartthattheirsubsequentmovementsareindependent.Thisphenomenonisoftencalledthebutterflyeffect:abutterflyflapsitswings,andamonthlatertheweatheriscompletelydifferentfromwhatitwouldotherwisehavebeen.ThephraseisgenerallycreditedtoLorenz.Hedidn’tintroduceit,butsomethingsimilarfeaturedinthetitleofoneofhislectures.However,someoneelseinventedthetitleforhim,andthelecturewasn’taboutthefamous1963article,butalesser-knownonefromthesameyear.

Whateverthephenomenoniscalled,ithasanimportantpracticalconsequence.Althoughchaoticdynamicsisinprincipledeterministic,inpracticeitbecomesunpredictableveryquickly,becauseanyuncertaintyintheexactinitialstategrowsexponentiallyfast.Thereisapredictionhorizonbeyondwhichthefuturecannotbeforeseen.Forweather,afamiliarsystemwhosestandardcomputermodelsareknowntobechaotic,thishorizonisafewdaysahead.FortheSolarSystem,itistensofmillionsofyearsahead.Forsimplelaboratorytoys,suchasadoublependulum(apendulumhungfromthebottomofanotherone)itisafewsecondsahead.Thelong-heldassumptionthat‘deterministic’and‘predictable’arethesameiswrong.Itwouldbevalidifthe

presentstateofasystemcouldbemeasuredwithperfectaccuracy,butthat’snotpossible.

Theshort-termpredictabilityofchaoscanbeusedtodistinguishitfrompurerandomness.Manydifferenttechniqueshavebeendevisedtomakethisdistinction,andtoworkouttheunderlyingdynamicsifthesystemisbehavingdeterministicallybutchaotically.

Chaosnowhasapplicationsineverybranchofscience,fromastronomytozoology.InChapter4wesawhowitisleadingtonew,moreefficienttrajectoriesforspacemissions.Inbroaderterms,astronomersJackWisdomandJacquesLaskarhaveshownthatthedynamicsoftheSolarSystemischaotic.IfyouwanttoknowwhereaboutsinitsorbitPlutowillbein10,000,000AD–forgetit.TheyhavealsoshownthattheMoon’stidesstabilisetheEarthagainstinfluencesthatwouldotherwiseleadtochaoticmotion,causingrapidshiftsofclimatefromwarmperiodstoiceagesandbackagain.Sochaostheorydemonstratesthat,withouttheMoon,theEarthwouldbeaprettyunpleasantplacetolive.ThisfeatureofourplanetaryneighbourhoodisoftenusedtoarguethattheevolutionoflifeonaplanetrequiresastabilisingMoon,butthisisanoverstatement.Lifeintheoceanswouldscarcelynoticeiftheplanet’saxischangedoveraperiodofmillionsofyears.Lifeonlandwouldhaveplentyoftimetomigrateelsewhere,unlessitgottrappedsomewherethatlackedalandroutetoaplacewhereconditionsweremoresuitable.Climatechangeishappeningmuchfasterrightnowthananythingthatachangeinaxialtiltcouldcause.

May’ssuggestionthatirregularpopulationdynamicsinanecosystemmightsometimesbecausedbyinternalchaos,ratherthanextraneousrandomness,hasbeenverifiedinlaboratoryversionsofseveralreal-worldecosystems.In1995ateamheadedbyAmericanecologistJamesCushingfoundchaoticdynamicsinpopulationsoftheflourbeetle(orbranbug)Triboliumcastaneum,whichcaninfeststoresofflour.2In1999,DutchbiologistsJefHuismanandFranzWeissingappliedchaostothe‘paradoxoftheplankton’,theunexpecteddiversityofplanktonspecies.3Astandardprincipleinecology,theprincipleofcompetitiveexclusion,statesthatanecosystemcannotcontainmorespeciesthanthenumberofenvironmentalniches,waystomakealiving.Planktonappeartoviolatethisprinciple:thenumberofnichesissmall,butthenumberofspeciesisinthethousands.Theytracedthistoaloopholeinthederivationoftheprincipleofcompetitiveexclusion:theassumptionthatpopulationsaresteady.Ifthepopulationscanchangeovertime,thenthemathematicalderivationfromthe

usualmodelfails,andintuitivelydifferentspeciescanoccupythesamenichebytakingturns–notbyconsciouscooperation,butbyonespeciestemporarilytakingoverfromanotherandundergoingapopulationboom,whilethedisplacedspeciesdropstoasmallpopulation,Figure59.

Fig59Sixspeciessharingthreeresources.Thebandsarecloselyspacedchaoticoscillations.CourtesyofJefHuismanandFranzWeissing.

In2008,Huisman’steampublishedtheresultsofalaboratoryexperimentwithaminiatureecologybasedononefoundintheBalticSea,involvingbacteriaandseveralkindsofplankton.Asix-yearstudyrevealedchaoticdynamicsinwhichpopulationsfluctuatedwildly,oftenbecoming100timesaslargeforatimeandthencrashing.Theusualmethodsfordetectingchaosconfirmeditspresence.Therewasevenabutterflyeffect:thesystem’spredictionhorizonwasafewweeks.4

Thereareapplicationsofchaosthatimpingeoneverydaylife,buttheymostlyoccurinmanufacturingprocessesandpublicservices,ratherthanbeingincorporatedintogadgets.Thediscoveryofthebutterflyeffecthaschangedthewayweatherforecastsarecarriedout.Insteadofputtingallofthecomputationaleffortintorefiningasingleprediction,meteorologistsnowrunmanyforecasts,makingdifferenttinyrandomchangestotheobservationsprovidedbyweatherballoonsandsatellitesbeforestartingeachrun.Ifalloftheseforecastsagree,thenthepredictionislikelytobeaccurate;iftheydiffersignificantly,theweatherisinalesspredictablestate.Theforecaststhemselveshavebeenimprovedbyseveralotheradvances,notablyincalculatingtheinfluenceoftheoceansonthestateoftheatmosphere,butthemainroleofchaoshasbeento

warnforecastersnottoexpecttoomuchandtoquantifyhowlikelyaforecastistobecorrect.

Industrialapplicationsincludeabetterunderstandingofmixingprocesses,whicharewidelyusedtomakemedicinalpillsormixfoodingredients.Theactivemedicineinapillusuallyoccursinverysmallquantities,andithastobemixedwithsomeinertsubstance.It’simportanttogetenoughoftheactiveingredientineachpill,butnottoomuch.Amixingmachineislikeagiantfoodprocessor,andlikethefoodprocessor,itsdynamicsisdeterministicbutchaotic.Themathematicsofchaoshasprovidedanewunderstandingofmixingprocessesandledtosomeimproveddesigns.Themethodsusedtodetectchaosindatahaveinspirednewtestequipmentforthewireusedtomakesprings,improvingefficiencyinspring-andwire-making.Thehumblespringhasmanyvitaluses:itcanbefoundinmattresses,cars,DVDplayers,evenballpointpens.Chaoticcontrol,atechniquethatusesthebutterflyeffecttokeepdynamicbehaviourstable,isshowingpromiseinthedesignofmoreefficientandlessintrusiveheartpacemakers.

Overall,though,themainimpactofchaoshasbeenonscientificthinking.Inthefortyyearsorsosinceitsexistencestartedtobewidelyappreciated,chaoshaschangedfromaminormathematicalcuriosityintoabasicfeatureofscience.Wecannowstudymanyofnature’sirregularitieswithoutresortingtostatistics,byteasingoutthehiddenpatternsthatcharacterisedeterministicchaos.Thisisjustoneofthewaysinwhichmoderndynamicalsystemstheory,withitsemphasisonnonlinearbehaviour,iscausingaquietrevolutioninthewayscientiststhinkabouttheworld.

17TheMidasformula

Black–ScholesEquation

Whatdoesitsay?

Itdescribeshowthepriceofafinancialderivativechangesovertime,basedontheprinciplethatwhenthepriceiscorrect,thederivativecarriesnoriskandnoonecanmakeaprofitbysellingitatadifferentprice.

Whyisthatimportant?

Itmakesitpossibletotradeaderivativebeforeitmaturesbyassigninganagreed‘rational’valuetoit,sothatitcanbecomeavirtualcommodityinitsownright.

Whatdiditleadto?

Massivegrowthofthefinancialsector,evermorecomplexfinancialinstruments,surgesineconomicprosperitypunctuatedbycrashes,theturbulentstockmarketsofthe1990s,the2008–9financialcrisis,andtheongoingeconomicslump.

Sincetheturnofthecenturythegreatestsourceofgrowthinthefinancialsectorhasbeeninfinancialinstrumentsknownasderivatives.Derivativesarenotmoney,noraretheyinvestmentsinstocksandshares.Theyareinvestmentsininvestments,promisesaboutpromises.Derivativestradersusevirtualmoney,numbersinacomputer.Theyborrowitfrominvestorswhohaveprobablyborroweditfromsomewhereelse.Oftentheyhaven’tborroweditatall,notevenvirtually:theyhaveclickedamousetoagreethattheywillborrowthemoneyifiteverbecomesnecessary.Buttheyhavenointentionoflettingitbecomenecessary;theywillsellthederivativebeforethathappens.Thelender–hypotheticallender,sincetheloanwillneveroccur,forthesamereason–probablydoesn’tactuallyhavethemoneyeither.Thisisfinanceincloudcuckooland,yetithasbecomethestandardpracticeoftheworld’sbankingsystem.

Unfortunately,theconsequencesofderivativestradingdo,ultimately,turnintorealmoney,andrealpeoplesuffer.Thetrickworks,mostofthetime,becausethedisconnectwithrealityhasnonotableeffect,otherthanmakingafewbankersandtradersextremelyrichastheysiphonoffrealmoneyfromthevirtualpool.Untilthingsgowrong.Thenthepigeonscomehometoroost,bearingwiththemvirtualdebtsthathavetobepaidwithrealmoney.Byeveryoneelse,naturally.

Thisiswhattriggeredthebankingcrisisof2008–9,fromwhichtheworld’seconomiesarestillreeling.Lowinterestratesandenormouspersonalbonuspaymentsencouragedbankersandtheirbankstobeteverlargersumsofvirtualmoneyonevermorecomplexderivatives,ultimatelysecured–sotheybelieved–inthepropertymarket,housesandbusinesses.Asthesupplyofsuitablepropertyandpeopletobuyitbegantodryup,thefinancialworld’sleadersneededtofindnewwaystoconvinceshareholdersthattheywerecreatingprofit,inordertojustifyandfinancetheirbonuses.Sotheystartedtradingpackagesofdebt,alsoallegedlysecured,somewheredowntheline,onrealproperty.Keepingtheschemegoingdemandedthecontinuedpurchaseofproperty,toincreasethepoolofcollateral.Sothebanksstartedsellingmortgagestopeoplewhoseabilitytorepaythemwasincreasinglydoubtful.Thiswasthesubprimemortgagemarket,‘subprime’beingaeuphemismfor‘likelytodefault’.Whichsoonbecame‘certaintodefault’.

Thebanksbehavedlikeoneofthosecartooncharacterswhowandersofftheedgeofacliff,hoversinspaceuntilhelooksdown,andonlythenplungestothe

ground.Itallseemedtobegoingnicelyuntilthebankersaskedthemselveswhethermultipleaccountingwithnon-existentmoneyandovervaluedassetswassustainable,wonderedwhattherealvalueoftheirholdingsinderivativeswas,andrealisedthattheydidn’thaveaclue.Exceptthatitwasdefinitelyalotlessthanthey’dtoldshareholdersandgovernmentregulators.

Asthedreadfultruthdawned,confidenceplummeted.Thisdepressedthehousingmarket,sotheassetsagainstwhichthedebtsweresecuredstartedtolosetheirvalue.Atthispointthewholesystembecametrappedinapositivefeedbackloop,inwhicheachdownwardrevisionofvaluecausedittoberevisedevenfurtherdownward.Theendresultwasthelossofabout17trilliondollars.Facedwiththeprospectofthetotalcollapseoftheworldfinancialsystem,trashingdepositors’savingsandmakingtheGreatDepressionof1929looklikeagardenparty,governmentswereforcedtobailoutthebanks,whichwereonthevergeofbankruptcy.One,LehmanBrothers,wasallowedtogounder,butthelossofconfidencewassogreatthatitseemedunwisetorepeatthelesson.Sotaxpayersstumpedupthemoney,andalotofitwasrealmoney.Thebanksgrabbedthecashwithbothhands,andthentriedtopretendthatthecatastrophehadn’tbeentheirfault.Theyblamedgovernmentregulators,despitehavingcampaignedagainstregulation:aninterestingcaseof‘It’syourfault:youletusdoit.’

Howdidthebiggestfinancialtrainwreckinhumanhistorycomeabout?Arguably,onecontributorwasamathematicalequation.

Thesimplestderivativeshavebeenaroundforalongtime.Theyareknownasfuturesandoptions,andtheygobacktotheeighteenthcenturyattheDojimariceexchangeinOsaka,Japan.Theexchangewasfoundedin1697,atimeofgreateconomicprosperityinJapan,whentheupperclasses,thesamurai,werepaidinrice,notmoney.Naturallythereemergedaclassofricebrokerswhotradedriceasthoughitweremoney.AstheOsakamerchantsstrengthenedtheirgriponrice,thecountry’sstaplefood,theiractivitieshadaknock-oneffectonthecommodity’sprice.Atthesametime,thefinancialsystemwasbeginningtoshifttohardcash,andthecombinationproveddeadly.In1730thepriceofricedroppedthroughthefloor.

Ironically,thetriggerwaspoorharvests.Thesamurai,stillweddedtopaymentinrice,butwatchfulofthegrowthofmoney,startedtopanic.Theirfavoured‘currency’wasrapidlylosingitsvalue.Merchantsexacerbatedtheproblembyartificiallykeepingriceoutofthemarket,squirrellingawayhuge

quantitiesinwarehouses.Althoughitmightseemthatthiswouldincreasethemonetarypriceofrice,ithadtheoppositeeffect,becausethesamuraiweretreatingriceasacurrency.Theycouldnoteatanythingremotelyapproachingtheamountofricetheyowned.Sowhileordinarypeoplestarved,themerchantsstockpiledrice.Ricebecamesoscarcethatpapermoneytookover,anditquicklybecamemoredesirablethanricebecauseitwaspossibleactuallytolayhandsonit.SoontheDojimamerchantswererunningwhatamountedtoagiganticbankingsystem,holdingaccountsforthewealthyanddeterminingtheexchangeratebetweenriceandpapermoney.

