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8/20/2019 The Magic Of Mathematics-slicer.pdf

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DiscoveringtheSpellof othematics

,z

lJM*L«*

authorofTheMathematicsCalendars

TheJoyofMathematics


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THEMAGIC

DiscoveringtheSpellofMathematics

THEONIPAPPAS

WIDEWORLDPUBLISHING/TETRA


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Copyright © 1994byTheoniPappas.

Allrightsreserved.Nopartofthiswork maybereproducedor

copiedinanyformorbyanymeanswithoutwrittenpermission

fromWideWorldPublishing/Tetra.

Portionsofthisbook haveappearedinpreviouslypublished

works,butweretooessentialtonotbeincluded.

WideWorldPublishing/TetraP.O.Box476

SanCarlos,CA94070

PrintedintheUnitedStatesofAmerica.

SecondPrinting,October1994.

LibraryofCongressCataloging-in-PublicationData

Pappas,Theoni,

Themagicofmathematics:discoveringthespellofmathematics /TheoniPappas.

p.cm.

Includesbibliographicalreferencesandindex,

ISBN0-933174-99-3

1. Mathematics--Popularworks. I.Title.

QA93,P3681994

510-dc20 94-11653

CIP


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Thisbook isdedicatedto

mathematicians

whohavecreatedand

arecreating

themagicofmathematics.


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CONTENTS

PREFACE 1

MATHEMATICSINEVERYDAYTHINGS3

MAGICALMATHEMATICALWORLDS33

MATHEMATICS&ART63

THEMAOICOFNUMBERS97

MATHEMATICALMAGICINNATURE119

MATHEMATICALMAGICFROMTHEPAST

143

MATHEMATICSPLAYSITSMUSIC173

THEREVOLUTIONOFCOMPUTERS189

MATHEMATICS&THEMYSTERIESOFLIFE223

MATHEMATICSANDARCHITECTURE243

THESPELLOFLOGIC,RECREATION&GAMES265

SOLUTIONS311

BIBLIOGRAPHY315

INDEX321


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THEMAGiCOfMATHEMATICS1

PREFACE

Youdon'thavetosolveproblemsor

beamathematiciantodiscoverthe

magicofmathematics.Thisbook IsacollectionofIdeas —ideas

withanunderlyingmathematicaltheme.Itisnotatextbook.Do

notexpecttobecomeproficientinatopicorfindanidea

exhausted.TheMagicofMathematicsdelvesintotheworldof

ideas,exploresthespellmathematicscastsonourlives,andhelps

youdiscovermathematicswhereyouleastexpectit.

Manythink ofmathematicsasarigidfixedcurriculum.Nothing

couldbefurtherfromthetruth.Thehumanmindcontinually

createsmathematicalideasandfascinatingnewworlds

independentofourworld — andprestotheseideasconnecttoour

worldalmostasIfamagicwandhadbeenwaved.Thewayin

whichobjectsfromonedimensioncandisappearintoanother,a

newpointcanalwaysbefoundbetweenanytwopoints,numbers

operate,equationsaresolved,graphsproducepictures,infinity

solvesproblems,formulasaregenerated —allseemtopossessa

magicalquality.

Mathematicalideasarefigmentsoftheimagination.Itsideasexistinalienworldsanditsobjectsareproducedbysheerlogicand

creativity.Aperfectsquareorcircleexistsinamathematical

world,whileourworldhasonlyrepresentationsofthings

mathematical.

Thetopicsandconceptswhicharementionedineachchapterare

bynomeansconfinedtothatsection.Onthecontrary,examplescaneasilycrossoverthearbitraryboundariesofchapters.Evenif

itwerepossible,itwouldbeundesirabletorestrictamathematical

ideatoaspecificarea.Eachtopicisessentiallyself-contained,

andcanbeenjoyedindependently.Ihopethisbook willbea

steppingstoneintomathematicalworlds.


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PrintGallerybyM.C.Escher. © 1994M.C.Escher/CordonArt-Baarn-Holland,Allrightsreserved.


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MATHEMATICSIN

EVERYDAYTHINGS

THEMATHEMATICSOFFLYING

THEMATHEMATICSOFATELEPHONECALL

PARABOLICREFLECTORS&YOUR

HEADLIGHTS

COMPLEXITYANDTHEPRESENT

MATHEMATICS&THECAMERA

RECYCLINGTHENUMBERS

BICYCLES,POOLTABLES&ELLIPSES

THERECYCLINGNUMBERS

LOOKOUTFORTESSELLATIONS

STAMPINGOUTMATHEMATICS

MOUSE' STALE

AMATHEMATICALVISIT

THEEQUATIONOFTIME

WHYAREMANHOLESROUND?


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4THEMAGICOFMATHEMATICS

Thereisnobranchofmathematics,howeverabstract,

whichmaynotsomedaybeappliedtophenomenaoftherealworld. — NikolaiLobachevsky

Somanythingswithwhichwecomeintocontactinourdaily

routineshaveamathematicalbasisorconnection.Theserange

fromtakingaplaneflighttotheshapeofamanhole.Oftenwhen

oneleastexpects,onefindsmathematicsisinvolved.Hereisa

randomsamplingofsuchcases.


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THEMATHEMATICS

OFFLYINC

Thegraceandeaseoftheflightofbirdshavealwaystantalized

human'sdesiretofly.Ancientstoriesfrommanyculturesattestto

interestinvariousflyingcreatures.Viewinghanggliders,one

realizesthattheflightofDaedalusandIcaruswasprobablynot

justaGreek myth.Todayenormoussizedalrcraftsliftthemselves

andtheircargointothedomainofthebird.Thehistoricalstepsto

achieveflight,aswenowknowIt,hasliterallyhadItsupand

downs.Throughouttheyears,scientists.Inventors,artists,

mathematiciansandotherprofessionshavebeenIntriguedbytheIdea

offlyingandhavedevelopeddesigns,prototypes,andexperiments

ineffortstobeairborne.

HereIsacondensedoutlineofthehistoryofflying:

•KiteswereinventedbytheChinese(400-300B.C.).

•LeonardodaVinciscientificallystudiedtheflightofbirdsand

sketchedvariousflyingmachines(1500).


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6 THEMAGICOFMATHEMATICS

•ItalianmathematicianGiovanniBorelliprovedthathuman

musclesweretooweak tosupportflight(1680).

•FrenchmenJeanPilatredeRozierandMarquisd!Arlandes

madethefirsthotairballoonascent(1783).

•Britishinventor.SirGeorgeCayley, designedtheairfoil

(cross-section)ofawing,builtandflew(1804)thefirstmodel

glider,andfoundedthescienceofaerodynamics.

•Germany' sOttoLilienthaldevisedasystemtomeasurethelift

producedbyexperimentalwingsandmadethefirstsuccessful

mannedgliderflightsbetween1891-1896.

•In1903OrviUeandWilburWrightmadethefirstengine

poweredpropellerdrivenairplaneflights.Theyexperimented

withwindtunnelsandweighingsystemstomeasurethelift

anddragofdesigns.Theyperfectedtheirflyingtechniquesandmachinestothepointthatby1905theirflightshadreached38

minutesinlengthcoveringadistanceof20miles!

Here'showwegetofftheground:Inordertofly,thereareverticalandhorizontalforcesthatmust

bebalanced.Gravity(thedownwardverticalforce)keepsus

earthbound.Tocounteractthepullofgravity,lift(avertical

upwardforce)mustbecreated.Theshapeofwingsandthedesignof

airplanesIsessentialIncreatinglift.Thestudyofnature'sdesign

ofwingsandofbirdsInflightholdsthekey.Itseemsalmost

sacrilegioustoquantifytheeleganceoftheflightofbirds,butwithout

themathematicalandphysicalanalysesofthecomponentsof

flying, today'sairplaneswouldneverhavelefttheground.Onedoes

not

alwaysthink ofairasasubstance,sinceitisinvisible.Yetair

isamedium,aswater.Thewingofanairplane,aswellasthe

airplaneitself,dividesorslicestheairasitpassesthroughit.

SwissmathematicianDanielBernoulli(1700-1782)discovered

thatasthespeedofgasorfluidincreasesitspressuredecreases.

Bernoulli'sprinciple1explainshowtheshapeofawingcreates

theliftforce.Thetopofthewingiscurved.Thiscurveincreases


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thespeedofairandtherebydecreasestheairpressureoftheair

passingoverit.Sincethebottomofthewingdoesnothavethis

curve,thespeedoftheairpassingunderthewingisslowerand

thusitsairpressureishigher.Thehighairpressurebeneaththe

wingmovesorpushestowardthelowpressureabovethewing,

andthusliftstheplaneintotheair.Theweight(thepullof

gravity)istheverticalforcethatcounteractstheliftoftheplane.

Thewing'sshapemakesthedistance

overthetoplonger,whichmeansair

musttravelover

thetopfaster,makingthe

pressureonthetopofthewinglowerthanunderthe

wing.Thegreaterpressurebelowthe

wingpushesthe

wingup.

Whenthewingisatasteeperangle,thedistanceover

thetopiseven

longer,therebyincreasingthe

liftingforce.

Dragandthrustarethehorizontalforceswhichentertheflying

picture.Thrustpushestheplaneforwardwhiledragpushesit

backwards.Abirdcreatesthrustbyflappingitswings,whilea

planereliesonitspropellersor jets.Foraplanetomaintainalevel

andstraightflightalltheforcesactingonitmustequalizeone

another,i.e.bezero.Theliftandgravitymustbezero,whilethe

thrustanddragmustbalance.Duringtakeoffthethrustmustbe

greaterthanthedrag,butinflighttheymustbeequal,otherwise

theplane'sspeedwouldbecontinuallyincreasing.

Viewingbirdsswoopinganddivingrevealstwootherflyingfactors.


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8 THEMACICOFMATHEMATICS

WhenthespeedofairoverthetopofthewingIsIncreased,thelift

willalsoIncrease.ByIncreasingthewing'sangletothe

approachingair,calledtheangleofattack,thespeedoverthetopofthewingcanbefurtherIncreased.IfthisangleIncreasesto

approximately15ormoredegrees,theliftcanstopabruptlyandthe

birdorplanebeginstofallInsteadofrising.Whenthistakesplace

itiscalledtheangleofstall.Theangleofstallmakestheairform

vortexesonthetopofthewing.Thesevibratethewingcausingthe

lifttoweakenandtheforceofgravitytooverpowertheliftforce.

Nothavingbeenendowedwiththeflyingequi pmentofbirds,

humanshaveutilizedmathematicalandphysicalprinciplestolift

themselvesandotherthingsofftheground.Engineeringdesigns

andfeatures2havebeencontinuallyadaptedtoImprovean

aircraft'sperformance.

lLawsgoverningtheflowofairforairplanesapplytomanyotheraspectsInourlives,suchasskyscrapers,suspensionbridges,certaincomputerdisk drives,waterandgaspumps,andturbines.