Eventuallythegovernmentrealisedthatthisarrangementhandedfartoomuchpowertothericemerchants,andreorganisedtheRiceExchangealongwithmostotherpartsofthecountry’seconomy.In1939theRiceExchangewasreplacedbytheGovernmentRiceAgency.ButwhiletheRiceExchangeexisted,themerchantsinventedanewkindofcontracttoevenoutthelargeswingsinthepriceofrice.Thesignatoriesguaranteedtobuy(orsell)aspecifiedquantityofriceataspecifiedfuturedateforaspecifiedprice.Todaytheseinstrumentsareknownasfuturesoroptions.Supposeamerchantagreestobuyriceinsixmonths’timeatanagreedprice.Ifthemarketpricehasrisenabovetheagreedonebythetimetheoptionfallsdue,hegetsthericecheapandimmediatelysellsitataprofit.Ontheotherhand,ifthepriceislower,heiscommittedtobuyingriceatahigherpricethanitsmarketvalueandmakesaloss.

Farmersfindsuchinstrumentsusefulbecausetheyactuallywanttosellarealcommodity:rice.Peopleusingriceforfood,ormanufacturingfoodstuffsthatuseit,wanttobuythecommodity.Inthissortoftransaction,thecontractreducestherisktobothparties–thoughataprice.Itamountstoaformofinsurance:aguaranteedmarketataguaranteedprice,independentofshiftsinthemarketvalue.It’sworthpayingasmallpremiumtoavoiduncertainty.Butmostinvestorstookoutcontractsinricefutureswiththesoleaimofmakingmoney,andthelastthingtheinvestorwantedwastonsandtonsofrice.Theyalwayssolditbeforetheyhadtotakedelivery.Sothemainroleoffutureswastofuelfinancialspeculation,andthiswasmadeworsebytheuseofriceascurrency.Justastoday’sgoldstandardcreatesartificiallyhighpricesforasubstance(gold)thathaslittleintrinsicvalue,andtherebyfuelsdemandforit,sothepriceofricebecamegovernedbythetradingoffuturesratherthanthetradingofriceitself.Thecontractswereaformofgambling,andsoonthecontractsthemselvesacquiredavalue,andcouldbetradedasthoughtheywererealcommodities.Moreover,althoughtheamountofricewaslimitedbywhatthefarmerscouldgrow,therewasnolimittothenumberofcontractsforricethatcouldbeissued.

Theworld’smajorstockmarketswerequicktospotanopportunitytoconvertsmokeandmirrorsintohardcash,andtheyhavetradedfutureseversince.Atfirst,thispracticedidnotofitselfcauseenormouseconomicproblems,althoughitsometimesledtoinstabilityratherthanthestabilitythatisoftenassertedtojustifythesystem.Butaroundtheyear2000,theworld’sfinancialsectorbegantoinventevermoreelaboratevariantsonthefuturestheme,complex‘derivatives’whosevaluewasbasedonhypotheticalfuturemovementsofsomeasset.Unlikefutures,forwhichtheasset,atleast,wasreal,derivativesmightbebasedonanassetthatwasitselfaderivative.Nolongerwerebanksbuyingandsellingbetsonthefuturepriceofacommoditylikerice;theywerebuyingandsellingbetsonthefuturepriceofabet.

Itquicklybecamebigbusiness.In1998theinternationalfinancialsystemtradedroughly$100trillioninderivatives.By2007thishadgrowntoonequadrillionUSdollars.Trillions,quadrillions…weknowthesearelargenumbers,buthowlarge?Toputthisfigureincontext,thetotalvalueofalltheproductsmadebytheworld’smanufacturingindustries,forthelastthousandyears,isabout100trillionUSdollars,adjustedforinflation.That’sonetenthofoneyear’sderivativestrading.Admittedlythebulkofindustrialproductionhasoccurredinthepastfiftyyears,butevenso,thisisastaggeringamount.Itmeans,inparticular,thatthederivativestradesconsistalmostentirelyofmoneythatdoesnotactuallyexist–virtualmoney,numbersinacomputer,withnolinktoanythingintherealworld.Infact,thesetradeshavetobevirtual:thetotalamountofmoneyincirculation,worldwide,iscompletelyinadequatetopaytheamountsthatarebeingtradedattheclickofamouse.Bypeoplewhohavenointerestinthecommodityconcerned,andwouldn’tknowwhattodowithitiftheytookdelivery,usingmoneythattheydon’tactuallypossess.

Youdon’tneedtobearocketscientisttosuspectthatthisisarecipefordisaster.Yetforadecade,theworldeconomygrewrelentlesslyonthebackofderivativestrading.Notonlycouldyougetamortgagetobuyahouse:youcouldgetmorethanthehousewasworth.Thebankdidn’tevenbothertocheckwhatyourtrueincomewas,orwhatotherdebtsyouhad.Youcouldgeta125%self-certifiedmortgage–meaningyoutoldthebankwhatyoucouldaffordanditdidn’taskawkwardquestions–andspendthesurplusonaholiday,acar,plasticsurgery,orcratesofbeer.Bankswentoutoftheirwaytopersuadecustomerstotakeoutloans,evenwhentheydidn’tneedthem.

Whattheythoughtwouldsavethemifaborrowerdefaultedontheirrepaymentswasstraightforward.Thoseloansweresecuredonyourhouse.Housepricesweresoaring,sothatmissing25%ofequitywouldsoonbecome

real;ifyoudefaulted,thebankcouldseizeyourhouse,sellit,andgetitsloanback.Itseemedfoolproof.Ofcourseitwasn’t.Thebankersdidn’taskthemselveswhatwouldhappentothepriceofhousingifhundredsofbankswerealltryingtosellmillionsofhousesatthesametime.Nordidtheyaskwhetherpricescouldcontinuetorisesignificantlyfasterthaninflation.Theygenuinelyseemedtothinkthathousepricescouldrise10–15%inrealtermseveryyear,indefinitely.Theywerestillurgingregulatorstorelaxtherulesandallowthemtolendevenmoremoneywhenthebottomdroppedoutofthepropertymarket.

Manyoftoday’smostsophisticatedmathematicalmodelsoffinancialsystemscanbetracedbacktoBrownianmotion,mentionedinChapter12.Whenviewedthroughamicroscope,smallparticlessuspendedinafluidjigglearounderratically,andEinsteinandSmoluchowskidevelopedmathematicalmodelsofthisprocessandusedthemtoestablishtheexistenceofatoms.Theusualmodelassumesthattheparticlereceivesrandomkicksthroughdistanceswhoseprobabilitydistributionisnormal,abellcurve.Thedirectionofeachkickisuniformlydistributed–anydirectionhasthesamechanceofhappening.Thisprocessiscalledarandomwalk.ThemodelofBrownianmotionisacontinuumversionofsuchrandomwalks,inwhichthesizesofthekicksandthetimebetweensuccessivekicksbecomearbitrarilysmall.Intuitively,weconsiderinfinitelymanyinfinitesimalkicks.

ThestatisticalpropertiesofBrownianmotion,overlargenumbersoftrials,aredeterminedbyaprobabilitydistribution,whichgivesthelikelihoodthattheparticleendsupataparticularlocationafteragiventime.Thisdistributionisradiallysymmetric:theprobabilitydependsonlyonhowfarthepointisfromtheorigin.Initiallytheparticleisverylikelytobeclosetotheorigin,butastimepasses,therangeoflikelypositionsspreadsoutastheparticlegetsmorechancetoexploredistantregionsofspace.Remarkably,thetimeevolutionofthisprobabilitydistributionobeystheheatequation,whichinthiscontextisoftencalledthediffusionequation.Sotheprobabilityspreadsjustlikeheat.

AfterEinsteinandSmoluchowskipublishedtheirwork,itturnedoutthatmuchofthemathematicalcontenthadbeenderivedearlier,in1900,bytheFrenchmathematicianLouisBachelierinhisPhDthesis.ButBachelierhadadifferentapplicationinmind:thestockandoptionmarkets.ThetitleofhisthesiswasThéoriedelaspeculation(‘TheoryofSpeculation’).Theworkwasnotreceivedwithwildpraise,probablybecauseitssubject-matterwasfaroutsidethenormalrangeofmathematicsatthatperiod.Bachelier’ssupervisorwastherenownedandformidablemathematicianHenriPoincaré,whodeclaredthework

tobe‘veryoriginal’.Healsogavethegamewaysomewhat,byadding,withreferencetothepartofthethesisthatderivedthenormaldistributionforerrors:‘ItisregrettablethatM.Bachelierdidnotdevelopthispartofhisthesisfurther.’Whichanymathematicianwouldinterpretas‘thatwastheplacewherethemathematicsstartedtogetreallyinteresting,andifonlyhe’ddonemoreworkonthat,ratherthanonfuzzyideasaboutthestockmarket,itwouldhavebeeneasytogivehimamuchbettergrade.’Thethesiswasgraded‘honorable’,apass;itwasevenpublished.Butitdidnotgetthetopgradeof‘trèshonorable’.

Bachelierineffectpinneddowntheprinciplethatfluctuationsofthestockmarketfollowarandomwalk.Thesizesofsuccessivefluctuationsconformtoabellcurve,andthemeanandstandarddeviationcanbeestimatedfrommarketdata.Oneimplicationisthatlargefluctuationsareveryimprobable.Thereasonisthatthetailsofthenormaldistributiondiedownveryfastindeed:fasterthanexponential.Thebellcurvedecreasestowardszeroataratethatisexponentialinthesquareofx.Statisticians(andphysicistsandmarketanalysts)talkoftwo-sigmafluctuations,three-sigmaones,andsoon.Heresigma(σ)isthestandarddeviation,ameasureofhowwidethebellcurveis.Athree-sigmafluctuation,say,isonethatdeviatesfromthemeanbyatleastthreetimesthestandarddeviation.Themathematicsofthebellcurveletsusassignprobabilitiestothese‘extremeevents’,seeTable3.

Table3Probabilitiesofmany-sigmaevents.

TheupshotofBachelier’sBrownianmotionmodelisthatlargestockmarketfluctuationsaresorarethatinpracticetheyshouldneverhappen.Table3showsthatafive-sigmaevent,forexample,isexpectedtooccuraboutsixtimesinevery10milliontrials.However,stockmarketdatashowthattheyarefarmorecommonthanthat.StockinCiscoSystems,aworldleaderincommunications,hasundergoneten5-sigmaeventsinthelasttwentyyears,whereasBrownianmotionpredicts0.003ofthem.Ipickedthiscompanyatrandomandit’sinno

wayunusual.OnBlackMonday(19October1987)theworld’sstockmarketslostmorethan20%oftheirvaluewithinafewhours;aneventthisextremeshouldhavebeenvirtuallyimpossible.

ThedatasuggestunequivocallythatextremeeventsarenowherenearasrareasBrownianmotionpredicts.Theprobabilitydistributiondoesnotdiewayexponentially(orfaster);itdiesawaylikeapower-lawcurvex−aforsomepositiveconstanta.Inthefinancialjargon,suchadistributionissaidtohaveafattail.Fattailsindicateincreasedlevelsofrisk.Ifyourinvestmenthasafive-sigmaexpectedreturn,thenassumingBrownianmotion,thechancethatitwillfailislessthanoneinamillion.Butiftailsarefat,itmightbemuchlarger,maybeoneinahundred.Thatmakesitamuchpoorerbet.

Arelatedterm,madepopularbyNassimNicholasTaleb,anexpertinmathematicalfinance,is‘blackswanevent’.His2007bookTheBlackSwanbecameamajorbestseller.Inancienttimes,allknownswanswerewhite.ThepoetJuvenalreferstosomethingas‘ararebirdinthelands,andverylikeablackswan’,andhemeantthatitwasimpossible.Thephrasewaswidelyusedinthesixteenthcentury,muchaswemightrefertoaflyingpig.Butin1697,whentheDutchexplorerWillemdeVlaminghwenttotheaptlynamedSwanRiverinWesternAustralia,hefoundmassesofblackswans.Thephrasechangeditsmeaning,andnowreferstoanassumptionthatappearstobegroundedinfact,butmightatanymomentturnouttobewildlymistaken.YetanothertermcurrentisX-event,‘extremeevent’.

Theseearlyanalysesofmarketsinmathematicaltermsencouragedtheseductiveideathatthemarketcouldbemodelledmathematically,creatingarationalandsafewaytomakeunlimitedsumsofmoney.In1973itseemedthatthedreammightbecomereal,whenFischerBlackandMyronScholesintroducedamethodforpricingoptions:theBlack–Scholesequation.RobertMertonprovidedamathematicalanalysisoftheirmodelinthesameyear,andextendedit.Theequationis:

Itinvolvesfivedistinctquantities–timet,thepriceSofthecommodity,thepriceVofthederivative,whichdependsonSandt,therisk-freeinterestrater(thetheoreticalinterestthatcanbeearnedbyaninvestmentwithzerorisk,suchasgovernmentbonds),andthevolatilityσ2ofthestock.Itisalsomathematicallysophisticated:asecond-orderpartialdifferentialequationlikethewaveandheat

equations.Itexpressestherateofchangeofthepriceofthederivative,withrespecttotime,asalinearcombinationofthreeterms:thepriceofthederivativeitself,howfastthatchangesrelativetothestockprice,andhowthatchangeaccelerates.Theothervariablesappearinthecoefficientsofthoseterms.Ifthetermsrepresentingthepriceofthederivativeanditsrateofchangewereomitted,theequationwouldbeexactlytheheatequation,describinghowthepriceoftheoptiondiffusesthroughstock-price-space.ThistracesbacktoBachelier’sassumptionofBrownianmotion.Theothertermstakeadditionalfactorsintoaccount.

TheBlack–Scholesequationwasderivedasaconsequenceofanumberofsimplifyingfinancialassumptions–forinstance,thattherearenotransactioncostsandnolimitsonshort-selling,andthatitispossibletolendandborrowmoneyataknown,fixed,risk-freeinterestrate.Theapproachiscalledarbitragepricingtheory,anditsmathematicalcoregoesbacktoBachelier.ItassumesthatmarketpricesbehavestatisticallylikeBrownianmotion,inwhichboththerateofdriftandthemarketvolatilityareconstant.Driftisthemovementofthemean,andvolatilityisfinancialjargonforstandarddeviation,ameasureofaveragedivergencefromthemean.Thisassumptionissocommoninthefinancialliteraturethatithasbecomeanindustrystandard.

Therearetwomainkindsofoption.Inaputoption,thebuyeroftheoptionpurchasestherighttosellacommodityorfinancialinstrumentataspecifiedtimeforanagreedprice,iftheysowish.Acalloptionissimilar,butitconferstherighttobuyinsteadofsell.TheBlack–Scholesequationhasexplicitsolutions:oneformulaforputoptions,anotherforcalloptions.1Ifsuchformulashadnotexisted,theequationcouldstillhavebeensolvednumericallyandimplementedassoftware.However,theformulasmakeitstraightforwardtocalculatetherecommendedprice,aswellasyieldingimportanttheoreticalinsights.

TheBlack–Scholesequationwasdevisedtobringadegreeofrationalitytothefuturesmarket,whichitdoesveryeffectivelyundernormalmarketconditions.Itprovidesasystematicwaytocalculatethevalueofanoptionbeforeitmatures.Thenitcanbesold.Suppose,forinstance,thatamerchantcontractstobuy1000tonsofricein12months’timeatapriceof500perton–acalloption.Afterfivemonthsshedecidestoselltheoptiontoanyonewillingtobuyit.Everyoneknowshowthemarketpriceforricehasbeenchanging,sohowmuchisthatcontractworthrightnow?Ifyoustarttradingsuchoptionswithoutknowingtheanswer,you’reintrouble.Ifthetradelosesmoney,you’reopentotheaccusationthatyougotthepricewrongandyourjobcouldbeatrisk.So

whatshouldthepricebe?Tradingbytheseatofyourpantsceasestobeanoptionwhenthesumsinvolvedareinthebillions.Therehastobeanagreedwaytopriceanoptionatanytimebeforematurity.Theequationdoesjustthat.Itprovidesaformula,whichanyonecanuse,andifyourbossusesthesameformula,hewillgetthesameresultthatyoudid,providedyoudidn’tmakeerrorsofarithmetic.Inpractice,bothofyouwoulduseastandardcomputerpackage.

TheequationwassoeffectivethatitwonMertonandScholesthe1997NobelPrizeinEconomics.2Blackhaddiedbythen,andtherulesoftheprizeprohibitposthumousawards,buthiscontributionwasexplicitlycitedbytheSwedishAcademy.Theeffectivenessoftheequationdependedonthemarketbehavingitself.Iftheassumptionsbehindthemodelceasedtohold,itwasnolongerwisetouseit.Butastimepassedandconfidencegrew,manybankersandtradersforgotthat;theyusedtheequationasakindoftalisman,abitofmathematicalmagicthatprotectedthemagainstcriticism.Black–Scholesnotonlyprovidesapricethatisreasonableundernormalconditions;italsocoversyourbackifthetradegoesbelly-up.Don’tblameme,boss,Iusedtheindustrystandardformula.ThefinancialsectorwasquicktoseetheadvantagesoftheBlack–Scholesequationanditssolutions,andequallyquicktodevelopahostofrelatedequationswithdifferentassumptionsaimedatdifferentfinancialinstruments.Thethen-sedateworldofconventionalbankingcouldusetheequationstojustifyloansandtrades,alwayskeepinganeyeopenforpotentialtrouble.Butlessconventionalbusinesseswouldsoonfollow,andtheyhadthefaithofatrueconvert.Tothem,thepossibilityofthemodelgoingwrongwasinconceivable.ItbecameknownastheMidasformula–arecipeformakingeverythingturntogold.ButthefinancialsectorforgothowthestoryofKingMidasended.

Thedarlingofthefinancialsector,forseveralyears,wasacompanycalledLongTermCapitalManagement(LTCM).Itwasahedgefund,aprivatefundthatspreadsitsinvestmentsinawaythatisintendedtoprotectinvestorswhenthemarketgoesdown,andmakebigprofitswhenitgoesup.Itspecialisedintradingstrategiesbasedonmathematicalmodels,includingtheBlack–Scholesequationanditsextensions,togetherwithtechniquessuchasarbitrage,whichexploitsdiscrepanciesbetweenthepricesofbondsandthevaluethatcanactuallyberealised.InitiallyLTCMwasaspectacularsuccess,yieldingreturnsintheregionof40%peryearuntil1998.Atthatpointitlost$4.6billioninunderfourmonths,andtheFederalReserveBankpersuadeditsmajorcreditorstobailitouttothetuneof$3.6billion.Eventuallythebanksinvolvedgottheirmoneyback,butLTCMwaswoundupin2000.

Whatwentwrong?Thereareasmanytheoriesastherearefinancialcommentators,buttheconsensusisthattheproximatecauseofLTCM’sfailurewastheRussianfinancialcrisisof1998.WesternmarketshadinvestedheavilyinRussia,whoseeconomywasheavilydependentonoilexports.TheAsianfinancialcrisisof1997causedthepriceofoiltoslump,andthemaincasualtywastheRussianeconomy.TheWorldBankprovidedaloanof$22.6billiontoproptheRussiansup.

TheultimatecauseofLTCM’sdemisewasalreadyinplaceonthedayitstartedtrading.Assoonasrealityceasedtoobeytheassumptionsofthemodel,LTCMwasindeeptrouble.TheRussianfinancialcrisisthrewaspannerintheworksthatdemolishedalmostallofthoseassumptions.Somefactorshadabiggereffectthanothers.Increasedvolatilitywasoneofthem.Anotherwastheassumptionthatextremefluctuationshardlyeveroccur:nofattails.Butthecrisissentthemarketsintoturmoil,andinthepanic,pricesdroppedbyhugeamounts–manysigmas–inseconds.Becauseallofthefactorsconcernedwereinterrelated,theseeventstriggeredotherrapidchanges,sorapidthattraderscouldnotpossiblyknowthestateofthemarketatanyinstant.Eveniftheywantedtobehaverationally,whichpeopledon’tdoinageneralpanic,theyhadnobasisuponwhichtodoso.

IftheBrownianmodelisright,eventsasextremeastheRussianfinancialcrisisshouldoccurnomoreoftenthanonceacentury.Icanremembersevenfrompersonalexperienceinthepast40years:overinvestmentinproperty,theformerSovietUnion,Brazil,property(again),property(yetagain),dotcomcompanies,and…oh,yes,property.

Withhindsight,thecollapseofLTCMwasawarning.Thedangersoftradingbyformulainaworldthatdidnotobeythecosyassumptionsbehindtheformulaweredulynoted–andquicklyignored.Hindsightisallverywell,butanyonecanseethedangerafteracrisishasstruck.Whataboutforesight?Theorthodoxclaimabouttherecentglobalfinancialcrisisisthat,likethefirstswanwithblackfeathers,noonesawitcoming.

That’snotentirelytrue.

TheInternationalCongressofMathematiciansisthelargestmathematicalconferenceintheworld,takingplaceeveryfouryears.InAugust2002ittookplaceinBeijing,andMaryPoovey,professorofhumanitiesanddirectoroftheInstitutefortheProductionofKnowledgeatNewYorkUniversity,gavealecturewiththetitle‘Cannumbersensurehonesty?’3Thesubtitlewas

‘unrealisticexpectationsandtheUSaccountingscandal’,anditdescribedtherecentemergenceofa‘newaxisofpower’inworldaffairs:

Thisaxisrunsthroughlargemultinationalcorporations,manyofwhichavoidnationaltaxesbyincorporatingintaxhavenslikeHongKong.Itrunsthroughinvestmentbanks,throughnongovernmentalorganizationsliketheInternationalMonetaryFund,throughstateandcorporatepensionfunds,andthroughthewalletsofordinaryinvestors.Thisaxisoffinancialpowercontributestoeconomiccatastropheslikethe1998meltdowninJapanandArgentina’sdefaultin2001,anditleavesitstracesinthedailygyrationsofstockindexesliketheDowJonesIndustrialsandLondon’sFinancialTimesStockExchange100Index(theFTSE).

Shewentontosaythatthisnewaxisofpowerisintrinsicallyneithergoodnorbad:whatmattersishowitwieldsitspower.IthelpedtoraiseChina’sstandardofliving,whichmanyofuswouldconsidertobebeneficial.Italsoencouragedaworldwideabandonmentofwelfaresocieties,replacingthembyashareholderculture,whichmanyofuswouldconsidertobeharmful.AlesscontroversialexampleofabadoutcomeistheEnronscandal,whichbrokein2001.EnronwasanenergycompanybasedinTexas,anditscollapseledtowhatwasthenthebiggestbankruptcyinAmericanhistory,andalosstoshareholdersof$11billion.Enronwasanotherwarning,thistimeaboutderegulatedaccountinglaws.Again,fewheededthewarning.

Pooveydid.Shepointedtothecontrastbetweenthetraditionalfinancialsystembasedontheproductionofrealgoods,andtheemergingnewonebasedoninvestment,currencytrading,and‘complexwagersthatfuturepriceswouldriseorfall’.By1995thiseconomyofvirtualmoneyhadovertakentherealeconomyofmanufacturing.Thenewaxisofpowerwasdeliberatelyconfusingrealandvirtualmoney:arbitraryfiguresincompanyaccountsandactualcashorcommodities.Thistrend,sheargued,wasleadingtoacultureinwhichthevaluesofbothgoodsandfinancialinstrumentswerebecomingwildlyunstable,liabletoexplodeorcollapseattheclickofamouse.

Thearticleillustratedthesepointsusingfivecommonfinancialtechniquesandinstruments,suchas‘marktomarketaccounting’,inwhichacompanysetsupapartnershipwithasubsidiary.Thesubsidiarybuysastakeintheparentcompany’sfutureprofits;themoneyinvolvedisthenrecordedasinstantearningsbytheparentcompanywhiletheriskisrelegatedtothesubsidiary’s

balancesheet.Enronusedthistechniquewhenitchangeditsmarketingstrategyfromsellingenergytosellingenergyfutures.Thebigproblemwithbringingforwardpotentialfutureprofitsinthismanneristhattheycannotthenbelistedasprofitsnextyear.Theansweristorepeatthemanoeuvre.It’sliketryingtodriveacarwithoutbrakesbypressingeverharderontheaccelerator.Theinevitableresultisacrash.

Poovey’sfifthexamplewasderivatives,anditwasthemostimportantofthemallbecausethesumsofmoneyinvolvedweresogigantic.HeranalysislargelyreinforceswhatI’vealreadysaid.Hermainconclusionwas:‘Futuresandderivativestradingdependsuponthebeliefthatthestockmarketbehavesinastatisticallypredictableway,inotherwords,thatmathematicalequationsaccuratelydescribethemarket.’Butshenotedthattheevidencepointsinatotallydifferentdirection:somewherebetween75%and90%ofallfuturestraderslosemoneyinanyyear.

Twotypesofderivativewereparticularlyimplicatedincreatingthetoxicfinancialmarketsoftheearlytwenty-firstcentury:creditdefaultswapsandcollateraliseddebtobligations.Acreditdefaultswapisaformofinsurance:payyourpremiumandyoucollectfromaninsurancecompanyifsomeonedefaultsonadebt.Butanyonecouldtakeoutsuchinsuranceonanything.Theydidn’thavetobethecompanythatowed,orwasowed,thedebt.Soahedgefundcould,ineffect,betthatabank’scustomersweregoingtodefaultontheirmortgagepayments–andiftheydid,thehedgefundwouldmakeabundle,eventhoughitwasnotapartytothemortgageagreements.Thisprovidedanincentiveforspeculatorstoinfluencemarketconditionstomakedefaultsmorelikely.Acollateraliseddebtobligationisbasedonacollection(portfolio)ofassets.Thesemightbetangible,suchasmortgagessecuredagainstrealproperty,ortheymightbederivatives,ortheymightbeamixtureofboth.Theowneroftheassetssellsinvestorstherighttoashareoftheprofitsfromthoseassets.Theinvestorcanplayitsafe,andgetfirstcallontheprofits,butthiscoststhemmore.Ortheycantakearisk,payless,andbelowerdownthepeckingorderforpayment.

Bothtypesofderivativeweretradedbybanks,hedgefunds,andotherspeculators.TheywerepricedusingdescendantsoftheBlack–Scholesequation,sotheywereconsideredtobeassetsintheirownright.Banksborrowedmoneyfromotherbanks,sothattheycouldlendittopeoplewhowantedmortgages;theysecuredtheseloanswithrealpropertyandfancyderivatives.Sooneveryonewaslendinghugesumsofmoneytoeveryoneelse,muchofitsecuredonfinancialderivatives.Hedgefundsandotherspeculatorsweretryingtomakemoneybyspottingpotentialdisastersandbettingthattheywouldhappen.The

valueofthederivativesconcerned,andofrealassetssuchasproperty,wasoftencalculatedonamarktomarketbasis,whichisopentoabusebecauseitusesartificialaccountingproceduresandriskysubsidiarycompaniestorepresentestimatedfutureprofitasactualpresent-dayprofit.Nearlyeveryoneinthebusinessassessedhowriskythederivativeswereusingthesamemethod,knownas‘valueatrisk’.Thiscalculatestheprobabilitythattheinvestmentmightmakealossthatexceedssomespecifiedthreshold.Forexample,investorsmightbewillingtoacceptalossofamilliondollarsifitsprobabilitywerelessthan5%,butnotifitweremorelikely.LikeBlack–Scholes,valueatriskassumesthattherearenofattails.Perhapstheworstfeaturewasthattheentirefinancialsectorwasestimatingitsrisksusingexactlythesamemethod.Ifthemethodwereatfault,thiswouldcreateashareddelusionthattheriskwaslowwheninrealityitwasmuchhigher.

Itwasatraincrashwaitingtohappen,acartooncharacterwhohadwalkedamileofftheedgeofthecliffandremainedsuspendedinmid-aironlybecauseheflatlyrefusedtotakealookatwhatwasunderhisfeet.AsPooveyandotherslikeherhadrepeatedlywarned,themodelsusedtovaluethefinancialproductsandestimatetheirrisksincorporatedsimplifyingassumptionsthatdidnotaccuratelyrepresentrealmarketsandthedangersinherentinthem.Playersinthefinancialmarketsignoredthesewarnings.Sixyearslater,weallfoundoutwhythiswasamistake.