T̂heflapsandslotsarechangesadaptedtothewingwhichenhancelift.TheflapIsahingedsectionthatwhenengagedchangesthecurvatureofthewingandaddstotheliftforce.SlotsareopeningsInthewingthat

delaythestallforafew

degrees.


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MATHEMATICSINEVERYDAYTHINGSS

Everytimeyoupick-upthe

telephonereceivertoplaceacall,

sendafax,ormodemInformation

THEMATHEMATIC

OFATELEPHONI

CALI

—

youareenteringaphenomenally

complicatedandenormousnetwork.Thecommunicationnetthat

encompassestheglobeisamazing.ItisdifficulttoImaginehow

manycallsarefieldedanddirectedeachdayoverthisnetwork.

Howdoesasystemwhichis"broken-up"byvariedsystemsof

differentcountriesandbodiesofwateroperate?Howdoesasingle

phonecallfinditswaytosomeoneinyourcity,stateoranother

country?

Intheearlyyearsofthetelephone,onepicked-upthereceiverand

crankedthephonetogetanoperator.Alocaloperatorcameon

thelinefromthelocalswitchboardandsaid"numberplease",

andfromthereconnectedyouwiththepartyyouweretryingto

reach.Todaytheprocesshasmushroomedashavethevarious


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methodsusedtoconvertanddirectcalls.MathematicsInvolving

sophisticatedtypesoflinearprogramming,coupledwithbinary

systemsandcodes,makesenseoutofapotentiallyprecarious

situation.

40

3l̂̂.16' J976

/*U*X~W*A~*-

Howdoesyourvoicetravel?Yourvoiceproducessoundswhichare

convertedInthereceivertoelectricalsignals.Todaythese

electrical impulsescan

becarriedand

convertedIna

varietyofways.

Theymaybe

changedtolaser

lightsignals

whicharethen

carriedalong

fiberoptics

cables1,they

maybe

convertedto

radiosignals

andtransmitted

overradioor

microwavelinks

fromtowerto

toweracrossa

country,orthey

mayremainas

electrical

signalsalongthephone

lines.Mostof

thecalls

connectedinthe

USAaredoneby

anautomatic

ûft

N

*****kX**4-u4u-fcwv̂Û-fri,tf̂.c-4L~-~iTt-t. ê4r*y**f **7

4tt~ZÊce.i ?k7\ lf~«Zio-t


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MATHEMATICSINEVERYPAYTHIKICS11

switchingsystem.Presentlytheelectronicswitchingsystemisthe

fastest.Itssystemhasaprogramwhichcontainstheneeded

informationforallaspectsoftelephoneoperationswhilekeeping

track ofwhichtelephonesarebeingusedandwhichpathsare

available.Callscanbetransmittedbyelectriccurrentsatdifferent

frequenciesorconvertedtodigitalsignals.Eithermethodenables

multipleconversationstobetransmittedalongthesamewires.

ThemostmodernsystemsconvertcallsIntodigitalsignalswhich

arethenencodedwithabinarynumbersequence.TheIndividual

callscanthustravel"simultaneously"alongthelinesinaspecified

orderuntiltheyaredecodedfortheirdestinations.

Whenacallisplaced,thesystemchoosesthebestpathforthecall

andsendsachainofcommandstocompletethecircuitry.The

entireprocesstakesafractionofasecond.Ideallyitwouldtakea

directroutetotheotherparty—thatwouldbedesirablefromthe

viewpointoftheeconomicsofdistanceandtime.Butifthedirect

lineisatcapacityservicing

othercalls,thenew

callmustbesent

alongthebestof

thealternative

routes.Here

iswhere

linear

programming2comesInto

the

picture.Visualizethe

telephonerouting

problemasacomplex

geometricsolidwithmillionsoffacets.Eachvertexrepresentsa

possiblesolution.Thechallengeistofindthebestsolutionwithout

havingtocalculateeveryone.In1947,mathematicianGeorgeB.


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Danzigdevelopedthesimplexmethodtofindthesolutionto

complexlinearprogrammingproblems.Thesimplexmethod.In

essence,runsalongtheedgesofthesolid,checkingonecorner

afteranother,whilealwaysheadingforthebestsolution.Aslong

asthenumberofpossibilitiesisnomorethan15,000to20,000,

thismethodmanagestofindthesolutionefficiently.In1984,

mathematicianNarendraKarmarkardiscoveredamethodthat

drasticallycutsdownthetimeneededtosolveverycumbersome

linearprogrammingproblems,suchasthebestroutesfor

telephonecallsoverlongdistances.TheKarmarkaralgorithm

takesashort-cutbygoingthroughthemiddleofthesolid.After

selectinganarbitraryinteriorpoint,thealgorithmwarpsthe

entirestructuresothatitreshapestheproblemwhichbringsthe

chosenpointexactlyIntothecenter.Thenextstepistofinda

newpointInthedirectionofthebestsolutionandtowarpthe

structureagain,andbringthenewpointIntothecenter.Unless

thewarpingisdone,thedirectionthatappearstogivethebest

i mprovementeachtimeisanillusion.Theserepeated

transformationsarebasedonconceptsofprojectivegeometry

andleadrapidlytothebestsolution.

Today,theoldtelephonesalutation"numberplease"takesona

doublemeani ng.Theoncesimpleprocessofpickingupyour

telephonereceiverandplacingacall,nowsetsIntomotionavast

andcomplicatednetwork thatreliesonmathematics.

D̂ependingonthetypeoflinesused,thenumberof"simultaneous"conversationscanrangefrom96toover13000.Fiberopticsystemscan

carryevenmoreinformationthanthetraditionalcopper/aluminumcables.

L̂inearprogrammingtechniquesareusedtosolveavarietyofproblems.Usuallytheproblemsentailmanyconditionsandvariables.Asimplecasemaybeanagriculturalproblem:Afarmerwantstodecidehowtomosteffectivelyusehis/herlandtomaximizeproductionandprofitConditionsandvariableswouldinvolvesuchthingsasconsideringdifferentcrops,howmuchlandeachcroprequires,howmuchyieldeach

producesperacre,andhowmuchrevenueeachbringswhensold.Tosolvesuchaproblem,onewriteslinearInequalitiesand/orequationsforeachconditionandlooksata2-dimensionalgraphofapolygonalregionforthesolution.


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PARABOLKWhenyouflick theswitchofyour

REFLECTORS&YOUI

headlightsfrombrighttodim, HEADLIOHTmathematicsisatwork.Tobe

specific,theprinciplesofaparabolado

thetrick.Thereflectorsbehind

the

headlightsareparabolic

Inshape.Infact,theyare

paraboloids(3-

dimensionalparabolas

formedbyrotatinga

parabola1aboutitsaxisof

symmetry).Thebrightbeam

iscreatedbya

lightsourcelocatedatthefocalpointoftheparabolicre

flectors.Thus,thelightraystraveloutparalleltotheparabola's

axisofsymmetry.Whenthelightsaredi mmed,thelightsource

changeslocation.Itisnolongeratthefocus,andasaresultthe

lightraysdonottravelparalleltotheaxis.Thelowbeamsnow

pointdownandup.Thosepointingupareshielded,sothatonly

thedownwardlowbeamsarereflectedashorterdistancethanthe

highbeams.

Theparabolaisanancientcurvethatwas

discoveredbyMenaechmus(circa375-325B.C.)

whilehewastryingtoduplicatethecube.Over

thecenturies,newusesanddiscoveries

involvingtheparabolahavebeenmade.For

example,itwasGalileo(1564-1642)whoshowed

thatatrajectile'spathwasparabolic.Todayonecangointoa

hardwarestoreandfindahighlyenergyefficientparabolicelectric

heaterwhichusesonly1000wattsbutproducesthesamenumber

ofBTUthermalunitsasaheaterthatoperateson1500watts.

1Parabolaisthesetofallpointsinaplanewhichareequidistantfroma

fixedpointcalleditsfocusandafixedlinecalleditsdirectrix.


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COMPLEXITY&

THEPRESENT"Thehoursfromseven'tilnearly

midnightarenormallyquietoneson

thebridge....Beginningalmost

exactlyatseveno'clock,...it justlookedasifeverybodyin

ManhattanwhoownedamotorcarhaddecidedtodriveoutonLong

Islandthatevening."

Asthisexcerptfrom

TheLawbyRobertM,

CoatesIllustrates,

sometimesthings

justseemtotake

placewithno

apparentreason.

NorIsthere%

awarning

thata

particular

eventIs

abouttotake

place.Wehave

allexperienced

sucheventsand

usually

attributed

themto

"coincidence",

sincethere

wereno

apparent

indicatorsto

predictotherwise


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MATHEMATICSINEVERYDAYTHIN6S15

ComplexityIsanemergingsciencewhichmayholdanswersorat

leastexplanationstosuchquestionsas:

Howisitthat

•theuniverseemergedoutofthevoid?

•cellsknowwhichorgansandpartstobecomeandwhen?

•onJanuary17,1994LosAngelessufferedan

earthquakeofunexpectedmagnitudeanddestruction?

•

theSovietUnion'slongreignoveritssatellitecountriescollapsedinsuchashorttime?

•Yugoslaviawasthrownsuddenlyintosevereinternal

wars?

•aspeciesthathasnotchangedformillionsofyears

suddenlyexperiencesamutation?

•fornoapparentreasonthestock marketsurgesupward

orplungesdownward?

Thelistisendless.Theunderlyingcommonfactoroftheseevents

isthateachrepresentsaverycomplexsystem.Asystemgoverned

byanenormousnumberanddiversityoffactors,whichare

delicatelybalanced,titteringbetweenstabilityandchaos.The

factorswhichactonsuchasystemareevergrowingand

changing. Consequently,acomplexsystemisalwaysInastateof

potentialchaosi.e.attheedgeofchaos.Thereseemstobea

continual tugofwarbetweenorderandchaos.Spontaneousself-

organizingdynamicsareanessentialpartofacomplexsystem.It

isthemeansbywhichthesystemregainsequilibriumby

changing andadaptingItselftoconstantlychangingfactors/

circumstances.Thosestudyingthisnewsciencedrawona

hostofmathematicalandscientificideas,suchaschaostheory,fractals,

probability,artificialintelligence,fuzzylogic,etc.Thesescientists

andmathematiciansfeelthattoday'smathematics,alongwith

othertoolsandhightechinnovations,arecapableofcreatinga

complexityframework thatcanimpactmajoraspectsofourglobal

world,especiallyeconomics,theenvironment,andpolitics.


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MATHEMATICS&

THECAMERAEverwonderaboutthef-stop

numberofacamera?WheredidIt

getItsname?HowIsIt

determined?"fstandsforthemathematicaltermfactor.The

brightnessofthephotographicimageonfilmdependsonthe

apertureand

focallengthof

thelens.