Perhapsthereisabetterway.TheBlack–Scholesequationchangedtheworldbycreatingabooming

quadrillion-dollarindustry;itsgeneralisations,usedunintelligentlybyasmallcoterieofbankers,changedtheworldagainbycontributingtoamultitrillion-dollarfinancialcrashwhoseevermoremaligneffects,nowextendingtoentirenationaleconomics,arestillbeingfeltworldwide.Theequationbelongstotherealmofclassicalcontinuummathematics,havingitsrootsinthepartialdifferentialequationsofmathematicalphysics.Thisisarealminwhichquantitiesareinfinitelydivisible,timeflowscontinuously,andvariableschangesmoothly.Thetechniqueworksformathematicalphysics,butitseemslessappropriatetotheworldoffinance,inwhichmoneycomesindiscretepackets,tradesoccuroneatatime(albeitveryfast),andmanyvariablescanjumperratically.

TheBlack–Scholesequationisalsobasedonthetraditionalassumptionsofclassicalmathematicaleconomics:perfectinformation,perfectrationality,marketequilibrium,thelawofsupplyanddemand.Thesubjecthasbeentaught

fordecadesasifthesethingsareaxiomatic,andmanytrainedeconomistshaveneverquestionedthem.Yettheylackconvincingempiricalsupport.Onthefewoccasionswhenanyonedoesexperimentstoobservehowpeoplemakefinancialdecisions,theclassicalscenariosusuallyfail.It’sasthoughastronomershadspentthelasthundredyearscalculatinghowplanetsmove,basedonwhattheythoughtwasreasonable,withoutactuallytakingalooktoseewhattheyreallydid.

It’snotthatclassicaleconomicsiscompletelywrong.Butit’swrongmoreoftenthatitsproponentsclaim,andwhenitdoesgowrong,itgoesverywrongindeed.Sophysicists,mathematicians,andeconomistsarelookingforbettermodels.Attheforefrontoftheseeffortsaremodelsbasedoncomplexityscience,anewbranchofmathematicsthatreplacesclassicalcontinuumthinkingbyanexplicitcollectionofindividualagents,interactingaccordingtospecifiedrules.

Aclassicalmodelofthemovementofthepriceofsomecommodity,forexample,assumesthatatanyinstantthereisasingle‘fair’price,whichinprincipleisknowntoeveryone,andthatprospectivepurchaserscomparethispricewithautilityfunction(howusefulthecommodityistothem)andbuyitifitsutilityoutweighsitscost.Acomplexsystemmodelisverydifferent.Itmightinvolve,say,tenthousandagents,eachwithitsownviewofwhatthecommodityisworthandhowdesirableitis.Someagentswouldknowmorethanothers,somewouldhavemoreaccurateinformationthanothers;manywouldbelongtosmallnetworksthattradedinformation(accurateornot)aswellasmoneyandgoods.

Anumberofinterestingfeatureshaveemergedfromsuchmodels.Oneistheroleoftheherdinstinct.Markettraderstendtocopyothermarkettraders.Iftheydon’t,anditturnsoutthattheothersareontoagoodthing,theirbosseswillbeunhappy.Ontheotherhand,iftheyfollowtheherdandeveryone’sgotitwrong,theyhaveagoodexcuse:it’swhateveryoneelsewasdoing.Black–Scholeswasperfectfortheherdinstinct.Infact,virtuallyeveryfinancialcrisisinthelastcenturyhasbeenpushedovertheedgebytheherdinstinct.Insteadofsomebanksinvestinginpropertyandothersinmanufacturing,say,theyallrushintoproperty.Thisoverloadsthemarket,withtoomuchmoneyseekingtoolittleproperty,andthewholethingcomestobits.SonowtheyallrushintoloanstoBrazil,ortoRussia,orbackintoanewlyrevivedpropertymarket,orlosetheircollectivemarblesoverdotcomcompanies–threekidsinaroomwithacomputerandamodembeingvaluedattentimestheworthofamajormanufacturerwitharealproduct,realcustomers,andrealfactoriesandoffices.Whenthatgoesbelly-up,theyallrushintothesubprimemortgagemarket…

That’snothypothetical.Evenastherepercussionsoftheglobalbankingcrisisreverberatethroughordinarypeople’slives,andnationaleconomiesflounder,therearesignsthatnolessonshavebeenlearned.Arerunofthedotcomfadisinprogress,nowaimedatsocialnetworkingwebsites:Facebookhasbeenvaluedat$100billion,andTwitter(thewebsitewherecelebritiessend140-character‘tweets’totheirdevotedfollowers)hasbeenvaluedat$8billiondespiteneverhavingmadeaprofit.TheInternationalMonetaryFundhasalsoissuedastrongwarningaboutexchangetradedfunds(ETFs),averysuccessfulwaytoinvestincommoditieslikeoil,gold,orwheatwithoutactuallybuyingany.Allofthesehavegoneupinpriceveryrapidly,providingbigprofitsforpensionfundsandotherlargeinvestors,buttheIMFhaswarnedthattheseinvestmentvehicleshave‘allthehallmarksofabubblewaitingtoburst…reminiscentofwhathappenedinthesecuritisationmarketbeforethecrisis’.ETFsareverylikethederivativesthattriggeredthecreditcrunch,butsecuredincommoditiesratherthanproperty.ThestampedeintoETFshasdrivencommoditypricesthroughtheroof,inflatingthemoutofallproportiontotherealdemand.Manypeopleinthethirdworldarenowunabletoaffordstaplefoodstuffsbecausespeculatorsindevelopedcountriesaretakingbiggamblesonwheat.TheoustingofHosniMubarakinEgyptwastosomeextenttriggeredbyhugeincreasesinthepriceofbread.

ThemaindangeristhatETFsarestartingtoberepackagedintofurtherderivatives,likethecollateraliseddebtobligationsandcreditdefaultswapsthatburstthesubprimemortgagebubble.Ifthecommoditiesbubblebursts,wecouldseearerunofthecollapse:justdelete‘property’andinsert‘commodities’.Commoditypricesareveryvolatile,soETFsarehigh-riskinvestments–notagreatchoiceforapensionfund.Soonceagaininvestorsarebeingencouragedtotakeevermorecomplex,andevermorerisky,bets,usingmoneytheydon’thavetobuystakesinthingstheydon’twantandcan’tuse,inpursuitofspeculativeprofits–whilethepeoplewhodowantthosethingscannolongeraffordthem.

RemembertheDojimariceexchange?

Economicsisnottheonlyareatodiscoverthatitsprizedtraditionaltheoriesnolongerworkinanincreasinglycomplexworld,wheretheoldrulesnolongerapply.Anotherisecology,thestudyofnaturalsystemssuchasforestsorcoralreefs.Infact,economicsandecologyareuncannilysimilarinmanyrespects.Someoftheresemblanceisillusory:historicallyeachhasoftenusedtheothertojustifyitsmodels,insteadofcomparingthemodelswiththerealworld.Butsomeisreal:theinteractionsbetweenlargenumbersoforganismsareverylike

thosebetweenlargenumbersofstockmarkettraders.Thisresemblancecanbeusedasananalogy,inwhichcaseitisdangerous

becauseanalogiesoftenbreakdown.Oritcanbeusedasasourceofinspiration,borrowingmodellingtechniquesfromecologyandapplyingtheminsuitablymodifiedformtoeconomics.InJanuary2011,inthejournalNature,AndrewHaldaneandRobertMayoutlinedsomepossibilities.4Theirargumentsreinforceseveralofthemessagesearlierinthischapter,andsuggestwaysofimprovingthestabilityoffinancialsystems.

HaldaneandMaylookedatanaspectofthefinancialcrisisthatI’venotyetmentioned:howderivativesaffectthestabilityofthefinancialsystem.Theycomparetheprevailingviewoforthodoxeconomists,whichmaintainthatthemarketautomaticallyseeksastableequilibrium,withasimilarviewin1960secology,thatthe‘balanceofnature’tendstokeepecosystemsstable.Indeed,atthattimemanyecologiststhoughtthatanysufficientlycomplexecosystemwouldbestableinthisway,andthatunstablebehaviour,suchassustainedoscillations,impliedthatthesystemwasinsufficientlycomplex.WesawinChapter16thatthisiswrong.Infact,currentunderstandingindicatesexactlytheopposite.Supposethatalargenumberofspeciesinteractinanecosystem.Asthenetworkofecologicalinteractionsbecomesmorecomplexthroughtheadditionofnewlinksbetweenspecies,ortheinteractionsbecomestronger,thereisasharpthresholdbeyondwhichtheecosystemceasestobestable.(Herechaoscountsasstability;fluctuationscanoccurprovidedtheyremainwithinspecificlimits.)Thisdiscoveryledecologiststolookforspecialtypesofinteractionnetwork,unusuallyconducivetostability.

Mightitbepossibletotransfertheseecologicaldiscoveriestoglobalfinance?Therearecloseanalogies,withfoodorenergyinanecologycorrespondingtomoneyinafinancialsystem.HaldaneandMaywereawarethatthisanalogyshouldnotbeuseddirectly,remarking:‘Infinancialecosystems,evolutionaryforceshaveoftenbeensurvivalofthefattestratherthanthefittest.’Theydecidedtoconstructfinancialmodelsnotbymimickingecologicalmodels,butbyexploitingthegeneralmodellingprinciplesthathadledtoabetterunderstandingofecosystems.

Theydevelopedseveraleconomicmodels,showingineachcasethatundersuitablecircumstances,theeconomicsystemwouldbecomeunstable.Ecologistsdealwithanunstableecosystembymanagingitinawaythatcreatesstability.Epidemiologistsdothesamewithadiseaseepidemic;thisiswhy,forexample,theBritishgovernmentdevelopedapolicyofcontrollingthe2001foot-and-

mouthepidemicbyrapidlyslaughteringcattleonfarmsnearanythatprovedpositiveforthedisease,andstoppingallmovementofcattlearoundthecountry.Sogovernmentregulators’answertoanunstablefinancialsystemshouldbetotakeactiontostabiliseit.Tosomeextenttheyarenowdoingthis,afteraninitialpanicinwhichtheythrewhugeamountsoftaxpayers’moneyatthebanksbutomittedtoimposeanyconditionsbeyondvaguepromises,whichhavenotbeenkept.

However,thenewregulationslargelyfailtoaddresstherealproblem,whichisthepoordesignofthefinancialsystemitself.Thefacilitytotransferbillionsattheclickofamousemayallowever-quickerprofits,butitalsoletsshockspropagatefaster,andencouragesincreasingcomplexity.Bothofthesearedestabilising.Thefailuretotaxfinancialtransactionsallowstraderstoexploitthisincreasedspeedbymakingbiggerbetsonthemarket,atafasterrate.Thisalsotendstocreateinstability.Engineersknowthatthewaytogetarapidresponseistouseanunstablesystem:stabilitybydefinitionindicatesaninnateresistancetochange,whereasaquickresponserequirestheopposite.Sothequestforevergreaterprofitshascausedanevermoreunstablefinancialsystemtoevolve.

Buildingyetagainonanalogieswithecosystems,HaldaneandMayoffersomeexamplesofhowstabilitymightbeenhanced.Somecorrespondtotheregulators’owninstincts,suchasrequiringbankstoholdmorecapital,whichbuffersthemagainstshocks.Othersdonot;anexampleisthesuggestionthatregulatorsshouldfocusnotontherisksassociatedwithindividualbanks,butonthoseassociatedwiththeentirefinancialsystem.Thecomplexityofthederivativesmarketcouldbereducedbyrequiringalltransactionstopassthroughacentralisedclearingagency.Thiswouldhavetobeextremelyrobust,supportedbyallmajornations,butifitwere,thenpropagatingshockswouldbedampeddownastheypassedthroughit.

Anothersuggestionisincreaseddiversityoftradingmethodsandriskassessment.Anecologicalmonocultureisunstablebecauseanyshockthatoccursislikelytoaffecteverythingsimultaneously,inthesameway.Whenallbanksareusingthesamemethodstoassessrisk,thesameproblemarises:whentheygetitwrong,theyallgetitwrongatthesametime.Thefinancialcrisisaroseinpartbecauseallofthemainbankswerefundingtheirpotentialliabilitiesinthesameway,assessingthevalueoftheirassetsinthesameway,andassessingtheirlikelyriskinthesameway.

Thefinalsuggestionismodularity.Itisthoughtthatecosystemsstabilise

themselvesbyorganising(throughevolution)intomoreorlessself-containedmodules,connectedtoeachotherinafairlysimplemanner.Modularityhelpstopreventshockspropagating.Thisiswhyregulatorsworldwidearegivingseriousconsiderationtobreakingupbigbanksandreplacingthembyanumberofsmallerones.AsAlanGreenspan,adistinguishedAmericaneconomistandformerchairmanoftheFederalReserveoftheUSAsaidofbanks:‘Ifthey’retoobigtofail,they’retoobig.’

Wasanequationtoblameforthefinancialcrash,then?Anequationisatool,andlikeanytool,ithastobewieldedbysomeonehow

knowshowtouseit,andfortherightpurpose.TheBlack–Scholesequationmayhavecontributedtothecrash,butonlybecauseitwasabused.Itwasnomoreresponsibleforthedisasterthanatrader’scomputerwouldhavebeenifitsuseledtoacatastrophicloss.Theblameforthefailureoftoolsshouldrestwiththosewhoareresponsiblefortheiruse.Thereisadangerthatthefinancialsectormayturnitsbackonmathematicalanalysis,whenwhatitactuallyneedsisabetterrangeofmodels,and–crucially–asolidunderstandingoftheirlimitations.Thefinancialsystemistoocomplextoberunonhumanhunchesandvaguereasoning.Itdesperatelyneedsmoremathematics,notless.Butitalsoneedstolearnhowtousemathematicsintelligently,ratherthanassomekindofmagicaltalisman.

WhereNext?Whensomeonewritesdownanequation,thereisn’tasuddenclapofthunderafterwhicheverythingisdifferent.Mostequationshavelittleornoeffect(Iwritethemdownallthetime,andbelieveme,Iknow).Buteventhegreatestandmostinfluentialequationsneedhelptochangetheworld–efficientwaystosolvethem,peoplewiththeimaginationanddrivetoexploitwhattheytellus,machinery,resources,materials,money.Bearingthisinmind,equationshaverepeatedlyopenedupnewdirectionsforhumanity,andactedasourguidesasweexplorethem.

Ittookalotmorethanseventeenequationstogetuswherewearetoday.Mylistisaselectionofsomeofthemostinfluential,andeachofthemrequiredahostofothersbeforeitbecameseriouslyuseful.Buteachoftheseventeenfullydeservesinclusion,becauseitplayedapivotalroleinhistory.Pythagorasledtopracticalmethodsforsurveyingourlandsandnavigatingourwaytonewones.Newtontellsushowplanetsmoveandhowtosendspaceprobestoexplorethem.Maxwellprovidedavitalcluethatledtoradio,TV,andmoderncommunications.Shannonderivedunavoidablelimitstohowefficientthosecommunicationscanbe.

Often,whatanequationledtowasquitedifferentfromwhatinteresteditsinventor/discoverers.Whowouldhavepredictedinthefifteenthcenturythatabaffling,apparentlyimpossiblenumber,stumbleduponwhilesolvingalgebraproblems,wouldbeindeliblylinkedtotheevenmorebafflingandapparentlyimpossibleworldofquantumphysics–letalonethatthiswouldpavetheroadtomiraculousdevicesthatcansolveamillionalgebraproblemseverysecond,andletusinstantlybeseenandheardbyfriendsontheothersideoftheplanet?HowwouldFourierhavereactedifhehadbeentoldthathisnewmethodforstudyingheatflowwouldbebuiltintomachinesthesizeofapackofcards,abletopaintextraordinarilyaccurateanddetailedpicturesofanythingtheyarepointedat–incolour,evenmoving,withthousandsofthemcontainedinsomethingthesizeofacoin?

Equationstriggerevents,andevents,toparaphraseformerBritishPrimeMinisterHaroldMacmillan,arewhatkeepusawakeatnight.Whenarevolutionaryequationisunleashed,itdevelopsalifeofitsown.Theconsequencescanbegoodorbad,evenwhentheoriginalintentionwasbenevolent,asitwasforeveryoneofmyseventeen.Einstein’snewphysicsgaveusanewunderstandingoftheworld,butoneofthethingsweuseditfor

wasnuclearweapons.Notasdirectlyaspopularmythclaims,butitplayeditspartnonetheless.TheBlack–Scholesequationcreatedavibrantfinancialsectorandthenthreatenedtodestroyit.Equationsarewhatwemakeofthem,andtheworldcanbechangedfortheworseaswellasforthebetter.

Equationscomeinmanykinds.Somearemathematicaltruths,tautologies:thinkofNapier’slogarithms.Buttautologiescanstillbepowerfulaidstohumanthoughtanddeed.Somearestatementsaboutthephysicalworld,whichforallweknowcouldhavebeendifferent.Equationsofthiskindtellusnature’slaws,andsolvingthemtellsustheconsequencesofthoselaws.Somehavebothelements:Pythagoras’sequationisatheoreminEuclid’sgeometry,butitalsogovernsmeasurementsmadebysurveyorsandnavigators.Somearelittlebetterthandefinitions–butiandinformationtellusagreatdeal,oncewehavedefinedthem.

Someequationsareuniversallyvalid.Somedescribetheworldveryaccurately,butnotperfectly.Somearelessaccurate,confinedtomorelimitedrealms,yetoffervitalinsights.Somearebasicallyplainwrong,yettheycanactasstepping-stonestosomethingbetter.Theymaystillhaveahugeeffect.

Someevenopenupdifficultquestions,philosophicalinnature,abouttheworldweliveinandourownplacewithinit.Theproblemofquantummeasurement,dramatisedbySchrödinger’shaplesscat,isonesuch.Thesecondlawofthermodynamicsraisesdeepissuesaboutdisorderandthearrowoftime.Inbothcases,someoftheapparentparadoxescanberesolved,inpart,bythinkinglessaboutthecontentoftheequationandmoreaboutthecontextinwhichitapplies.Notthesymbols,buttheboundaryconditions.Thearrowoftimeisnotaproblemaboutentropy:it’saproblemaboutthecontextinwhichwethinkaboutentropy.

Existingequationscanacquirenewimportance.Thesearchforfusionpower,asacleanalternativetonuclearpowerandfossilfuels,requiresanunderstandingofhowextremelyhotgas,formingaplasma,movesinamagneticfield.Theatomsofthegasloseelectronsandbecomeelectricallycharged.Sotheproblemisoneinmagnetohydrodynamics,requiringacombinationoftheexistingequationsforfluidflowandforelectromagnetism.Thecombinationleadstonewphenomena,suggestinghowtokeeptheplasmastableatthetemperaturesneededtoproducefusion.Theequationsareoldfavourites.

Thereis(ormaybe)oneequation,aboveall,thatphysicistsandcosmologistswouldgivetheireyeteethtolayhandson:aTheoryofEverything,whichinEinstein’sdaywascalledaUnifiedFieldTheory.Thisisthelong-sought

equationthatunifiesquantummechanicsandrelativity,andEinsteinspenthislateryearsinafruitlessquesttofindit.Thesetwotheoriesarebothsuccessful,buttheirsuccessesoccurindifferentdomains:theverysmallandtheverylarge.Whentheyoverlap,theyareincompatible.Forexample,quantummechanicsislinear,relativityisn’t.Wanted:anequationthatexplainswhybotharesosuccessful,butdoesthejobofbothwithnologicalinconsistencies.TherearemanycandidatesforaTheoryofEverything,thebestknownbeingthetheoryofsuperstrings.This,amongotherthings,introducesextradimensionsofspace:sixofthem,seveninsomeversions.Superstringsaremathematicallyelegant,butthereisnoconvincingevidenceforthemasadescriptionofnature.Inanycase,itisdesperatelyhardtocarryoutthecalculationsneededtoextractquantitativepredictionsfromsuperstringtheory.

Forallweknow,theremaynotbeaTheoryofEverything.Allofourequationsforthephysicalworldmayjustbeoversimplifiedmodels,describinglimitedrealmsofnatureinawaythatwecanunderstand,butnotcapturingthedeepstructureofreality.Evenifnaturetrulyobeysrigidlaws,theymightnotbeexpressibleasequations.

Evenifequationsarerelevant,theyneednotbesimple.Theymightbesocomplicatedthatwecan’tevenwritethemdown.The3billionDNAbasesofthehumangenomeare,inasense,partoftheequationforahumanbeing.Theyareparametersthatmightbeinsertedintoamoregeneralequationforbiologicaldevelopment.Itis(barely)possibletoprintthegenomeonpaper;itwouldneedabouttwothousandbooksthesizeofthisone.Itfitsintoacomputermemoryfairlyeasily.Butit’sonlyonetinypartofanyhypotheticalhumanequation.

Whenequationsbecomethatcomplex,weneedhelp.Computersarealreadyextractingequationsfrombigsetsofdata,incircumstanceswheretheusualhumanmethodsfailoraretooopaquetobeuseful.Anewapproachcalledevolutionarycomputingextractssignificantpatterns:specifically,formulasforconservedquantities–thingsthatdon’tchange.OnesuchsystemcalledEureqa,formulatedbyMichaelSchmidtandHodLipson,hasscoredsomesuccesses.Softwarelikethismighthelp.Oritmightnotleadanywherethatreallymatters.

Somescientists,especiallythosewithbackgroundsincomputing,thinkthatit’stimeweabandonedtraditionalequationsaltogether,especiallycontinuumoneslikeordinaryandpartialdifferentialequations.Thefutureisdiscrete,itcomesinwholenumbers,andtheequationsshouldgivewaytoalgorithms–recipesforcalculatingthings.Insteadofsolvingtheequations,weshouldsimulatetheworlddigitallybyrunningthealgorithms.Indeed,theworlditself

maybedigital.StephenWolframmadeacaseforthisviewinhiscontroversialbookANewKindofScience,whichadvocatesatypeofcomplexsystemcalledacellularautomaton.Thisisanarrayofcells,typicallysmallsquares,eachexistinginavarietyofdistinctstates.Thecellsinteractwiththeirneighboursaccordingtofixedrules.Theylookabitlikeaneightiescomputergame,withcolouredblockschasingeachotheroverthescreen.

Wolframputsforwardseveralreasonswhycellularautomatashouldbesuperiortotraditionalmathematicalequations.Inparticular,someofthemcancarryoutanycalculationthatcouldbeperformedbyacomputer,thesimplestbeingthefamousRule110automaton.Thiscanfindsuccessivedigitsofπ,solvethethree-bodyequationsnumerically,implementtheBlack–Scholesformulaforacalloption–whatever.Traditionalmethodsforsolvingequationsaremorelimited.Idon’tfindthisargumentterriblyconvincing,becauseitisalsotruethatanycellularautomatoncanbesimulatedbyatraditionaldynamicalsystem.Whatcountsisnotwhetheronemathematicalsystemcansimulateanother,butwhichismosteffectiveforsolvingproblemsorprovidinginsights.It’squickertosumatraditionalseriesforπbyhandthanitistocalculatethesamenumberofdigitsusingtheRule110automaton.

However,itisstillentirelycrediblethatwemightsoonfindnewlawsofnaturebasedondiscrete,digitalstructuresandsystems.Thefuturemayconsistofalgorithms,notequations.Butuntilthatdaydawns,ifever,ourgreatestinsightsintonature’slawstaketheformofequations,andweshouldlearntounderstandthemandappreciatethem.Equationshaveatrackrecord.Theyreallyhavechangedtheworld–andtheywillchangeitagain.

IncredibleNumbersbyProfessorIanStewart

AvailablenowoniPad

incrediblenumbersapp.com

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NotesChapter1

1ThePenguinBookofCuriousandInterestingMathematicsbyDavidWellsquotesabriefformofthejoke:AnIndianchiefhadthreewiveswhowerepreparingtogivebirth,oneonabuffalohide,oneonabearhide,andthethirdonahippopotamushide.Induecourse,thefirstgavehimason,thesecondadaughter,andthethird,twins,aboyandagirl,therebyillustratingthewell-knowntheoremthatthesquawonthehippopotamusisequaltothesumofthesquawsontheothertwohides.Thejokegoesbackatleasttothemid-1950s,whenitwasbroadcastintheBBCradioseries‘MyWord’,hostedbycomedyscriptwritersFrankMuirandDenisNorden.

2Quotedwithoutreferenceon:http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_Pythagoras.html

3A.Sachs,A.Goetze,andO.Neugebauer.MathematicalCuneiformTexts,AmericanOrientalSociety,NewHaven1945.

4ThefigureisrepeatedforconvenienceinFigure60.

Fig60Splittingatriangleintotwowithrightangles.

Theperpendicularcutsthesidebintotwopieces.Bytrigonometry,onepiecehaslengthacosC,sotheotherhaslengthb-acosC.Lethbetheheightoftheperpendicular.ByPythagoras:

a2=h2+(acosC)2

c2=h2+(b-acosC)2

Thatis,

a2-h2=a2cos2C

c2-h2=(b-acosC)2=b2-2abcosC+a2cos2C

Subtractthefirstequationfromthesecond;nowtheunwantedh2cancelsout.Sodothetermsa2cos2C,andweareleftwith

c2-a2=b2-2abcosC

whichleadstothestatedformula.

Chapter2

1http://www.17centurymaths.com/contents/napiercontents.html

2QuotedfromaletterJohnMarrwrotetoWilliamLilly.

3ProsthapheiresiswasbasedonatrigonometricformuladiscoveredbyFrançoisViète,namely

Ifyouownedatableofsines,theformulaallowedyoutocalculateanyproductusingonlysums,differences,anddivisionby2.

Chapter3

1Keynesneverdeliveredthelecture.TheRoyalSocietyplannedtocommemorateIsaacNewton’stercentenaryin1942,butWorldWarIIintervened,sothecelebrationswerepostponedto1946.ThelecturerswerethephysicistsEdwarddaCostaAndradeandNielsBohr,andthemathematiciansHerbertTurnbullandJacquesHadamard.ThesocietyhadalsoinvitedKeynes,whoseinterestsincludedNewton’smanuscriptsaswellaseconomics.Hehadwrittenalecturewiththetitle‘Newton,theman’,buthediedjustbeforetheeventtookplace.HisbrotherGeoffreyreadthelectureonhisbehalf.

2ThisphrasecomesfromaletterthatNewtonwrotetoHookein1676.Itwasn’tnew:in1159JohnofSalisburywrotethat‘BernardofChartresusedtosaythatwearelikedwarfsontheshouldersofgiants,sothatwecanseemorethanthey.’Bytheseventeenthcenturyithadbecomeacliché.

3Divisionbyzeroleadstofallaciousproofs.Forexample,wecan‘prove’thatallnumbersarezero.Assumethata=b.Thereforea2=ab,soa2–b2=ab–b2.Factorisetoget(a+b)(a–b)=b(a–b).Divideby(a–b)todeducethata+b=b.Thereforea=0.Theerroristhedivisionby(a–b),whichis0becauseweassumeda=b.

4RichardWestfall.NeveratRest,CambridgeUniversityPress,Cambridge1980,p.425.

5ErikH.Hauri,ThomasWeinreich,AlbertoE.Saal,MalcolmC.Rutherford,andJamesA.VanOrman.Highpre-eruptivewatercontentspreservedinlunarmeltinclusions,ScienceOnline(26May2011)1204626.[DOI:10.1126/science.1204626].Theirresultsprovedcontroversial.

6However,it’snotcoincidence.Itworksforanydifferentiablefunction:onewithacontinuousderivative.Theseincludeallpolynomialsandallconvergentpowerseries,suchasthelogarithm,theexponential,andthevarioustrigonometricfunctions.

7Themoderndefinitionis:afunctionf(h)tendstoalimitLashtendstozeroifforanyε>0thereexistsδ>0suchthat|h|<δimpliesthat|f(h)–L|<ε.Usinganyε>0avoidsreferringtoanythingflowingorbecomingsmaller:itdealswithallpossiblevaluesinonego.

Chapter4

1ThebookofGenesisreferstothe‘firmament’.MostscholarsthinkthisderivesfromtheancientHebrewbeliefthatthestarsweretinylightsfixedtoasolidvaultofHeaven,shapedlikeahemisphere.Thisiswhatthenightskylookslike:thewayourvisualsensesrespondtodistantobjectsmakesthestarsappeartobeatmuchthesamedistancefromus.Manycultures,especiallyintheMiddleandFarEast,thoughtoftheheavensasaslowlyspinningbowl.

2TheGreatCometof1577isnotHalley’scomet,butanotherofhistoricalimportance,nowcalledC/1577V1.Itwasvisibletothenakedeyein1577AD.BraheobservedthecometanddeducedthatcometswerelocatedoutsidetheEarth’satmosphere.Thecometiscurrentlyabout24billionkilometresfromtheSun.

3Thefigurewasnotknownuntil1798,whenHenryCavendishobtainedareasonablyaccuratevalueinalaboratoryexperiment.Itisabout6.67×610–11newtonmetresquaredperkilogramsquared.