Photographers

usewhatIs

knownasthe

f-number

systemtorelatefocal

lengthand

aperture.The

f-stopIs

calculatedby

measuringthe

diameterofthe

apertureand

dividingItinto

thefocal

lengthofthe

lens.For

example,

f4=80mmlens/20mmaperture.

fl6=80mmlens/5mmaperture.

Weseethelensopeningissmaller(theaperturedecreases)asthe

f-stopnumberincreases.Workingwithf-stopnumbersand

shutterspeeds,youcanmanuallydecidehowmuchofthe

photographyouwantinfocus.


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Heremathematicalunitsand

symbolswereusedtogetthepoint

acrossaboutrecyclingpaper!

RECYCLINC

THENUMBER!

•Atonofvirginpaper*atonofrecycledpaper

•Atonofrecycledpaperuses4102kwhlessenergy.

•Atonofrecycledpaperuses7000gallonslesswaterto

produce.•Atonofrecycledpaperproduces60poundslessair

pollution.

•Atonofrecycledpaperproduces3cubicyardslesssolid

waste.

•Atonofrecycledpaperuseslesstaxmoneyforlandfill.

•Atonofrecycledpaperuses17fewerloggedtrees.

—thenumbersbehindrecyclingandlandfill —

•37%ofalllandfilliscomprisedofpaper.

•Only29%ofallnewspapersproducedarerecycledbythe

consumer.

•165millioncubicyardsoflandfillareneededforourpaper

wastesperyear.

•97%ofthevirginforestsofthecontinentalUSAhavebeen

cutdowninthepast200years.

IWasOnceATree...Newsletter,Spring1990,

AlonzoPrinting,HaywardCA.


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BICYCLES,

POOLTABLES&

ELLIPSES

Theellipse,alongwithotherconic

sectioncurves,wasstudiedbythe

Greeksasearlyasthe3rdcentury

Ifthisballishitthroughthelocationofthefocus,markedwithanX,itwillbounceoffthecushionandgotothe

otherfocuswherethepocketislocated.

B.C..Mostof

usassociate

theellipsewith

anangled

circleorthe

orbitalpathof

a

planet,but

elliptical

shapesand

properties

alsolend

themselvesto


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Anellipsehastwofoci,andthesumofthedistances

fromthefocitoanypointoftheellipsealwaysequalsthelengthofitsmajoraxis.le. \PF1\ + \PF2\.= \AB\.

contemporarynonscientificapplications.Whowouldhave

imaginedthatanellipsewouldfinditselfinthedesignofbicycle

gearsandpool tables?Todaysomebicycleshavebeen

manufacturedwithafrontellipticalgearandcircularreargears.

The

drawing,

onthepreviouspage,illustrateshowthis

designcan

utilizethedownwardthrustoflegpowerandaquick upward

return. Elliptipools,ellipticalshapedpooltables,aredesignedto

utilizethereflectionpropertyoftheellipse'stwofoci.Asillustrated

onthepreviouspage,theelliptipoolhasonepocketlocatedatone

ofthetwofocuspointsoftheellipse.Aballhitsothatitpasses

throughtheellipse'snon-pocketfocuswillbounceoffthesideof

thetableandtravelthereflectedpathovertothe

pocket(theother

focus).


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LOOKOUTFOR

TESSELLATIONS u„ThisEscher-liketransformationby

Mark SlmonsonIllustratestheuse

oftessellationsasaformofvisual

communication.Thisgraphic

appearedInTheUtneReaderandonthecoverofTransactions,a

MetropolitanTransportationCommunicationpublication.Reprinted courtesyofMark Slmonson.BlueskyGraphics,

Minneapolis,MN.


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STAMPINGOU1

MATHEMATICSOneusuallydoesn'texpectto

encountermathematicalIdeasona

triptothepostoffice,butherearea

fewofthestampsthathavebeen

printedwithmathematicalthemes.TheseandmanyotherIdeas

haveappearedonsuchpopularItemsasposters,television,

T-shirts,post-Its,mugs,bumperstickers,andstickers.7*̂**̂V

US10FORMULASMAT£MATKASODECAMBIABOHIAFUDELATIERS*

ThePythagoreantheorem —Nicaragua

ThePythagoreantheorem —Greece

Bolyai—Rumania Gauss—-GermanyMathematicalFormulas

—Israel


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THEMOUSE' STALE

'Furysaidtoamouse,That

hemet

inthe

house,'Letus

bothgotoInw.

/ will

proseoutoyou.—

Come,I'11takeno

denial;"Wemust

hnvca

Saidthomousoto

thaour,'Sucha

trial,dflursll',With111)

JjrjrorJudge,wuulilUo

WMllna-uururdKtll.

1I'IIll«

lil


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'wttftktw.tehmxit̂fŷHiiln

*$ipf$4'tfkljJmf,shir.̂2-

mtoptipi,Vtifa jury*•

'""

' *kk wtiptetm$4?'«tiAcondemn$mt£&&

(themouse'sbody)andalongthirdline(themouse'stall).Lastly,

theyfoundthatatall-rhymeIsapoeticstructuredefinedbyapair

ofrhyminglinesfollowedbyanotherlineofdifferentlength.Do

youthink LewisCarrollplannedallthisintentionally?

1EuclidandHisModemRivals,AnElementaryTreatiseonDeterminants,

AliceinWonderland,TheHuntingoftheSnark,PhantasmagoriaandOther

Poems,ThroughtheLookingGlassareofafewofDodgsonworks.


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AMATHEMATICAL

VISIT Notquitesurewhattoexpect,IrangthedoorbellAvoiceaskedmeto

pleasepushthefirstfivetermsof

theFibonaccisequence.

Fortunately,Ihaddonesome

researchaftermymagazineassignedmethestoryonthehomeof

therenownedmathematician,Selath.

Ipushed1,1,2,3,5andthedoorslowlyopened.AsIpassed

throughthedoorway,Iwasstruck bythecatenarystoneshaped

archwayindependentlysuspendedattheentrance.Aftera

minute,Selathenteredsaying,"MayIofferyousomethingafteryour

longdrive?'

"TdreaRy

appreciatea

glassofcold

water,"I

replied.

"Pleasecome

withme,"he

said,leading

theway.

AsIfollowed,

Icouldn'thelpnoticingthemanyuniqueandunusualobjects.Inthe

kitchen,wecametoapeculiartablewithmanylegs.Selathpulled

anequallyunusualbottlefromtherefrigerator.Imusthavehada

quizzicalexpression,forSelathbegan,"Whileyoudrtnk yourwater,

wemightaswellstartthetourhereinthekitchen."Asyounoticed

thistableandbottlearenotyoureverydayaccessories.Iuse


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tangramtablesfordiningbecauseitssevencomponentscanbe

rearrangedintoasmanyshapesasthetangrampuzzle.Hereinthe

kitchenitsmadeintoasquareshapetoday,whileTvearrangedthe

oneinthelivingroomintoatriangle,sinceIamexpectingtwoguests

fordinner.ThewatercontaineriswhatsknownasaKlein

bottle—itsinsideandoutsideareone.If

youlook attheflooryou'llnoticeonlytwo

shapesoftilesareused."

"Yes,"Ireplied,"

butthedesigndoesn't

seemtorepeatanywhere."

"Veryperceptive."Selathseemedpleased

withmyresponse."ThesearePenrose

tiles.Thesetwoshapescan

coveraplaneinanon-repeating

fashion."

"Pleasecontinue,"Iurged."

Tm

mostanxioustoseeallthe

mathematicalpartsofyour

home."

"Well,actuallyalmostallofmy

houseismathematical.

Anywhereyouseewallpaper

Tvedesignedspecial

tessellationpatternsforwallsalaEscher.LetsproceedtotheOp

room.Everyiteminhereisanopticalillusion.Infact,realityinthis

roomisanillusion.Furniture,fvdures,photos,everything!For

example,thecouchismadefrommodulocubesinblack andwhite

fabricstackedtogivethefeelingofanoscillatingillusion.The

sculptureinthemiddlewasdesignedtoshowconvergenceand

divergence,whilethetwostackedfiguresareexactlythesamesize.


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26THEMACICOFMATHEMATICS

Thislamp'sbase,viewedfromthis

location,makestheimpossibletribar."

"Fascinating!Icouldspendhours

discoveringthingsinthisroom,"I

repliedenthusiastically.

"Sinceweareonatightschedule,lets

movetothenextroom,"Selathsaidashe

ledtheway.

Weenteredadarkenedroom.

"Watchyourstephere.Come

thiswaytotheparabolic

screen,"Selathdirected.

AsIpeeredintothedisca

movingsceneappeared."Is

thisavideocamera?"Iasked.

"Oh,no,"Selathlaughed,"Icall

itmyantiquesurveillance

system.Thelensabovetheholecaptureslightinthedaytimeand

rotatestoprojectscenesoutsidemyhome,muchthesamewaya

camerawould.Itiscalledacameraobscura.Ihaveaspeciallens

fornightviewing."

Iwasbusilytakingnotes,realizingIwouldhavemuchadditional

researchtodobeforewritingmyarticle.

GlancingaroundInoted,'Yourfluorescentclock seemstobeoff."

Mywatchread5:30pmwhilehisread21:30.

"No,its justthatIhavethe24hoursofthedayarrangedinbase


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eightbecauseTmworkingoneighthourcyclesthisweek.So24:00

howswouldbe30:00hours,8:00wouldbe10:00,andsoon."

Selathexplained.

"Whateverworksbestforyou,"Ireplied,abitconfused.

"Now,let'sgotothemasterbedroom."

Andoffwewent,passingallsortsofshapesandobjectsrdnever

seeninahomebefore.

'Themasterbedroomhasasemi-sphericalskylightinadditionto


33/333


movablegeodesicskylights.Theyaredesignedtooptimizetheuse

ofsolarenergy."

"Marvelous,butwhereisthebed?'Iasked.

"Justpushthebuttononthiswoodencube,andyouwillseeabed

unfoldwithaheadboardandtwoendtables."

"Whatagreatwaytomakeabed,"Ireplied.

"Therearemanymorethingstosee,buttimeisshort.Letsgointhe

bathroomsoyoucanseethemirrorsoverthebasin.Comethisway.

Nowleanforward."

TomysurpriseIsawaninfinitenumberofimagesofmyself

repeated.Themirrorswerereflectingback andforthintoone

anotheradinfinitum.

"Nowturnaroundandnoticethismirror.Whatsdifferentaboutit?"

Selathasked.

"Mypartisonthewrongside,"Ireplied.

'Tothecontrarythismirror1letsyouseeyourselfasyouarereally

seenbyothers,"Selathexplained.

Justthenthedoorbellrang.Thedinnerguestshadarrived."Why

don'tyoustaytodinner?'Selathasked.'Youhaven'tseenthe

livingroomyet,andFmsureyou'llenjoymeetingmyguests."

Itwashardtoconcealbyenthusiasm."Butyourtableissetfor

three,"Iblurted.

"Noproblem.WiththetangramtableIcan justrearrangeafew

partsandwe'llhavearectangle.

1Madefromtwomirrorsplacedatrightanglestoeachother.Theright-angledmirrorsarethenpositionedsothattheywillreflectyourreflection.


34/333


THEEQUATION

OFTIM

Ifyouhaveeveruseda

sundial,youmayhave

noticedthatthetime

registeredonthesundial

differedslightlyfromthat

onyourwatch.This

differenceIstiedIntothelengthof

daylightduringtheyear.Inthe

15thcentury,JohannesKeplerformulated

threelawsthatgoverned

planetarymotion.Kepler

describedhowtheEarthtravels

aroundtheSunInanelliptical

orbit,andalsoexplainedthatthe

linesegment joiningtheSunand

theEarthsweepsout

equalareas

(sectors) InequalIntervalsof

timealongitsorbit.TheSunIs

locatedatoneofthefociofthe

ellipsetherebymakingeach

sector'sareaequalforafixed

timeIntervalandthearclengths

ofthesectorsunequal.ThustheEarth'sorbitspeedvariesalong

itspath.Thisaccountsforthe

variationsinthelengthsof,„,,,,,,„.,,,

,A10thcenturypocketsundialdaylightduringdifferenttimesof̂ m̂six̂ l̂istedoneach

year.Sundialsrelyondaylight,side-Astlck teplacedin.theholeof

thecolumnwiththecurrentmonth.


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anddaylightdependsonthetimeoftheyearandgeographic

location.Ontheotherhand,thetimeintervalsofourotherclocks

areconsistent.Thedifferencebetweenasundial'stimeandan

ordinaryclock isreferredtoastheequationoftime.Thealmanac

liststheequationoftimechart,whichindicateshowmanyminutes

fastorslowthesundialisfromtheregularclocks.Forexample,

thechartmaylook liketheonebelow.

EquationofTimeChart

(Thenegativeandpositivenumbersindicatetheminutesthe

sundialisslowerorfasterthananordinaryclock.Naturally

thetabledoesnottakeintoconsiderationdaylightdifferencewithintimezones.)

DATE

Jan

Feb

Mar

April

May

VARIATION

1 -3

15-9

1 -13

15-14

1 -3

15-9

1 -4

150

1 +3

15+4

// thesundialshows11:50on

May15, itstimeshouldbeincreasedby4minutesto11:54.


36/333


WHYARE

MANHOLES

ROUND?

WhyIstheshapeofamanholecircular?

Whynotasquare,rectangular,hexagonal,orelliptical

shape?

Isitbecauseacircle'sshapeIsmorepleasing?

ThereIsamathematicalreason.

Whatisyourexplanation?


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MetamorphosisWbyM.C.Esclwr. © 1994M.C.IDscher/CordonArt-Baam-HoUand.Allrightsreserved.


38/333

MACICAL

MATHEMATICAL

WORLDS

HOWMATHEMATICALWORLDS

AREFORMED

GEOMETRICWORLDS

NUMBERWORLDS

THEWORLDSOFDIMENSIONS

THEWORLDSOFINFINITIES

FRACTALWORLDS

MATHEMATICALWORLDSINLITERATURE


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Howcanitbethatmathematics,aproductofhuman

thoughtindependentofexperience,issoadmirably

adaptedtotheobjectsofreality.

—AlbertEinstein

Mathematicsislinkedandusedbysomanythingsinourworld,

yetdelvesinitsownworlds—worldssostrange,soperfect,so

totallyalientothingsofourworld.Acompletemathematical

worldcanexistonthepinpointofaneedleorintheinfinitesetof

numbers.Onefindssuchworldscomposedofpoints,equations,

curves,knots,fractals,

andsoon.Untilone

understandshow

mathematicalworldsandsystems

areformed,someofits

worldsmayseem

contradictory.Forexample,one

mightask howaninfinite

worldcanexistonlyona

tinylinesegment,ora

worldbecreatedusing %

onlythreepoints/This

chapterseekstoexplore

the

magicofsomeofthese

mathematicalworldsand _ —£

delveintotheirdomains.

XT

Asdiscussedlater,thecountingnumbers

formamathematicalworldin.themselves.


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MAGICALMATHEMATICALWORLPS35

Why,sometimesI'vebelieved

asmanyassiximpossiblethings

beforebreakfast.

—LewisCarroll

HOW

MATHEMATICA

WORLDSARE

FORMED

LittledidEuclidknowin300B.C.whenhebegantoorganize

geometricideasintoamathematicalsystemthathewasdeveloping

thefirstmathematicalworld.Mathematicalworldsandtheir

elementsabound — herewefindtheworldofarithmeticwithits

elementsthenumbers,worldsofalgebrawithvariables,theworld

ofEuclideangeometrywithsquaresandtriangles,topologywith

suchobjectsastheMobiusstripandnetworks,fractalswith

objectsthatcontinuallychange — allareIndependentworldsyet

areinterrelatedwithone

another.Allformtheuniverseofmathematics.Auniversethatcanexistwithoutanythingfromour

universe,yetauniversethatdescribesandexplainsthingsall

aroundus.

Everymathematicalworldexistsinamathematicalsystem.The

systemsetsthegroundrulesfortheexistenceoftheobjectsinits

world.It

explainshowits

objectsare

formed,how

theygeneratenewobjects,andhowtheyaregoverned.Amathematicalsystem

iscomposedofbasicelements,whicharecalledundefinedterms.

Thesetermscanbedescribed,sothatonehasafeelingofwhat

theymean,buttechnicallytheycannotbedefined.Why?Because

ittakestermstoformdefinitions,andyouhavetobeginwithsome

terms.Forthesebeginningwordstherearenoothertermsthat

existwhichcanbeusedtodefinethem.

Thebestwaytounderstandsuchasystemistolook atone.Here's

howafiniteminimathematicalworldmighttakeform.Assume

thisminiworld'sundefinedtermsarepointsandlines.Inaddition

toundefinedterms,amathematicalsystemalsohasaxioms,

theorems,anddefinitions.Axioms(alsocalledpostulates)areideas


41/333


weacceptasbeingtruewithoutproof.Definitionsarenewterms

wedescribe/defineusingundefinedtermsorpreviouslydefinedterms.TheoremsareIdeaswhichmustbeprovenbyusingexisting

axioms,definitionsortheorems.

Whattypeofdefinitions,theoremsandaxiomscanourminiworld

have?Herearesomethatmightevolve—

Undefinedterms:Pointsandlines.

Definition1:Asetofpointsiscolltnearifalinecontains

theset.

Definition2:Asetofpointsisnoncolltnearifalinecannot

containtheset.

Axiom1:Ourminiworldcontainsonly3distinctpoints,

whichdonotlieonaline.

Axiom2:Anytwodistinctpointsmakea

line.

Theorem1:Onlythreedistinctlinescan

existsInthis

world.

proof:Axiom1statesthatthereare3distinctpointsinthisworld.UsingAxiom2weknowthateverypairofthesepointsdeterminesaline.Hencethreelinesare

formedbythethreepoints

ofthisworld.

ThisexampleIllustrateshowamathematicalworldmightevolve.

AsnewIdeascometomind,oneaddsmoreundefinedterms,

axioms,definitions,andtheoremsandtherebyexpandstheworld.

Thefollowingsectionsintroduceyoutosomemathematicalworlds

andtheirinhabitants.


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MACICALMATHEMATICALWORLDS 37

GEOMETRIC

... Theuniversestands WORLDS

continuallyopentoourgaze,

butitcannotbeunderstood

unlessonefirstlearnsto

comprehendthelanguageandinterpretthecharactersin

whichitiswritten.Itiswritteninthelanguageof

mathematics,anditscharactersare...geometricfigures,withoutwhichitishumanlyimpossibletounderstanda

singlewordofit;withoutthese,oneiswanderingaboutina

dark labyrinth.—Galileo

Mathematicshas

manytypesof

geometries.These

includeEuclidean

andanalytic

geometriesandahost

ofnon-Euclidean

geometries.Herewe

findhyperbolic,elliptic,projective,

topological,fractal

geometries.Each

geometryformsa

mathematicalsystem

withitsown

undefinedterms,

axioms,theorems

anddefinitions.

Althoughthese

geometricworldsmay

usethesamenames

ThisisanabstractdesignofHenriPoincare's(1854-1912)hyperbolicworld.Hereacircleisthe

boundaryofthisworld.Thesizesoftheinhabitantschangeinrelationtotheirdistance

fromthecenter.Astheyapproachthecenter

theygrow,andastheymoveawayfromthecenter theyshrink.Thustheywillneverreachthe

boundary,andforallpurposes,theirworldis

infinitetothem.


43/333

38THEMAC\COFMATHEMATICS

fortheirelementsorproperties,theirelementspossessdifferent

characteristics.Forexample.InEuclideangeometrylinesare

straightandtwodistinctlinescaneitherintersectInonepoint,be

parallel,orbeskew.ButlinesInellipticgeometryarenotstraight

linesbutgreatcirclesofasphere,andthereforeanytwoofIts

distinctlinesalwaysIntersectintwopoints.

Considerthewordparallel.In

Euclideangeometryparallellines

arealwaysequidistantandnever

Intersect.Notsoinellipticor

hyperbolicgeometry.Why?

Becauseeverygreatcircleofa

sphereIntersectsanother.Thus,

ellipticgeometryhasnoparallel

lines.Inhyperbolicgeometryparallel

linesneverIntersect,buttheydonot

resembleEuclideanlines.Hyperbolic

parallellinescontinuallycomecloserand

closertogether,yetneverIntersect.They

arecalledasymptotic.Euclidean,hyperbolic,andelliptic

geometriescreatethreedramaticallydifferentworldswithlinesand

points,etc.,butwhose

propertiesareuniverses

apart.Eachof

theseworldsIsamathematicalsystemuntoItself,andeachhas

applicationsinouruniverse.

Theabovediagramshowstwogreatcircles,line1and2intersectingatpointsA&B.

L̂-

¦}Inhyperbolicgeometry,linesMandNarebothparalleltolineLandpassthroughpointP.MandNareasymptotictolineL.


44/333


NUMBERWORLD!

Numberscanbeconsideredthe

firstelementsofmathematics.

Theirearlysymbolswereprobably

marksdrawnintheearthto

indicateanumberofthings.Buteversincemathematicians

enteredthescenethesimpleworldofcountingnumbershas

neverbeenthesame.Manypeoplearefamiliarwithintegers,

StoneAgenumberpatternsfoundtnLaPdeta,Spain.

fractionsanddecimals,andusethesefortheirdaily

computations.Butnumberworldsalsoincludetherationaland

irrationalnumbers,thecomplexnumbers,thenever-ending

non-repeatingdecimals,transcendentalnumbers,transfinite

numbers,andmanymanysubsetsofnumbersthatarelinkedby

specificproperties,suchasperfectnumberswhoseproperfactors

totalthenumber,orpolygonalnumberswhoseshapesare

connectedtotheshapesofregularpolygons,andonandon.Itis


45/333


InterestingtodelveIntotheinterrelationshipofnumbers,surmise

howtheydeveloped,andexploretheirvariousproperties.