4JuneBarrow-Green.PoincaréandtheThreeBodyProblem,AmericanMathematicalSociety,Providence1997.

Chapter5

1In1535themathematiciansAntonioFiorandNiccolòFontana(nicknamed

Tartaglia,‘thestammerer’)engagedinapubliccontest.Theyseteachothercubicequationstosolve,andTartagliabeatFiorcomprehensively.Atthattime,cubicequationswereclassifiedintothreedistincttypes,becausenegativenumberswerenotrecognised.Fiorknewhowtosolvejustonetype;initiallyTartagliaknewhowtosolveonedifferenttype,butshortlybeforethecontesthefiguredouthowtosolvealltheothertypes.HethensetFioronlythetypesthatheknewFiorcouldnotsolve.Cardano,workingonhisalgebratext,heardaboutthecontest,andrealisedthatFiorandTartagliaknewhowtosolvecubics.Thisdiscoverywouldgreatlyenhancethebook,soheaskedTartagliatorevealhismethods.EventuallyTartagliadivulgedthesecret,laterstatingthatCardanohad

promisednevertomakeitpublic.ButthemethodappearedintheArsMagna,soTartagliaaccusedCardanoofplagiarism.However,Cardanohadanexcuse,andhealsohadagoodreasontofindawayroundhispromise.HisstudentLodovicoFerrarihadfoundhowtosolvequarticequations,anequallynovelanddramaticdiscovery,andCardanowantedthatinhisbook,too.However,Ferrari’smethodrequiredthesolutionofanassociatedcubicequation,soCardanocouldnotpublishFerrari’sworkwithoutalsopublishingTartaglia’s.ThenhelearnedthatFiorwasastudentofScipiodelFerro,whowas

rumouredtohavesolvedallthreetypesofcubic,passingjustonetypeontoFior.DelFerro’sunpublishedpaperswereinthepossessionofAnnibaledelNave.SoCardanoandFerrariwenttoBolognain1543toconsultdelNave,andinthepaperstheyfoundsolutionstoallthreetypesofcubic.SoCardanocouldhonestlysaythathewaspublishingdelFerro’smethod,notTartaglia’s.Tartagliastillfeltcheated,andpublishedalong,bitterdiatribeagainstCardano.Ferrarichallengedhimtoapublicdebateandwonhandsdown.Tartaglianeverreallyrecoveredhisreputationafterthat.

Chapter6

1SummarisedinChapter12of:IanStewart.MathematicsofLife,Profile,London2011.

Chapter7

1Yes,Iknowthisisthepluralof‘die’,butnowadayseveryoneusesitforthesingularaswell,andI’vegivenupfightingthistendency.Itcouldbeworse:someonejustsentmeane-mailcarefullyusing‘dice’forthesingularand‘die’fortheplural.

2TherearemanyfallaciesinPascal’sargument.Themainoneisthatitwouldapplytoanyhypotheticalsupernaturalbeing.

3Thetheoremstatesthatundercertain(fairlycommon)conditions,thesumofalargenumberofrandomvariableswillhaveanapproximatelynormaldistribution.Moreprecisely,if(x1,…,xn)isasequenceofindependentidenticallydistributedrandomvariables,eachhavingmeanμandvarianceσ2,thenthecentrallimittheoremstatesthat

convergestoanormaldistributionwithmean0andstandarddeviationσasnbecomesarbitrarilylarge.

Chapter8

1Lookatthreeconsecutivemasses,numberedn–1,n,n+1.Supposethatattimettheyaredisplaceddistancesun-1(t),un(t),andun+1(t)fromtheirinitialpositionsonthehorizontalaxis.ByNewton’ssecondlawtheaccelerationofeachmassisproportionaltotheforcesthatactonit.Makethesimplifyingassumptionthateachmassmovesthroughaverysmalldistanceintheverticaldirectiononly.Toaverygoodapproximation,theforcethatmassn–1exertsonmassnisthenproportionaltothedifferenceun–1(t)–un(t),andsimilarlytheforcethatmassn+1exertsonmassnisproportionaltothedifferenceun+1(t)–un(t).Addingthesetogether,thetotalforceexertedonmassnisproportionaltoun–1(t)–2un(t)+un+1(t).Thisisthedifferencebetweenun–1(t)–un(t)andun(t)–un+1(t),andeachoftheseexpressionsisalsothedifferencebetweenthepositionsofconsecutivemasses.Sotheforceexertedonmassnisadifferencebetweendifferences.Nowsupposethemassesareveryclosetogether.Incalculus,adifference

–dividedbyasuitablesmallconstant–isanapproximationtoaderivative.Adifferencebetweendifferencesisanapproximationtoaderivativeofaderivative,thatis,asecondderivative.Inthelimitofinfinitelymanypointmasses,infinitesimallyclosetogether,theforceexertedatagivenpointofthespringisthereforeproportionalto∂2u/∂x2,wherexisthespacecoordinatemeasuredalongthelengthofthestring.ByNewton’ssecondlawthisisproportionaltotheaccelerationatrightanglestothatline,whichisthesecondtimederivative∂2u/∂t2.Writingtheconstantofproportionalityasc2weget

whereu(x,t)istheverticalpositionoflocationxonthestringattimet.

2Forananimationseehttp://en.wikipedia.org/wiki/Wave_equation

3Insymbols,thesolutionsarepreciselytheexpressions

u(x,t)=f(x-ct)+g(x+ct)

foranyfunctionsfandg.

4Animationsofthefirstfewnormalmodesofacirculardrumcanbefoundathttp://en.wikipedia.org/wiki/Vibrations_of_a_circular_drumCircularandrectangulardrumanimationsareathttp://www.mobiusilearn.com/viewcasestudies.aspx?id=2432

Chapter9

1Supposethatu(x,t)=e-n2αtsinnx.Then

Thereforeu(x,t)satisfiestheheatequation.

2ThisisJFIFencoding,usedfortheweb.EXIFcoding,forcameras,alsoincludes‘metadata’describingthecamerasettings,suchasdate,time,andexposure.

Chapter10

1http://www.nasa.gov/topics/earth/features/2010-warmest-year.html

Chapter11

1DonaldMcDonald.Howdoesacatfallonitsfeet?,NewScientist7no.189(1960)1647–9.Seealsohttp://en.wikipedia.org/wiki/Cat_righting_reflex

2Thecurlofbothsidesofthethirdequationgives

Vectorcalculustellsusthattheleft-handsideofthisequationsimplifiesto

∇×∇×E=∇(∇.E)-∇2E)=-∇2E

wherewealsousethefirstequation.Here∇2istheLaplaceoperator.Using

thefourthequation,theright-handsidebecomes

Cancellingouttwominussignsandmultiplyingbyc2yieldsthewaveequationforE:

AsimilarcalculationyieldsthewaveequationforH.

Chapter12

1Specifically,

whereSAandSBaretheentropiesinstatesAandB.

2Thesecondlawofthermodynamicsistechnicallyaninequality,notanequation.I’veincludedthesecondlawinthisbookbecauseitscentralpositioninsciencedemandeditsinclusion.Itisundeniablyamathematicalformula,alooseinterpretationof‘equation’thatiswidespreadoutsidethetechnicalscientificliterature.TheformulaalludedtoinNote1ofthischapter,usinganintegral,isagenuineequation.Itdefinesthechangeinentropy,butthesecondlawtellsuswhatitsmostimportantfeatureis.

3BrownwasanticipatedbytheDutchphysiologistJanIngenhousz,whosawthesamephenomenonincoaldustfloatingonthesurfaceofalcohol,buthedidn’tproposeanytheorytoexplainwhathehadseen.

Chapter13

1IntheGranSassoNationalLaboratory,inItaly,isa1300-tonneparticledetectorcalledOPERA(oscillationprojectwithemulsion-trackingapparatus).Overtwoyearsittracked16,000neutrinosproducedatCERN,theEuropeanparticlephysicslaboratoryinGeneva.Neutrinosareelectricallyneutralsubatomicparticleswithaverysmallmass,andtheycanpassthroughordinarymatterwithease.Theresultswerebaffling:onaveragetheneutrinoscompletedthe730-kilometretrip60nanoseconds(billionthsofasecond)soonerthantheywouldhavedoneiftheyhadbeentravellingatthespeedoflight.Themeasurementsareaccuratetowithin10nanoseconds,butthereremainsthepossibilityofsomesystematicerrorin

thewaythetimesarecalculatedandinterpreted,whichishighlycomplex.Theresultshavebeenpostedonline:‘Measurementoftheneutrino

velocitywiththeOPERAdetectorintheCNGSbeam’bytheOPERACollaboration,http://arxiv.org/abs/1109.4897Thisarticledoesnotclaimtohavedisprovedrelativity:itmerelypresents

itsobservationsassomethingthattheteamcannotexplainwithconventionalphysics.Anon-technicalreportcanbefoundathttp://www.nature.com/news/2011/110922/full/news.2011.554.htmlApossiblesourceofsystematicerror,relatedtodifferencesintheforce

ofgravityatthetwolaboratories,isproposedathttp://www.nature.com/news/2011/111005/full/news.2011.575.htmlbuttheOPERAteamdisputesthissuggestion.Mostphysiciststhinkthat,despitethegreatcareexercisedbythe

researchers,somesystematicerrorisinvolved.Inparticular,previousobservationsofneutrinosfromasupernovaseemtoconflictwiththenewones.Theresolutionofthecontroversywillrequireindependentexperiments,andthesewilltakeseveralyears.Theoreticalphysicistsarealreadyanalysingpotentialexplanationsrangingfromminor,well-knownextensionsofthestandardmodelofparticlephysicstoexoticnewphysicsinwhichtheuniversehasmoredimensionsthantheusualfour.Bythetimeyoureadthis,thestorywillalreadyhavemovedon.

2AthoroughexplanationisgivenbyTerenceTaoonhiswebsite:http://terrytao.wordpress.com/2007/12/28/einsteins-derivation-of-emc2/Thederivationoftheequationinvolvesfivesteps:(a)Describehowspaceandtimecoordinatestransformwhentheframeof

referenceischanged.(b)Usethisdescriptiontoworkouthowthefrequencyofaphoton

transformswhentheframeofreferenceischanged.(c)UsePlanck’slawtoworkouthowtheenergyandmomentumofa

photontransform.(d)Applyconservationofenergyandmomentumtoworkouthowthe

energyandmomentumofamovingbodytransform.(e)Fixthevalueofanotherwisearbitraryconstantinthecalculationby

comparingtheresultswithNewtonianphysicswhenthevelocityofthebodyissmall.

3IanStewartandJackCohen.FigmentsofReality,CambridgeUniversity

Press,Cambridge1997,page37.

4http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence

5Afewdidn’tseeitthatway.HenryCourten,reanalysingphotographsofthe1970solareclipse,reportedtheexistenceofatleastsevenverytinybodiesincloseorbitsroundtheSun–perhapsevidenceofathinlypopulatedinnerasteroidbelt.Noconclusiveevidenceoftheirexistencehasbeenfound,andtheywouldhavetobelessthan60kilometresacross.Theobjectsseeninthephotographsmayjusthavebeenpassingsmallcometsorasteroidsineccentricorbits.Whatevertheywere,theyweren’tVulcan.

6Thevacuumenergyinacubiccentimetreoffreespaceisestimatedtobe10–15joules.Accordingtoquantumelectrodynamicsitshouldintheorybe10107joules–wrongbyafactorof10122.http://en.wikipedia.org/wiki/Vacuum_energy

7Penrose’sworkisreportedin:PaulDavies.TheMindofGod,Simon&Schuster,NewYork1992.

8JoelSmollerandBlakeTemple.AoneparameterfamilyofexpandingwavesolutionsoftheEinsteinequationsthatinducesananomalousaccelerationintothestandardmodelofcosmology.http://arxiv.org/abs/0901.1639

9R.S.MacKayandC.P.Rourke.Anewparadigmfortheuniverse,preprint,UniversityofWarwick2011.Formoredetailsseethepaperslistedonhttp://msp.warwick.ac.uk/~cpr/paradigm/

Chapter14

1TheCopenhageninterpretationisusuallysaidtohaveemergedfromdiscussionsbetweenNielsBohr,WernerHeisenberg,MaxBorn,andothers,inthemid-1920s.ItacquiredthenamebecauseBohrwasDanish,butnoneofthephysicistsinvolvedusedthetermatthetime.DonHowardhassuggestedthatthename,andtheviewpointthatitencapsulates,firstappearedinthe1950s,probablythroughHeisenberg.See:D.Howard.‘WhoInventedthe“CopenhagenInterpretation”?AStudyinMythology’,PhilosophyofScience71(2004)669–682.

2OurcatHarlequincanoftenbeobservedinasuperpositionofthestates‘asleep’and‘snoring’,butthatprobablydoesn’tcount.

3TwosciencefictionnovelsaboutthisarePhilipK.Dick’sTheManintheHighCastleandNormanSpinrad’sTheIronDream.ThrillerwriterLen

Deighton’sSSGBisalsosetinacounterfactualNazi-ruledEngland.

Chapter15

1SupposeIrolladice[seeNote1ofChapter7]andassignsymbolsa,b,clikethis:

aThedicerolls1,2,or3

bThedicerolls4or5

cThedicerolls6

Symbolaoccurswithprobability symbolbhasprobability ,andsymbolchasprobability .Thenmyformula,whateveritis,willassignaninformationcontentH( , , ).However,Icouldthinkofthisexperimentinadifferentway.FirstI

decidewhetherthedicerollssomethinglessthanorequalto3,orgreater.Callthesepossibilitiesqandr,sothat

qThedicerolls1,2,or3

rThedicerolls4,5,or6

Nowqhasprobability andrhasprobability .EachconveysinformationH( , ).Caseqismyoriginala,andcaserismyoriginalbandc.Icansplitcaserintobandc,andtheirprobabilitiesare and giventhatrhashappened.Ifwenowconsideronlythiscase,theinformationconveyedbywhicheverofbandcturnsupisH( , ).Shannonnowinsiststhattheoriginalinformationshouldberelatedtotheinformationinthesesubcaseslikethis:

SeeFigure61.

Fig61Combiningchoicesindifferentways.Theinformationshouldbethesameineithercase.

Thefactor infrontofthefinalHispresentbecausethissecondchoiceoccursonlyhalfthetime,namelywhenrischoseninthefirststage.There

isnosuchfactorinfrontoftheHjustaftertheequalssign,becausethisreferstoachoicethatisalwaysmade–betweenqandr.