Thecountingnumbersdateback toprehistorictimes.Consider

thesimplemarksoftheStoneAgenumberpatternsfromLaPileta

CaveInsouthernSpain,whichwasInhabitedover25,000years

agountiltheBronzeAge(1500B.C.).Thenumbernwasknown

overthreethousandyearsago,whenItwasusedinthe

calculationsofacircle'sareaandcircumference,andlatershown

tobeirrationalandtranscendental.Ancientcivilizationswere

awarethatfractionalquantitiesexisted.TheEgyptiansusedthe

glyphformouth,O,towritetheirfractions.

Forexample,̂ ŵas1/3,̂ ŵas1/10.

Irrationalnumberswereknownbytheancientmathematicians,

whodevisedfascinatingmethodsforapproximatingtheirvalues.

Infact,theGreeksdevelopedtheladdermethodtoapproxlmatle

the|/2whiletheBabyloniansusedanothermethod.

0= 1= 10= H= 100= 101=twotwo two two two two

0 13 4 5

HexagramsandtheirbinaryeqiduaLents.

Overthecenturiesdifferentcivilizationsdevelopedsymbolsand

countingsystemsfornumbers,andInthe20thcenturythebinary

numbersandbasetwohavebeenputtowork withthecomputer

revolution.GottfriedWilhelmLeibniz(1646-1716)firstwrote

aboutthebinarysystemInhispaperDeProgressioneDyadica

(1679).HecorrespondedwithPereJoachimBouvet,aJesuit

missionaryinChina.ItwasthroughBouvetthatLeibnizlearned

thattheIChlnghexagramswereconnectedtohisbinary

numerationsystem.HenoticedthatIfhereplacedzeroforeach


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MACICALMATHEMATICALWORLDS

brokenlineand1fortheunbrokenline,thehexagramsillustrated

thebinarynumbers.Centuriespriortothis,theBabyloniansdevelopedandImprovedupontheSumeriansexagimalsystemto

developabase60numbersystem.Butthissectiononnumber

worldsIsnotaboutnumbersystemsbutabouttypesofnumbers.

Let'stakeaglimpseatthefirsttypeofnumbers — thecounting

numbers.Intheworldofcountingnumberswefindtheundefined

termsarethenumbers1,2,3 —withsuchaxiomsasthe

orderinwhichtwocountingnumbersareaddeddoesnotaffectthe

sum(a+b=b+a,calledthecommutativepropertyforaddition);the

orderinwhichtwocountingnumbersaremultipliedtogetherdoes

notaffecttheproduct(axb=bxa,calledthecommutativepropertyfor

multiplication). — andsuchtheoremsasAnevennumberplusan

evennumberisalsoanevennumber.And,Thesumofanytwoodd

numbersisalwaysanevennumber.Buttheworldofcounting

numberswerenotenoughtosolvealltheproblemsthatwereto

evolveovertheyears.CanyouImaginetacklingaproblemwhose

solutionwasthevaluexfortheequationx+5=3andnotknowing

aboutnegativenumbers?Whatwouldhavebeensomereactions

— theproblemIsdefective,thereisnoanswer.Arabtexts

IntroducednegativenumbersInEurope,butmostmathematiciansof

thethe16thand17thcenturieswerenotwillingtoacceptthese

numbers.NicholasChuquet(15thcentury)andMichaelStidel

(16thcentury)referredtonegativenumbersasabsurd.Although

JeromeCardan(1501-1576)gavenegativenumbersassolutions

toequations,heconsideredthemasImpossibleanswers.Even

BlaisePascalsaid"Ihaveknownthosewhocouldnotunderstand

thattotakefourfromzerothereremainszero."


47/333

42THEMACl


48/333


•

THEWORLDSOl

DIMENSI

Let'slook attheworldswhichare

createdbytheIdeaofdimensions.A

mathematicalworldcanexistona

singlepoint,onasingleline,ona

plane.Inspace.Inahypercube

(tesseract).Eachhigherdimension

encompassesthosebeneathIt,yet

eachlowerdimensioncanbeaworld

InItself.Imagineyourworldand

yourlifeonaflat

plane.Youcannot

look upordown.Threedimensional

creaturescanInvadeyourworld

withoutyouevenknowingbysimply

enteringyourdomainfromaboveor

below.Mathematicians,writers,and

artistshaveusedvariousIdeastotry

tocapturetheessenceofdifferentdimensionsIntheirworks.

Dimensionsbeyondthethirdhave

alwaybeenIntriguing.Thecubewas

oneofthefirst3-Dobjectstobe

introducedIntothefourth

dimensionbybecominga

hypercube.Thestagesforarrivingat

ahypercubeareillustrated.

Computerprogramshaveevenbeen

devisedtoderiveglimpsesofthe

fourthdimensionbypicturing3-D

perspectivesofthevarious

facetsofthehypercube.


49/333


THEWORLDSOF

INFINITIES Toseetheworldina

grainofsand,

Andaheavenina

wildflower;

Holdinfinityinthepalmofyourhand,

Andeternityinanhour. —WilliamBlake

Infinityhasstimulatedimaginationsforthousandsofyears.Itis

anIdeadrawnuponbytheologians,poets,artists,philosophers,

writers,scientists,mathematicians — anideathathasperplexed

andIntrigued — anideathatremainsillusive.Infinityhastaken

ondifferentIdentitiesIndifferentfieldsof

thought.In

earlytimes,

theideaofInfinitywas,rightlyorwrongly,linkedtolarge

numbers.Peopleofantiquityexperiencedafeelingoftheinfinite

bygazingatstarsandplanetsoratgrainsofsandonabeach.

AncientphilosophersandmathematicianssuchasZeno,

Anaxagoras,Democrltus,Aristotle,Archimedespondered,posedandarguedtheideasthatinfinitypresented.

AristotleproposedtheIdeasofpotentialandactualinfinities.He

arguedthatonlypotentialinfinityexisted.1

InTheSandReckonerArchimedesdispelledtheideathatthe

numberofgrainsofsandonabeachareinfinitebyactually

determiningamethodforcalculatingthenumberonallthe

beachesoftheearth.

InfinityhasbeentheculpritInmanyparadoxes.Zeno'sparadoxes

ofAchillesandthetortoiseandtheDichotomy2haveperplexedreadersforcenturies.Galileo'sparadoxes3dealingwith

segments, points,andinfinitesetsshouldalsobenoted.


50/333


ThisfieldofsunflowersintheSpanishcountrysidegivestheillusionofinfinity.

Thelistofmathematicianswiththeirdiscoveriesandusesor

misusesofInfinityextendsthroughthecenturies.Euclid(circa300

B.C.)showedthattheprimenumberswereInfinitebyshowing

therewasnolastprime.Headwayintherealmoftheinfinitewas

madebyBernhardBolzano(1781-1848),GottfriedW.Leibniz

(1646-1716),andJ.W.R.Dedeklnd(1831-1916).Butthe

phenomenalwork ofGeorgCantor(1845-1918)onsettheorywasa

majorbreakthrough.Building,creatingandrefiningIdeas,Cantor

foundanewwaytoorganizemathematicsbyuseofthenotionofa

set.HedeterminedawaytocompareInfinitesetsbydevelopingtransflnltenumbers-numbersthatdaredtocrosstherealmof

thefinite.Usingtheideaofequivalentsetsandcountabillty,he

determinedwhichInfinitesetshadthesamenumberofobjectsandassignedthematransflnltenumber.Hiswork andproofson

thesetopicsareingenious.


51/333


Inadditiontoteasingourminds,InfinityIsanIndispensable

mathematicaltool.IthasplayedacrucialroleInmanymathematicaldiscoveries.WefinditusedIn:determiningtheareasand

volumesbothingeometryandcalculus — calculatingapproximations

for7i,andeandotherirrationalnumbers—trigonometry-

calculus — half-lives — infinitesets — self-perpetuatinggeometric

objects — limits — series — dynamicsymmetry — andmore.

Otherpartsofthisbook explorevariousnotionsofInfinity,suchIdeas

asinfinitegeneratingfractals,thechaostheory,thecontinual

searchforalargerprimenumber,transfinitenumbersandothers

adInfinitum.

T̂hecountingnumbersarepotentiallyinfinite,sinceonecanbeaddedtoanynumbertogetthenext,buttheentiresetcannotbeactuallyattained.

InTheDichotomyParadoxZenoarguesthatatravelerwalkingtoa

specificdestinationwillneverreachthedestinationbecausethetravelermustfirstwalk halfthedistance.Reachingthishalfwaypoint,thetravelerthenhastowalk halftheremainingdistance.Thenhalfofthe

partthatremains.Sincetherewillalwaysbehalfofthepartthatremainstowalk andaninfinitenumberofhalfwaypointstopass,thetravelerwillneverreachthedestination.

3InGalileo's1634work,DialoguesConcerningTwoNewSciences,hediscussesinfinityinrelationtothepositiveintegersandthesquaresofthepositiveintegers.Heevendealswithone-to-onecorrespondence

betweenthesetwoinfinitesets.Buthereachestheconclusionthattheconceptsofequality,greaterthan,andlessthanwereonlyapplicabletofinitesets.Galileobelievedtheprinciplethatthewholeisalwaysgreaterthanitspartshadtoapplytobothfiniteandinfinitesets.Threehundredyearslater,Cantorshowedthisprincipledidnotholdforinfinitesetsandusedtheideaofone-to-onecorrespondencetorevisethetraditionalnotionsofequality,greaterthan,andlessthanwhendealingwithinfinitesets.Cantor'smodificationsdidawaywithmanyparadoxesinvolvinginfinitesetsandthewholeisalwaysgreaterthanitsparts.