2SeeChapter2of:C.E.ShannonandW.Weaver.TheMathematicalTheoryofCommunication,UniversityofIllinoisPress,Urbana1964.

Chapter16

1Ifthepopulationxtisrelativelysmall,sothatisclosetozero,then1–xtiscloseto1.Thenextgenerationwillthereforehaveasizeclosetokxt,whichisktimesaslargeasthecurrentone.Asthesizeofthepopulationincreases,theextrafactor1–xtmakestheactualgrowthratesmaller,anditdropstozeroasthepopulationapproachesitstheoreticalmaximum.

2R.F.Costantino,R.A.Desharnais,J.M.Cushing,andB.Dennis.Chaoticdynamicsinaninsectpopulation,Science275(1997)389–391.

3J.HuismanandF.J.Weissing.Biodiversityofplanktonbyspeciesoscillationsandchaos,Nature402(1999)407–410.

4E.Benincà,J.Huisman,R.Heerkloss,K.D.Jöhnk,P.Branco,E.H.VanNes,M.Scheffer,andS.P.Ellner.Chaosinalong-termexperimentwithaplanktoncommunity,Nature451(2008)822–825.

Chapter17

1Thevalueofacalloptionis

C(s,t)=N(d1)S-N(d2)Ke-r(T-t)

where

Thepriceofacorrespondingputoptionis

HereN(dj)isthecumulativedistributionfunctionofthestandardnormaldistributionforj=1,2,andT–tisthetimetomaturity.

2Strictly,aSverigesRiksbankPrizeinEconomicSciencesinMemoryof

AlfredNobel.

3M.Poovey.Cannumbersensurehonesty?UnrealisticexpectationsandtheU.S.accountingscandal,NoticesoftheAmericanMathematicalSociety50(2003)27–35.

4A.G.HaldaneandR.M.May.Systemicriskinbankingecosystems,Nature469(2011)351–355.

IllustrationCreditsThefollowingfiguresarereproducedwiththepermissionofthenamedcopyrightowners:Figure9:JohanHidding.Figures11,41:WikimediaCommons.ReproducedunderthetermsoftheGNUFreeDocumentationLicense.

Figures15,16,17:WangSangKoon,MartinLo,ShaneRossandJerroldMarsden.

Figure31:AndrzejStasiak.Figure43:BMWSauberFormula1Team.Figure53:WillemSchaap.Figure59:JefHuismanandFranzWeissing,Nature402(1999)407-410.

IndexThesuffix‘n’indicatesanote.ItalicpagereferencesrefertoFigures.

Aabstractalgebra101,103,276,279–80acceleration,defined39–40aerodynamics86,168,170,172airtravel167aircraftdesign172(Taqi)al-DinMuhammadibnMa’rufal-Shamial-Asadi199al-Katibi,Najmal-Dinal-Qazwini57Alexander,JamesWaddell101,103–4algebraabstract101,103,276,279–80Cardanoand75,77–8notationof78relationtogeometry9

algorithms157–60,264,320Alhazen57‘alternateworlds’258AmericanPsychologicalAssociation125analoguecomputers172,270analysis51,82seealsocalculusanalysisofvariancemethod125angularmomentum49animalpopulationdynamics286–7,291–2,330nApollomissions65–6,168Arago,François233arbitragepricingtheory304,306Archimedes3,57,169Argand,Jean-Robert81–2,87AristarchusofSamos56Aristotle46–7arithmetic,rulesof77

Arnold,Vladimir289–90Arp,Halton241arrowoftime208,211,213,318Aryabhata56astronomychaoticdynamicsin290–1geocentricuniverse8interpretationsofthenightsky55–6lasersin263testsofrelativity233–5useoflogarithms29

atomicbomb229atomictheory.seemolecules‘averageman’116

BBabylonianmathematics6–7,80Bachelier,Louis302–4bankingsystemacceptanceofderivatives297modularityand314

base,logarithms27–8,30beats143,144Belbruno,Edward69,70Bell,AlexanderGraham183‘bellcurve’seenormaldistributionTheBellCurve(book)123–5,127Berkeley,George,BishopofCloyne44–6,50–1Bernoulli,Daniel169,204–5Bernoulli,Jacob112–14Bernoulli,John137–9,141–3,145,154Besselfunctions146BigBang210,235–9,241binarynotation271–3binomialcoefficients113–14binomialdistribution120

bioinformatics281Black,Fischer304,305blackbodyradiation248blackholes235–6Black–ScholesEquation295,304–6,314,318,320critiqueof310–11variantsof306,309

‘blackswanevents’303blip-likeFourieranalysis161–2bloodflow174–5Boltzmann,Ludwig206–9,247–9,281Bolzano,Bernard51Bombelli,Rafael75,78,80–1Boulliau,Ismaël(Bullialdus)59,61boundaryconditions212,214,318Boyle,Robert200Boyle’slaw205–6Brahe,Tycho12,58braids103Branca,Giovanni199Briggs,Henry24–5,27–8Brownianmotion(RobertBrown)208asfinancialsystemmodel301,303,304,307

bubonicplague37–8‘butterflyeffect’290,292–3

Ccalculation,Napieron24–5calculuscomplexanalysis82,84history38–42Leibnizand39,42–5,50logicalflaws44–6,51Newtonand37–8,41–6,50partialderivatives139–40utility50,51–2

Cantor,Georg156cardesign172,174carbondioxide176–7,209–10Cardano,Girolamo75,77–80,109–10,323n–324nCarnotcycle(NicolasLéonardSadiCarnot)202–4Cartesiancoordinates(RenéDescartes)13cellularautomata320centrallimittheorem119–20,324nCFD(computationalfluiddynamics)170,173chaostheory64–5,69,70breadmakinganalogy289–90deterministicchaos283,285,291,293logisticequation283,286populationdynamicsand286–7,291–2,330nsensitivitytoinitialconditions290topologyand92,105weatherand175,292–3

Charles,JacquesAlexandreCésar200circlesaerofoilsfrom86equationsfor14knotsas100linesofforceas85

Clapeyron,Émile201Clausius,Rudolf203,205–6climatechange175–7,209clusteringandstatistics127codingtheory276–9Cohen,Jack229,328ncointosses110,112–16,272–4,288collateraliseddebtobligations309,312colouring,knotdiagrams102cometsGiacobini–Zinner69GreatCometof157758,323nOterma66–7

commodityprices298,304,310–12communicationchannelscapacity275noise269

commutativelaw87,279complexanalysis82,100complexequations86complexfunctions85–6complexnumbers81,82,250–2definitionof87logarithmsof29seealsoimaginarynumbers

complexityscienceandeconomicmodels310compressiblefluid,airas168compressionalgorithms157–60computationalfluiddynamics(CFD)170,173computers,marketgrowth271confidence,statistical122conicsections57ellipses41,57–9,61,63parabolas41,48,57

conservationlaws48,49,225conservationofenergyfirstlawofthermodynamicsand200,201–2kinetictheoryand205lawsofmotionand48,68specialrelativityand225

conservationofmomentum171,225continuityintopology92,96–7continuousfunctions46,156,274continuummathematics310,320continuummechanics174continuummodels171convergenceissues155–6coordinategeometry13,161Copenhageninterpretation252–62,328n

Copernicus,Nicolaus57–8‘cosinerule’11cosines10,83–4,142–3,152–4cosmologicalconstant236,237cosmologicalmicrowavebackground236,241creditdefaultswaps308–9,312Crick,Francis105crownandanchorgame111cryptography269,280cuberoots29,64,79cubes,andtopology91cubicequations79–80,323n–324ncurl,Maxwell’sequations187–8,326ncurvedsurfacegeometry15–19,231–2

Dd’Alembert,JeanLeRond139–42,145darkenergy237,238,239,240,242darkmatter237–8,239,240,242datacompressiondigitalphotography157–60fingerprints162medicalimaging163

Daubechies,Ingrid161,162Davy,Humphry182deBroglie,Louis-Victor249decibelunit34decoherence257–8,264Dellnitz,Michael72derivatives(calculus)43–5,47,50–2partialderivatives139–40,170

derivatives(financialinstruments)297–8,308–9volumestraded300,308–9

Descartes,René13,91,95determinismdistinguishedfrompredictability290–1,293

distinguishedfromrandomness288deterministicchaos283,285,291,293differentialequations47,52complexfunctionsand85–6entropy204inevitabilityofchaosand286–7,289Schrödinger’swaveequation250seealsopartialdifferentialequations

differentialgeometry95,231differentiation,introduced43diffraction223,250,257X-rays106,157

digitalphotography157–60dimensionsheatequation152multidimensionalgeometry276,278–9Navier-Stokesequation171SolarSystemdescription66,96superstringtheory105,319topology100vectors186waveequations144–5

Diophantus78discontinuousfunctions143,154,155discretecosinetransform159diseaseclusters128divergence(Maxwell’sequations)187divisionbyzero45,51,322nDNAencodedinformation281–2,319Fouriertransforms157topologyand105–6

dodecahedra91Dopplereffect236–7,241dotcomcompanies307,311drugtesting122,127

drums144–6

EE=mc2219,227–9,327n–328ne(constant)30,84Earthinteriorof147Mooneffectson291shapeof8–9sizeof11,12,56viewsof167–8

earthquakes30,144–7,157,221economics,classical310ecosystemsandchaos286–7,291–2modelling,financeanalogy312–14

edges,intopology58,98,99Edison,Thomas183efficiency,inerrordetection270,279eggsassembling214unboiling211–12

Egyptianmathematics8,11eigenfunctions251–2,255–9Einstein,Albert219–20andatomicweapons229–30Brownianmotion208E=mc2219,227–9,327n–328nfieldequations231,235,240–1photoelectriceffect249specialrelativity224–5

electromagneticfield185–6,221electromagneticinduction186electromagneticspectrum189,192–3electromagnetismimpactoneverydaylife181magnetohydrodynamicsand318

electrostatics85,151,242Elements(Euclid)4,14–15ellipses41,57–8,59,61,63Hohmanntransferellipses65,69

energyconservationof48,68,200–2,205,225darkenergy237defined48–9kineticenergy48–9,68,202,203,205,229potentialenergy48–9,68renewableresources209–10

energylandscapes68–72engineers’useoflogarithms29,30EnronCorporation308entropy203–4,206,209,211,213–14as‘missinginformation’281

enzymes106equalssignviiequationsalternativesto319–20describinglinesonplanes13–14E=mc2asarchetype219,227–9,327n–328n‘mostbeautiful’84Pythagoras’sTheorem5quadratic78–9symmetryof211typesofvi–vii,318widerinfluenceof317–18seealsodifferentialequations

EratosthenesBatavus12Eratosthenes(ofCyrene)11error-detectionand-correctioncodes267–8,270,276,278,280errorschannelcapacityand275normaldistribution115,116,119

‘ether’221–3,247

EuclidofAlexandriaonconicsections57fifthaxiom14–15proofofPythagoras’sTheorem4onregularsolids4,91

eugenicsmovement124Euler,Leonhard69,78,84,160fluiddynamics168–9,170formulaforpolyhedra89,92–4,97

Eulercharacteristic95,97,101Everett,Hugh,Jr.258–60evolutionofsensoryperception33–4evolutionarycomputing319exchangetradedfunds(ETFs)311–12‘expectation’andprobability111exponentialfunctions30complexnumbers83–4heatequation152,154

extremeevents302–4,306–7

Ffaces,regularsolids58Faraday,Michaelvii,181–6fattails302–3,306,309Fechner,Gustav33Fermat,Pierrede39,44,111Feynman,Richard257,261Feynmandiagrams105fields,inelectromagnetism85–6,96,182,185–7‘financialecosystems’313–14financialinstruments297,299,306,308seealsoderivatives;futuresandoptions

financialtransactionstax313fingerprints162Fisher,RonaldAylmer122,128five-bodydynamics71

fluiddynamicsbeforeEuler169magnetohydrodynamics318

fluidsapplicationofNewton’slawsofmotion170mathematicsofmagnetismand185

fluxions42,46,50FoodandDrugsAdministration(US)174forcesinlawsofmotion47,55linesofforce85,185,187relatedtocurvature18,20seealsogravity

Fourier,Joseph151Fourieranalysis155Fouriertransform149blips160datacompression159errordistributionsand119Schrödinger’sequation251uses149,156

freewill116friction47,170,201–2,212–13FukushimaDai-ichipowerplant31functionscomplexfunctions85–6exponentialfunctions30,83–4,152,154

futuresandoptions299,305,308,330nfuzzyboundarytheory69

Ggalacticrotationcurves237–8,241GalileoGalilei39–42,47–8,224,230Galoisfields(ÉvaristeGalois)280Galton,Francis119–21gamesofchance109–11

gaslaws200–1,205–6Gauss,CarlFriedrichelectromagnetism183imaginarynumbers81–2leastsquaresmethod118magnetism96andnon-Euclideanspace16–17,18–19notation78‘Gaussiancurvature’18

generalrelativity220,231,240testsof233–5,242,243

geocentricuniverse8,56geometrycoordinategeometry13curvedsurfacegeometry12,15–19,231–2multidimensional276,278–9non-Euclidean15–16relationtoalgebra9topology91–100,105

globalwarming175–7GPS(GlobalPositioningSystem)13,219,242–3gravitationalconstant59,323ngravityGalileo’sinvestigations39–41generalrelativityand220,231–2inversesquarelaw37,61–2Newton’slawof53,55,60–1,65,230three-bodyproblem63

Greene,Brian255,259greenhousegases176Greenspan,Alan314groups(algebra)101

HHaldane,Andrew312–14half-lives,radioisotopes31–2

Hamilton,WilliamRowan86–7Hamiltonianoperators250,251Hamming,Richard276,277,279Hardy,GodfreyHarold184harmonics143harmonies134–5,137,139,143–4hearing,human33–4heatandtemperature203–4time-reversalscenario212

heatequation(Fourier)151,156financialanalogue302,304

heatflow198hedgefunds306,309heightsofchildren121ofstructures10–11

Heisenberg,Werner250heliocentrictheories56–8Henry,Joseph182herdinstinct311heredity120–1HeroofAlexandria11,198Hertz,Heinrich190Hitler,Adolph259–61Hohmanntransferellipses65,69hole-through-a-hole-in-a-hole95–6,97holesintori94–7,99HOMFLYpolynomial104Hooke,Robert59–61,137,322nHubble,Edwin236Huffmancode159Huisman,Jef291–2humangenome281,319hyperbolicspace16–17hypercubes276–7

hypotenuse,defined5hypothesistesting122–3

Ii(imaginaryunit).seesquarerootofminusoneicosahedra91images.seedatacompression;photographyimaginarynumbers75–6,82seealsocomplexnumbers