52/333


FRACTAL

WORLDS / coinedfractalfromtheLatin

adjectivefracfus.The

correspondingLatinverb

frageremeans'tobreak':to

createirregularfragments....howappropriateforour

needs! — that,inadditionto'fragmented'(asinfractionor

refraction)fractusshouldalsomean'irregular1,both

meani ngspreservedinfragment. —BenoitMandelbrot

FractalsaremagnificentobjectswhichcomeInInfinitelymany

shapes.ErnestoCesaro(Italianmathematician1859-1906)wrote

thisaboutthegeometricfractal,theKochsnowflakecurve— What

strikesmeaboveallaboutthecurveisthatanypartissimilartothe

whole.Totrytoimagineitascompletelyaspossible,itmustbe

realizedthateachsmalltriangleintheconstructioncontainsthe

wholeshapereducedbyanappropriatefactor.Andthiscontainsa

reducedversionofeachsmalltrianglewhichinturncontainsthe

IIIIllll llllIIIIiiii miiiiimi miiiii

mi nn

In1883,CantorconstructedthisfractalcalledtheCantorsetStartingwiththesegmentoflengththeunitintervalonthenumberline,Cantorremovedthemiddleonethirdandgotstage1.Thentoeachremaining3rds,removedthemiddleone-third,therebycreatingthe2ndstage.Repeatingtheprocessadinfinitum,theinfinitesetofpointsthatremainsiscalledtheCantorsetHerearethefirststagesoftheCantorset


53/333


€

tr

a

wholeshapereducedeven

furtherandsoonto

Infinity... Itisthisself-

similarityinallitspart,

howeversmall,thatmakes

thecurveseemsso

wondrous.Ifitappearedin

realityitwouldnotbe

possibletodestroyitwithout

removingitaltogether,for

otherwiseitwould

ceaselesslyriseupagainfrom

thedepthsofitstriangles

likethelifeoftheuniverse

itself.Thisistheessenceoffractals.Ifaportionofitremains,

thatportionretainstheessenceofthefractal—whichinturncan

regenerateitself.SowhatIsafractal?Perhapsmathematicians

havepurposelyavoidedgivingadefinitiontonotrestrictorinhibit

ThefirstthreestagesofthePeanoCurve.ThePeanocuruevoasmodein.the1890's,

byrepeatedlyapplyingsuccessive

generationtoasegment.

ThefirstfourstagesoftheKochsnowjlake.TheKochsnowjlakeisgenerated bystartingwithanequilateraltriangle.Divideeachsideintothirds,deletethemiddlethird,andconstructapointoffthatlengthoutfromthedeletedside.

thecreativityoffractalcreationsandideasthatareformulatingIn

thisverynewfieldofmathematics.Withthisnewfield,Ideassuch

asfractionaldimensions,iterationtheory,turbulenceappllca-


54/333


?Ations,self-similarityhave

evolved. Applicationsfor

fractalsrangefromacid

raintozeolites,from

astronomytomedicine,from

cinematographytocartography

toeconomics,andonand

on.

ThefirstfourstagesoftheSlerplnksitriangle.Beginwithanequilateraltriangle.DivideitintoJourcongruenttrianglesasshownandremovethemiddleone.Repeatthisprocesstothesmallertrianglesformed

adinfinitum.Theresultingfractalhasinfiniteperimeterandzeroareal

Â\ K̂ Mathematicallyspeaking,a

ÂĴ /K?̂fractalisaformwhichbeginswithanobject —such

asasegment,apoint,a

triangle—thatisconstantly

beingalteredbyreapplying

aruleadinfinitum.The

rulecanbedescribedbya

mathematicalformulaorbywords.Thepreviousdiagrams

illustratefouroftheearliestfractalsmade.

Onecanthink ofafractalasanevergrowingcurve.Toviewa

fractal,youmustreallyviewitinmotion.Itisconstantly

developing.Todaywearefortunatetohavecomputerscapableofgeneratingfractalsbeforeoureyes.Itwasequallyfortunatethat

BenoitMandelbrot,inthesamespiritoftheearlymathematicians,

studiedandexpandedtheideasandapplicationsoffractals

almostsinglehandedlyfrom1951-75.Infact,hecoinedtheword

fractalHowastonishedtheadventurousmathematicians*ofthe

19thcentury,whofirstdaredtolook attheseideasmost

consideredmonstrous2andpsychopathic,wouldbetoseethewondrousgeometryoffractalsinmotion.

Whenweviewanillustrationorphotographofafractal,weare

seeingitforonemomentintime — itisfrozenataparticularstage

ofitsgrowth.Inessenceitisthisideaofgrowthorchangethat

linksfractalsdramaticallytonature.Forwhatisthereinnature


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thatIsnotchanging?Evenarock ischangingonamolecularlevel.

Fractalscanbedesignedtosimulatealmostanyshapeyoucan

imagine.Fractalsarenotnecessarilyconfinedtoonerule,buta

seriesofrulesandstipulationscanbetherule.Trycreatingyour

ownfractal.Pick asimpleobjectanddesignaruletoapplytoIt.

Thefirst jivestagesofacomputergeneratedgeometricfractal

1MathematiciansGeorgCantor,HelgevonKoch,KarlWelerstrass,Dubois Reymond,GulseppePeano,WaclawSlerplnskl,FelixHaussdorff,A.S.Besicovitch(HaussdorffandBesicovitchworkedonfractional

dimensions),GastonJulia,PierreFatou(JuliaandFatouworkedonIterationtheory),LewisRichardson(workedonturbulenceandself-similarity)

— spanningtheyearsfrom1860'stoearly20thcentury—exploredIdeas

dealingwiththe"monsters".

2These"monsters"wereneitheracceptedorconsideredworthexploringbyconservativemathematiciansofthetime.Itwasfeltthatfractalscontradictedacceptedmathematicsbecausesomewerecontinuousfunctionsthatwerenotdlfferentlable,somehadfiniteareasandinfinite

perimeters,andsomecouldcompletelyfillspace.


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THEPARABLEOFTHEFRACTAL

' Wake-upFractal!Youmustgettowork,"thevoiceproddedthe

sleepingFractal.

"Notagainandsoearly,"pleadedFractal"IJustgotmydimensions

inorder."

"Wake-upFractal!Comedownfromthatcloudyoumade,"thevoiceproddedthesleepingFractal.

'You'reneededattheGeologicalSurvey—anothercoastlineneeds

tobedescribed,"thevoicecontinued.

"WhenwillIgetabreak?'questionedFractal

'YouVehaditeasyforcenturies —

nowit'stimetogettowork,"the

voicereplied.


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52 THEMAC \COFMATHEMATICS

' Work,u>ork,work.Whydon'ttheycallonSquare,Circle,Polygon,

oranyotherEuclideanfigure?Whyme7'askedFractal

"Firstyoucomplainedatbeingignored,andthattheycalledyoua

monster.Nowthatthey'refinallyunderstandingyou,youwantto

retire.Justbethankfulyouaresopopular,"thevoicerebutted.

"Popularisonething,buttheywon'tletmerest.Itsneverbeenthe

samesincethatMandelbrotchristenedmeandgavememydebut,"

repliedFractal."Mathematiciansweretediouslystrugglingwith

me.TmsuremyfractionaldimensionsthrewthemoffforawhUe.

Thosepoorsoulsfromthe19thcenturyhadnocomputerstohelp

them.Mostmathematicianswouldnotacceptme,forIdidnotfitor

followtheirmathematicalrules.Butsomemathematicianswere

stubborn.NowhereIam,beingdesignedandusedinsomany

areas— computerscertainlywereaboon.Onemomentthescreen

displaysafragmentorbeginningpartofafractalandthenext

momentthescreenisbeingfilledwithitsgenerations—ever

growing.Theyarenowusingmeinalmosteverything—Icon

describeroots,vegetables,trees,popcorn,clouds,scenery...I

mustsayitsisveryexcitingtostretchmylimits.Ilovetodo

coastlinesbecauseitstill

bafflesmanypeopletolearnlean

enclosedaregionwhoseareaIsfinitewhilemyperimeterIsinfinite.

Fmservingformodelingmanyoftheworld'sphenomena.For

example—populationwithPeanocurves,fractalcurvesforcreating

sceneinmovies,fractalsfordescribingastronomy,meteorology,

economics,ecology,etcetera,etcetera,etcetera.Tmsobusyand

involvedthatthingsarebeginningtogetabitchaotic,especially

sincetheymixedmeIntothechaostheory,"Fractalsaidsoundingverytired.

Thevoicestartedagain."Stopcomplaining!Chaostheoryoffers

yousomevariety.Withoutityou'dJustbecontinuallyrepeating

thesameruleandgeneratingthesameoldshapeoverandover,at


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Thebeginningstagesofafractalcloud.

leastwhensomeinitialinputisslightlyvaried,somethingtotally

differentcanevolve."

"Isupposeyou'reright,"Fractalsighed.

"OfcourseI'mright.Justthink howboringitwouldbetobethe

sameshapeforever,likeapoorsquareoracircle,"thevoice

asserted.

"Well,atleasttherearenosurprisesforasquareorcircle."Fractal

countered.

"That'spreciselyit.Lifeisfullofsurprises;that'swhytheyare

callingonyousooften.Youaremorelikelife."Thevoiceseemed

complementingFractal.

'YoumeanFmhuman?'Fractalasked.

"Iwouldn'tgothatfar.Andbesidesalllifeisn'thuman.Let'ssay

you're justdifferent,andyou'renon-Euclidean!"Andwiththat

commentthevoicedriftedoff.


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54 THEMAC \COFMATHEMATICS

FINDINGTHEAREAOFASNOWFLAKECURVE

Thisbeautifulgeometricfractalwascreatedin1904byHelgevon

Koch.TogenerateaKochsnowflakecurve,beginwithan

equilateraltriangle.Divideeachsideintothirds.Deletethe

middlethird,andconstructapointofthatlengthoutfromthe

deletedside.Repeattheprocessforeachresultingpointad

infinitum.

A

Twofascinatingproperties,whichseem

contradictory,are—

•theareaofthesnowflakecurveis8/5oftheareaof j

theoriginaltrianglethatgeneratesit;»the

perimeterofthesnowflakecurveisinfinite,g

Hereisaninformal

proofthattheareaofthe

snowflakecurveis8/5ofitsgeneratingtriangle.

I.AssumetheareaofequilateralAABCisk.

II.DivideAABCintoninecongruentequilateral

trianglesofarea,a,asshown.Thusk=9a.

Nowconcentrateondeterminingthelimitofthearea

ofoneofthe6initialpointsofthesnowflakecurve.Weknowtheareaofthelargepointisa,sinceitsone *.,"

oftheninetriangles.Thenextsetofpointsgenerated

fromithavearea(a)(1 /9)each,Justliketheoriginal

trianglehadbeendividedinto9congruenttrianglesitalsois.In

fact,eachsuccessivepointisbrokendownintoninecongruent

triangleswithtwotrianglesspringingfromit.

STEPIIIshowsthesummationofthevariousareasofthispoint.

STEPIV:Now,byaddinguptheareascreatedbyeachofthe6

pointsplusthehexagonintheinterioroftheoriginalgenerating

triangle,wegetexpressionIV.

STEPIVischangedtoSTEPV.Theresultingseriesinthe


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MACICALMATHEMATICALWORLDS

III.

,9*9Noticethereare8ofthis

stagepoints.

•̂9»9,Noticethereare32ofthis

stage-points.

IV.a+2-+29)19*9

14+29.9.9

42+219.9.9.9

22*42*4Z!•¥H+-+ —Z-+5—+-r-+.

9 92 93 94 9l9

n-2

6a+6a

bracketsisageometricserieswithratio4/9and2/9asitsinitial

term,sowecancalculateitslimit.(2/9)/(l-(4/9))=2/5.

STEPVI.Substitutingthe2/5forthelimitoftheseries,weget

[l+2/5]6a+6a=72a/5

Nowweneedtoexpresstheareaofthesnowflakecurveintermsof

k,theareaoftheoriginalgeneratingtriangle.Sincek=9a,weget

a=k/9.Substitutingthisforain72a/S,weget(72/S)(k/9)=

(8/5)k.


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MONSTERCURVES

6464 XS6**56

Thestagesofthe

Sierpinskttriangle.

Supposetheareaofthe

initialgenerating

equilateraltriangleis1

squareunit.Thesumoftheareasoftheblack

andwhitetrianglesare

indicatedthroughthefirst

fivegenerations.Supposetheblack

trianglerepresentsremovalofarea.Notice

howthevalueforthe

whitetrianglesis

continuallydecreasing,meani ngthewhiteareais

approachingzero.Thus

theareafortheSierpinskt

triangleapproaches0,UJhlleitsperimeter

approachesinfinity.

UntilBenoltMandelbrotcoinedtheterm"fractal"Inthelate

1970's,thesecurveswerereferredtoasmonsters.19thcentury

conservativemathematiciansconsideredthesemonstercurves

pathological. Theyneitheracceptedorconsideredthemworth

exploringbecausetheycontradictedacceptedmathematicalideas.

Forexample,somewerecontinuousfunctions(functionswithout

anygaps)thatwerenotdlfferentlable,somehadfiniteareasand

infiniteperimeters,andsomecouldcompletelyfillspace.The

SierpinskitriangleJ (alsocalledSierpinsktgasket)hasanInfinite

perimeterandafinitearea.Theillustrationabovetriestopoint

outhowtheSierpinskttriangle'sareaiszero.

EarnedaftermathematicianWaclawSierpinski(1882-1920).


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MANDELBROTSETCONTROVERSY

Inthe17thcenturyanumberofprominent

mathematicians(Galileo,Pascal,Torricelli,

Descartes,Fermat,Wren,Wallis,Huygens,

JohannBernoulli,Leibniz,Newton)were

intentondiscoveringthepropertiesofthe

cycloid.Evenasthereweremanydiscoveries

atthisperiodoftime,therewerealsomanyargumentsaboutwhohaddiscoveredwhat

first,accusationsofplagiarism,and

minimizationofoneanother'swork.Asaresult,

thecycloidhasbeenlabeledtheappleof

discordandtheHelenofgeometry.20thcenturymathematicians

nowseemtohaveanewHelenofgeometry—theMandelbrotset.

WhofirstdiscoveredtheMandelbrotset1?Thisisaveryheated

questionamongpresentdaymathematicians.Thecontenders

are:

— BenoitMandelbrotisoftendescribedasapioneerbecauseofhisinitial

work onfractalsinthe1970s.Mandelbrotswork showingvariantsoftheMandelbrotsetwaspublishedDecember26,1980inAnnalsofthe

NewYork AcademyofSciences.Hiswork ontheactualMandelbrotset

was

publishedin1982.

—JohnHHubbardofCornellUniversityandAdrienDouadyofthe

UniversityofParisnamedthesetMandelbrotinthe1980swhile

workingonproofsofvariousaspectsofthesetIn1979,HubbardsayshemetwithMandelbrot,andshowedMandelbrothowtoprograma

computertoplotiterativefunctions.HubbardadmitsthatMandelbrotlaterdevelopedasuperiormethodforgeneratingtheimagesoftheset

— RobertBrooksandJ.PeterMatelskiclaimtheyindependentlydiscoveredanddescribedthesetpriortoMandelbrotalthoughtheir

work wasnotpublisheduntil1981.

—PierreFatoudescribedJuliasets'unusualpropertiesaround1906,and

GastonJulia'swork onJuliasetspredatesFatou's.(Juliasetsactedas

springboardsforMandelbrotsets.)Whogetsthecredit?Perhapsall.

iTheillustrationaboveisthemostfamiliarfractalformfromtheMandelbrotset.TheMandelbrotsetisatreasuretroveoffractals,whichcontainsaninfinitenumberoffractals.Thesetisgeneratedbyaniterativeequation,e.g.ẑ+c,wherezandcarecomplexnumbersandc

producesvaluesthanareconfinedtoacertain

boundary.


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MATHEMATICAL

WORLDSIN

LITERATURE

Thereisanastonishing

imaginationevenin

thescienceofmathematics.

Isthetesseractthefigmentofamathematicalimagination?Isthe

only"real"dimensionthe3rddimension?WelearninEuclidean

geometrythatapointonlyshowslocation,anditcannotbeseen

sinceithaszerodimension.Yetwecanseealinesegment

composedoftheseinvisiblepoints.Alineisinfiniteinlength,yetdoes

suchafigureexistintherealmofourlives?Whataboutaplane?

Infiniteintwodimensionsandonlyonepointthick.Whatisa

planeinourworld?Considerthepseudosphereofhyperbolic

geometry;asymptoticlinesofexponentialfunctions,infinitiesoftransfinitenumbers.Considertheimaginarynumbers,the

complexnumberplane,fractals,andevenacircle.Onewondersif

thesecanexistinourworld.Althoughthereisnodoubtoftheir

existenceintheirrespectivemathematicalsystems,these

conceptsareonlymodelsinourworld.

Manywriters,artistsandmathematicianshaveingeniouslyusedtheseconceptstodescribeworldswheretheseideascometolife.

SuchwritersasDante,ItaloCalvino,JorgeLuisBorges,and

MadeleineL'Englehavedrawnonmathematicstoenhancetheir

creations.

Inthe19thcentury,mathematicianHenriPoincarecreateda

modelofa

hyperbolicworldcontainedintheinteriorofacircle.

Heretoallthingsandinhabitants,theircircularworldwas

infinite.Unbeknownsttothesecreatures,everythingwouldshrink

asitmovedawayfromthecenterofthecircle,whilegrowingasit

approachedthecenter.Thismeantthecircle'sboundarywas

nevertobereached,andhencetheirworldappearedinfiniteto


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CircleLimitIV(HeavenSiHell)byM.C.EscherdepictsaworldreminscentofHenri.Poincare'shyperbolicwodd.

© 1994M.C.Escher/CordonArt-Baam-HoEand.ARrightsreserved.

them.In1958,artistM.C.Eschercreatedaseriesofwoodcuts,

entitledCircleLimitI,II,III,IVwhichconveyafeelingofwhat

Poincarehaddescribed.Escherdescribedaworldas"thebeautyof

this

infiniteworld-in-an-enclosed

plane."

1

Forhernovel,AWrinkleinTime,MadeleineL'Engleusesthetesse-

ractandmultipledimensionsasmeansofallowinghercharacters

totravelthroughouterspace,"...forthe5thdimensionyou'd

squarethefourthandaddthattotheotherfourdimensionsandyou

cantravelthroughspacewithouthavingtogothelongway


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around...Inotherwordsastraightlineisnottheshortestdistance

betweentwopoints."

ItaloCavinodescribesaworldthatexistsinasinglepointInhis

shortstoryAHAtOnePoint.Hisingeniouscreativitymakesone

believesuchazerodimensionalworldactuallyexists."Naturally,

wewereallthere, —OldQfwfgsaid,—whereelsecouldwehave

been?Nobodyknewthenthattherecouldbespace.Ortimeeither:

whatusedidwehave

fortime,packedintherelikesardines?Isay

"packedlikesardines,"usingaliteraryimage:inrealitythere

wasn'tevenspacetopack usinto.Everypointofeachofus

coincidedwitheverypointofeachoftheothersinasinglepoint,whichwaswhereweallwere."

Lookingback totheMiddleAgesandDante'sTheDivineComedy,wefindEuclidean

geometricobjectswerethebasesforDante's

hell.Thecone'sshapewasusedtoholdpeopleinstagesofhell.

Withinit,Dantehadninecircularcross-sectionsthatactedas

platformswhichgroupedpeoplebysinscommitted.

FromDante'sTheDivineComedy.Theplanofconcentricspheres,whichshowstheEtarthinthesphere(bearingtheepicycle)oftheMoon,andthesearealsoenclosedinthesphere(bearingtheepicycle)ofMercury.


66/333

MAGICALMATHEMATICALWORLDS 61

Inthe1900'sinfinitywasfeaturedinJorgeLuisBorges'TheBook

ofSand.Herethemaincharacteracquiresa"marvelous"book.

"Thenumberofpagesinthisbook isnomoreorlessthaninfinite.

Noneisthefirstpage,nonethelast.Idon'tknowwhythey're

numberedinthisarbitraryway.Perhapstosuggestthetermsofan

infiniteseriesadmitanynumber."Thisbook adverselychangeshis

lifeandhisoutlook onthings,untilherealizeshemustfindaway

todisposeofit— "Jthoughtoffire,butIfearedthattheburningofan

infinitebook mightlikewiseproveinfiniteandsuffocatetheplanet

withsmoke."Whatwouldyoursolutionbe?Youmightwantto

readthebook tofindhowtheheroresolvedhisdilemma.

SciencefictionwritershaveutilizedmathematicalIdeastohelp

createtheirworlds.Forexample,inanepisodeofStarTrek —The

NextGeneration,thestarshipisbeingpulledbyan"invisible"force

towardablack hole.Onlywhentheship'sschematicmonitor

changesperspectivedoesthecrewrealizetheunknownforceisa

2-dimensionalworldofminutelifeforms.

MathematicsIsfullofIdeasthatmakeone'simaginationchurn

andwonder—Aretheyreal?Tomathematicianstheyarereal.

MathematiciansarefamiliarwiththeworldsinwhichtheseIdeas

reside —

perhapsnotwithinourrealm,butrealintheirown

nonetheless!

M.C.Escher,HarryN.Abrams.Inc,NewYork,1983.


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68/333

MATHEMATICS

&ART

ART,THE4THDIMENSIONS.

NON-PERIODICTILING

MATHEMATICS&SCULPTURE

MATHEMATICALDESIGNS&ART

MATHEMATICS&

THEARTOFM.C.ESCHER

PROJECTIVEGEOMETRY&ART

MIXINGMATHEMATICS&ART

OFABRECHTPURER

COMPUTERART


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Themostbeautifulthingwecanexperienceisthe

mysterious.Itisthesourceofalltrueartandscience.

—AlbertEinstein

Linkingmathematicsandartmayseemalientomanypeople.Butmathematicalworldsofgeometries, algebra, dimensions,

computershaveprovidedtoolsforartiststoexplore,enhance,

simplify,andperfecttheirwork.Overthecenturies,artistsand

theirworkshavebeenInfluencedbytheknowledgeanduseof

mathematics.TheancientGreek sculptor,Phidias,Issaidtohave

usedthegoldenmeanIntheproportionsofmanyofhisworks.