Indiaastronomy56GreatTrigonometricSurvey12–13

infiniteseries83–4,142–3,145,153–4,161infinitesimals45,50inflation237,238,239–40,242informationasameasurablequantity268,269–70asnegativeentropy213redundant158

informationcontent,H265,273–4,329ninstantaneousratesofchange41,43integralcalculusentropyand204introduced43magnetismand96

integrationtheory156intelligenceandIQtesting124–7InternationalMonetaryFund307,311InterplanetarySuperhighway72,105intervals134–6invariants,combinatorial96invariants,topological89,93–4,96–7,101,103–4inversesquarelawsexpectedforelectromagnetism185gravity37,61–2,230

IQtesting124–7

irrationalnumbers155ISEE(InternationalSun-EarthExplorer)-369

JJapaneseearthquake,201130–1Jonespolynomial(VaughanJones)103–4,105,106Joukowskitransformation86JPEG(JointPhotographicExpertsGroup)158Jupiterorbitalresonances67,71tubesnear70,71,72

KKelvin,WilliamThomson,Lord247Kepler,Johannes39,41,58–9,63Keynes,JohnMaynard38,322nkineticenergy48–9,68,202–3,205,229kinetictheoryofgases200,202,204–6Kleinbottle(FelixKlein)98knotdiagrams101–2,104–5knots100–5KoonWang-Sang67,70,71Krönig,August205

LLagrange,Joseph-Louis69Lagrangepoints67,69–71Laplace,PierreSimonde29,119–20Laplacians145–6,326nLargeHadronCollider219,242lasers263–4,268lawofgravity(Newton)53,60–1,65,230lawoflargenumbers113,114lawsofconservation.seeconservationlawslawsofmotion(Newton)45,46–7,60applicationtofluids170

MONDand240lawsofplanetarymotion(Kepler)39,58–9,61lawsofthermodynamics200,210–11firstlaw200,201–2secondlaw197,204,209–10,213,326n–327nzerothlaw203

LeVerrier,Urbain233–4leastsquaresmethod117–18,125Lebesgue,Henri156Legendre,Adrien-Marie117–18Leibniz,GottfriedWilhelm39,42–5,50,81Lemaître,Georges236life,andthesecondlawofthermodynamics210lightasanelectromagneticwave189,220particulatebehaviour227–8,249wave-particleduality249–50seealsospeedoflight

lightbulbs247lightcones227limits,conceptof51,323nlinearequations141linkingnumbers(electromagnetism)96Listing,Johann96,98–9logarithmicmultiplicationequation21,29logarithms24–9ofcomplexfunctions86humanperceptionand33–4natural30innature33–4sliderulesand30oftrigonometricfunctions29,31

logisticequation283,286–8LongTermCapitalManagement(LTCM)306–7Lorenz,Edward289,290Lorenz,Hendrik223–4

Lorenz-FitzGeraldcontraction223–6

MMacKay,Robert241magnetism85–6,96magnetohydrodynamics318manifolds18–19,231,240mannedflight167many-worldsinterpretation258–61mapsdisc-shapedEarth9trigonometryand11,12–13

Marconi,Guglielmo191‘marktomarketaccounting’308,309mathematicalmodelsBrownianmotion208,301climateforecasting176cosmological56,241financial301,306gaslaws200statisticalmodels113,172vibratingstrings137–8warningsabout127,310

matrices19,231Maxwell,JamesClerk181,183,206,208Einsteinand224

Maxwell-Boltzmanndistribution206,247May,Robert(LordMayofOxford)286,288,291,312–14mean,defined115meanfreepaths205measurement,quantummechanics252,256medicalimaging162–3medicalresearch174medicaluseoflasers263Meitner,Lise230memory213–14

Mercury233Merton,Robert304,305Michelson,Albert222–3,224microwaves192–3,236‘Midasformula’305Milgrom,Mordehai240Millikan,Robert248mines199–200Minkowski,Hermann227–8,232mixingandchaos290,293Möbius,August96,98modes,vibrational139,145,147,248molecules,inthermodynamics200,205–6momentum49,171,249conservationof171,225

MOND(modifiedNewtoniandynamics)240MooneffectsonEarth291origins49,323nMorley,Edward222–3,224

Morse,Samuel183mortgagesbettingondefaults309self-certified301subprime298,311–12

motion,Newton’slawsof45,46–7,60,170,240multidimensionalgeometry276,278–9multiplicationequationforlogarithmic21relativecomplexityof25–6,27–8

music133–6

NNapier,John24–9naturallogarithms30natureasessentiallymathematical38instancesofchaosin285

logarithmsin33–4Navier,Claude-Louis169,170Navier–StokesEquation165,169,171,175,177negativeentropy213Neugebauer,Otto7neutrinos226,327nNewcomen,Thomas199Newton,Isaaccalculusand37–8,41–6,50Hookeand59–61,322ninfluence38–9,63,65–6Keplerand59,61lawsofmotion45,46–8Leibnizand39,42–4,50physicsof220

nightsky,interpretation55–6NobelPrizeinEconomics305,330nnodes143noisemeasurement34noiseremoval157,163,269non-Euclideangeometries15–16normaldistribution(bellcurve)definition115effectofcombining120errors115,116,119financialmodelsand301–2formulafor107kinetictheoryofgases206nullhypothesesand123socialsciences116,119

normalmodes145,147,248,251,325nnotationalgebraic78LeibnizandNewton42,43originofnumbersymbols23originoftheequalssignvii

nuclearaccidents31,32–3

nullhypotheses122–3numbersymbols,origin23

O

observationallimitations,quantummechanics252,256octahedra91octaves135–6,143odds,statistical110–11oilprice306oilprospecting148operatoralgebras103options,futuresand299,305,308,330norbitalresonances,Jupiter67,71OrdnanceSurveyofGreatBritain12Ørsted,HansChristian182orthogonality160

Pparabolas41,48,57paralleluniverses258–61partialderivatives139–40,170partialdifferentialequations141,151,169,172,320Black–Scholesas304,310

Pascal,Blaise111,113,324nPenrose,Roger239,328nphilosophicalquestionsactionatadistance221heliocentrism57mathematicsandrealityvii–viii,82,127quantummechanics252,254,318statusofrelativity224time’sarrow208,211

photoelectriceffect,Einsteinand249photography157–60photons228,249,257

pianos136pictureframetopology93–4π(constant)84,320Planck,Max(andPlanck’sconstant)247–9,250planetarymotionlaws39,58–9,61planktonparadox291–2,330nplayingcardsillustrations147,206–7,211,214Plimpton,GeorgeArthur7Poincaré,Henri224,302chaostheoryand63–4,105,285,289

polyhedraEuler’sformula89,92–4,97seealsoregularsolids

polynomialsAlexanderpolynomial103,104calculusapplicationto44,323nHOMFLYpolynomial104Jonespolynomial103–4,105,106Reed-Solomonencoding280

Poovey,Mary307–8,310,330npopulationdynamicsandchaos286–7,291–2,330npositivefeedback298potentialenergy48–9,68power,‘newaxisof’307pressure,inidealgases205Principia(Newton)absenceofcalculus46,50astronomicalsignificance55,59influenceof38,55,151

prioritydisputes42,44,60–1,118probabilityandwavefunctions253probabilitytheorycrownandanchorgame111drugtestingand127financialmodelsand301messageencodingand272–4

origins111riskand128–9seealsonormaldistribution

projectiveplane98property,assecurity297–8,301,311prosthapheiresis27,322nPtolemy(ClaudiusPtolemaeus)42,56–7,134pumps,steam199–200Pythagoras3,8Pythagoras’sTheoremcodingtheoryand276consequences9curvedsurfacesand17,19distancecalculationsfrom14originsandantecedents4,6proofs4,6specialrelativityand225–7

Pythagoreantriples5Pythagoreanworldview3,56,58,134–7,143

Qquadraticequations78–80quantumcomputers264quantumdots264quantumfieldtheoryandtopology92quantummechanicsandtheclassicalworld255–9,260–1imaginarynumbersin76,250–1many-worldsinterpretation258–61observationallimitations252,256practicalutility261–4Schrödinger’swaveequation250vacuumenergydiscrepancy328n

quarticequations79–80Quetelet,Adolphe116–17,119,121,124quincunxes120

RraceandIQ124radar192radio190radioactivedecay31–3,267randomwalks301–2randomnesschaosdistinguishedfrom285distinguishedfromdeterminism288patternsin109

Recorde,RobertviiReed-Solomoncodes280regressiontothemean121regularsolidsDescartesand91,95Euclidand4,91Euler’sformula89,92–4,97Keplerand58

Reidemeistermoves(KurtReidemeister)101–3relativitygeneralrelativity220,231,233–5,240,242–3misleadinglynamed219–20specialrelativity220,224–9,242–3

renewableenergy209–10ricetrading,Dojimaexchange298–9,312Riemann,GeorgBernhard18–20,95,156,231,276Riemannianmetric19,231,277riskderivativesand309–10,312financial,assessment309,314probabilitytheoryand128–9

Rourke,Colin241RoyalInstitution,London181–2Rule110automaton320

Russianfinancialcrisis,1998306–7

SSachs,Abraham7satnavsandrelativity219,242Scholes,Myron304,305Schrödinger,Erwin213,250–1,253–4Schrödinger’scat253–6,258–60,318Schwarzschild,Karl235sciencefiction66,329nseismicwaves144–7,157,221semiconductors191–2,261–2sensoryperceptiondatacompressionand159–60logarithmsand33–4

Shannon,Claude269–70,272–4,279,329nshipdesign172Shockley,William192,262short-selling304simplification,intopology93,101sinecurves137–8sines10,83–4,142–3,152–4singularities100,235sliderules30Smale,Stephen289–90Smoluchowski,Marian208Snellius,Willebrord12Snell’slaw257,261SnowC(harles)P(ercy)197–8,209socialsciencesmathematicalmodels127normaldistributions116,121

SolarSystempredictionhorizons290–1soundlevels34soundrecordings157,162soundwaves133–6,146spacemissionsApolloprogramme65–6,168

Hitenprobe69interplanetarytravel67–8,71–2Newton’slawofgravity65viewofEarthand167–8Voyager1and2267,276

space-timecurvature231–2geometries227–8,235,237inflationand240–1Lorenz-FitzGeraldcontraction225topologyand105

Spearman,Charles125–7specialrelativity220,224–9,242–3speculation,financial300,302,309,312speedoflighttheetherand222GranSassoneutrinos327nMaxwell’sequations188–9specialrelativityand224–6

spheresfromdeformationofsolids92–4,97hypercubesand278–9

sphericalgeometry12,16,17squarerootofminusone73,77,250–1seealsocomplexnumbers;imaginarynumbers

squarerootsinBabylonianmathematics7dualityof76–7solvingcubicequations79–80usinglogarithms29

squarewaves153,155squawonthehippopotamusjoke3–4,321nstandarddeviationdefined115andextremeevents302–3

statisticalmechanics206

statisticslawoflargenumbers113,114ofrandomness109statisticalsignificance122

steadystate(populations)287steadystate(universe)241steamengines198–200stents174stockmarketfluctuations302Stokes,GeorgeGabriel169,170subduction147subprimemortgagemarket298,311–12Sundman,KarlFritiof64–5superpositionprincipleappliedtocats253–5,258,328nappliedtotheuniverse258linearequations141,153Schrödinger’sequation251–2,257

supersonicflight173–4superstringtheory105,319surfaces,intopology98–100surveying11–12symbolicdynamics288symmetrictensors19,231symmetry,ofequationsandsolutions211

TT-waves193Taleb,NassimNicholas303tangents10‘tangles’106Taqial-DinMuhammadibnMa’rufal-Shamial-Asadi199telephonyFouriertransforms156–7progressin268

temperature

definitionandmeasurement202–3inidealgases205–6

tensors19,231–2Tesla,Nikola191tetrahedra91TheoryofEverything319thermodynamicsclassical200,204defined198,200seealsolawsofthermodynamics

three-bodyproblem63–5,69,320time,arrowof208,211,213,318time-reversalsymmetry211,214–15topologicaldynamics65,289topology91–100,105tori94–5,97–9transistors191–2,261–2trefoilknots102–3,106trianglesnon-right-angled9,11,321nright-angled5,8–10

triangulation12–13trigonometricfunctionscomplexnumbers83,84heatequation152–4logarithmsof29,31vibrationalfrequencies146seealsocosines;sines

trigonometry9–12Trojanpoints69tsunamis31,147,221tubes66,70–2turbulence171–2,192TychoBrahe12,58

U

ultravioletcatastrophe248–9uncertainty290,299seealsoprobabilitytheory

UnifiedFieldTheory319universallawofgravity61–2universeageof241expansionof236,241‘heatdeath’209matterdistribution210,240–1paralleluniverses258

unknot,the100–5

Vvacuumenergy239,328n‘valueatrisk’calculations309variance115,121,125–6vectorcalculus188,193,326nvectorfields187vectorsdefined186velocitydefined39asavector186

velocityfields170Venusmissions72verticeshypercubes276regularsolids91–6

vibratingstrings134–5,137–8,140,143,325nandtimeintervals40

vibrationalfrequencies145–6vibrationalmodes(blackbodyradiation)248violinstringvibration133–4,137vision,human159–60volcaniceruptions147,177Vulcan(hypotheticalplanet)233–4,328n

WWallis,John39,44,81–2WangSangKoon67,70,71Warson,James105water,andfluiddynamics168Watt,James198waveequation131,139,141–2,144animations325nheatequationcompared152–3Maxwelland188–9multi-dimensional144Schrödinger’s250–1,257

wavefunctions250–1,252–3,255–6,258–60wavenumber143wave-particleduality/wavicles249–50wavephenomena133wavelet/scalarquantization(WSQ)method162wavelets161–2,163waves,analysis86weatherforecasting289,292–3Weber-Fechnerlaw33–4Weierstrass,Karl51Weissing,Franz291,292Wessel,Caspar81,82,87windtunnels172–3Wolfram,Stephen320

XX-events(extremeevents)302–4,306–7X-raydiffraction106,157X-rays192

Zzero,approaching50,51zero,divisionby45,51,322n

TheGreatMathematicalProblems

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