AlbrechtDfireremployedconceptsfromprojectivegeometryto

achieveperspective,andgeometricconstructionsplayedavital

roleInhis

typographyof

Romanletters.

Sincereligious

doctrine

prohibitedthe

useofanimate

objectsIn

fĵ JN̂AWH*il»'rHHfi«{»$*«$* fc3h

¦&

AsketchfromoneofLeonardodaVincCsnotebooks

Illustratinglinesconvergingtoavanishingpoint

Moslemart,Moslemartistshadtorelyonmathematicsasan

avenuefortheirartisticexpression,thusleadingthemtocreatea

wealthoftessellationdesigns.LeonardodaVincifelt"...nohuman

inquirycanbecalledscienceunlessit

pursuesits

paththroughmathematicalexpositionanddemonstration."Leonardo's

sculpturesandpaintingsillustratehisstudyofthegoldenrectangle,

proportions,andprojectivegeometry,whilehisarchitectural

designsshowhiswork ingeometricstructuresandknowledgeof

symmetry.Thetopicsinthissectionareafewexamples

illustratingtheconnectionbetweenmathematicsandart.


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MATHEMATICS&ART65

ART,THE4THDIMENSION

NON-PERIODI

TILING

Mathematicstakesusintothe

regionofabsolutenecessity,to

whichnotonlytheactual

world,buteverypossibleworld,

mustconform.—AlbertEinstein

Onacanvas,theartistIsrestrictedtotwo-dimensionsto

communicatethefeelingofotherdimensions.Iconartistsofthe

Byzantineperioddepictedthree-dimensionalreligiousscenesin

onlytwo-dimensions,givingthesubjectmatteramystical

appearance. DuringtheRenaissance,artistsusingtheconceptsof

projectivegeometrytransformedtheirflatcanvasintothethree-

dimensionalworldtheywantedtoconvey.Today,mathematics

playsanactiveroleInprovidingInspirationandtoolsforthe

creationandcommunicationofanartist'sIdeas.Artistsuse

mathematicalIdeastoescapeintohigherdimensions.The

hypercube1,forexample,hasbeenusedbyartiststotakeastepIntothefourth-dimension.Intheearly1900'sarchitectClaude

Bragdonadaptedthehypercubealongwithotherfour-

dimensionaldesignsinhiswork2.Intriguedbythehypercube,

SalvadorDali3delvedIntomathematicsforhismodelofan

unfoldedhypercubewhichisthefocalpointinhispaintingThe

Crucifixion4(1954).

Today,thereareanumberofartistspursuingartInconnection

withmathematicalIdeas — inparticular,mathematicsof

non-periodictiling,multi-dimensionsandcomputerrenditions.In

fact, computerrenditionsofthehypercube,createdby

mathematicianThomasBrancroftandcomputerscientistCharles

StraussofBrownUniversity,producevisualizationofthe

hypercube movinginandoutofa3-Dspace.Variousimagesofthe

hypercubeinthe3-Dworldaretherebycapturedonthecomputer


71/333


>¦*'

TheunfoldedhypercubewastheInspirationforSalvadorDall'sTheCrucifixion(1954).MetropolitanMuseumofArt,GiftoftheChesterDale

Collection,1955.(55.5)


72/333

MATHEMATICS&ART67

monitor.Introducedtothispartofmathematics,artistTonyRob-

binhascreated3-Dcanvasrepresentationsofthehypercubewith

thecanvasactingasaplaneintersectingthehypercube.Non-

periodictilings,Penrosetiles,quasicrystalgeometryandfivefold

symmetry5,haveaidedRobblnincreatingfascinatingstructures,whichchangedramaticallyaccordingtotheperspectiveofthe

viewer.OnemomentoneviewsaseriesoftriangleswhileInthe

nextpositioninterlacedpentagonalstarsappear. The

combinationofnon-periodictilingofboth2and3-Dforms

intertwinedinanunusualtypeofsymmetrycreateanalmost

contradictoryimage.

*Alsoknownasthetesseract— a4-dimensionalrepresentationofacube.

2AtthesametimeBragdonusedmagiclinesinarchitecturalornamentsandgraphicdesignsofbooksandtextiles.3DalicontactedthemathematicsdepartmentatBrownUniversityforfurtherinformation.

*JesusChristisnailedtoacrossrepresentedbytheunfolded

fourth-dimensionalhypercube.

5Non-periodictilingistessellatingwithtilesorshapeswhichcreatedesignswhichhavenopattern.

n-Joldsymmetry:Ifapatternispreservedwhenrotated360'/n,itissaidtohaven-foldsymmetry.Therefore,apatternhasfivefoldsymmetryifarotationof72'retainsthepattern.

Quasicrystalsareanewlydiscoveredstateofsolidmatter.Solidmatterwasthoughttoexistonlyintwostates,amorphorousorcrystalline.In

amorphoroustheatoms(ormolecules)arearrangedrandomly,whilein

crystallinethearrangementistheperiodicrepetitionofaunitcell

buildingblock.Thediscoveryofquasicrystalsrevealedanewstateinwhichthearrangementofunitsisnon-periodicandwithanunusual

symmetry,e.g.fivefold,notpresentinamorphorousorcrystallinematter.


73/333


MATHEMATICS&SCULPTURE

Dimensions,space,centerof

gravity, symmetry, geometric

objects,andcomplementarysets

areallmathematicalideas

whichcomeintoplay

whenasculptorcreates.

Spaceplaysaprominent

roleinasculptor'sworks.

Someworkssimply

occupyspaceinthesameway

weandotherlivingthings

do.Intheseworksthe

centerofgravity1isapointwithinthesculpture.

Theseareobjectsthatare

anchoredtotheground

andoccupyspaceina

mannerwithwhichweare

comfortableor

accustomed.Forexample

Michelangelo's David,the

Discobolusbytheancient

Greek artist,Myron,or

BeniaminoBufano'sSt.

FrancisonHorseback all

havetheircenterof

gravitywithinthemassoftheirsculpture.Somemodernartsculptures

playwithspaceanditsthreedimensionsinunconventionalways.

Theseusespaceasanintegralpartofthework.Consequentlythe

centerofgravitycanbeapointofspaceratherthanapointofthe

mass,asillustratedbysuchworksasRedCubebyIsamuNog-

uchi,theEclpsebyCharlesPerry,andtheVaillancourtFountain

TheDiscobolus(circa450B.C.)byMyron,castinbronze,capturesamomentinmotion.


74/333

MATHEMATICS&ART69

byLouisVaillancourt.

Othersculptures

dependontheir

interactionwithspace.Here

thespacearoundthe

artwork (the

complementarysetofpoints

ofthemass)isas,or

equally,importantas

thesculpture.

ConsiderZincZincPlainby

CarlAndre.This

sculpture isstagedina

roomdevoidofany

otherworksor

objects.Theplaneiscreatedby36smallsquares

formingasquare

whichliesflatonthe

„ _- SanFrancisco'scontroversialVaillancourtFountaintloor.ineroomrepre-hasasitscenterofgravityapointofspace.sentsspace,thesetof

allpoints,andhedescribeshiswork as

"acutofspace".2

Some

worksseemtodefygravity.Theseincludesuchsculpturesasthe

mobilesofAlexanderCalderwiththeirexquisitebalanceand

symmetryandIsamu'sNoguchi' sRedCubebalancing

mysteriouslyonitsvertex.Thereareevensculptureswhichuse

theEarthitselfasanintegralpartoftheartanditsstatement,e.g.

TheRunningFencebyCrista,CarlAndre'sSecant,andthe

mysteriousgeometricgrasstheoremsappearinginEngland.

Oftenthephysicalnatureofanartist'sconceptionalwork requires

mathematicalunderstandingandknowledgetomakethework

possible.LeonardodaVincimathematicallyanalyzedmostofhis

creationsbeforeundertakingawork.IfM.C.Escherhadnot

mathematicallydissectedtheideasoftessellationandoptical


75/333


Illusions,hisworkswouldnothaveevolvedwiththeeasewith

whichhewasabletoundertakethemonceheunderstoodthe

mathematicsoftheseideas.

Todaytherearemany

examplesofsculptorslooking

atmathematicalideasto

expandtheirart.Tony

Robbinusesthestudyof

quasicrystalgeometry,4th

dimensionalgeometry,and

computersciencetodevelop

andexpandhisart.Inhis

Easteregggiantsculpture

RonaldDaleReschhadtouse

Intuition,

ingenuity,mathematics,andthecomputeras

ThissketchbyLeonardoshowshis

analysisofthehorse'sanatomy.

AuthorInfrontofContinuumbyCharles

Perry.NationalAir&SpaceMuseum,

WashingtonD.C.

wellashishandsto

completehiscreation.

Andartist-

mathematicianHelaman

R.P.Fergusonuses

traditionalsculpting,thecomputerand

mathematicalequations

tocreatesuchworksas

WideSphereandKlein

BottlewithCross-cap&

Vector.Consequentlyit

isnotsurprisingtofindmathematicalmodels

doublingasartistic

models.Amongthesewe

findthecube,the

polycube,thesphere,the


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MATHEMATICS&ART71

torus,thetrefoilknot,theM6biusstrip,polyhedrons,the

hemisphere,knots,squares,circles,triangles,pyramids,prisms MathematicalobjectsfromEuclideangeometryand

topologyhave

playedimportant

rolesinthe

sculpturesofsuch

artistsasIsamu

Noguchi,David

Smith,Henry

Moore,SolLeWitt.

Regardlessofthe

sculpture,

mathematicsis

inherentinit.Itmay

havebeen

conceivedand

createdwithouta

mathematical

thought,

nevertheless

mathematics

existsinthat

work, justasit

existsinnatural

creations.

AnAlexanderCaldermobile.EastBuildingoftheNationalGalleryofArt,Washington,D.C..

T̂hecenterofgravityIsthepointonwhichanobjectcanbebalanced.Forexample,thecenterofgravityorcentroldofatrianglecanbedeterminedbydrawingthattriangle'smedians.ThepointwherethethreemediansIntersecthappenstobethecenterofgravity.

2Art&Physics,LeonardShlaln,WilliamMorrow&Co.NY,1981.


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72 THEMACICOFMATHEMATICS

PUTTINGMATHEMATICSINTOSTONE

EineKleineRock Musik HI

PhotographybyEdBernik.FromHelamanFerguson:MathematicsinStoneandBronzebyClaireFerguson.MeridianCreativeGroup,Copyright © 1994.

Trefoilknots — torus

— spheres — vectors

— flow— movement

— thesearesomeof

themathematical

ideasinherentInthe

sculpturesof

HelamanFerguson.

Wehaveoftenheard

ofartistsusing

mathematicalideas

toenhancetheir

work.

Mathematician-artist

HelamanR.P.

Fergusonconveysthe

beautyof

mathematicsinhis

phenomenalsculptures.Ashestates

"Mathematicsisbothan

artformandascience...!

believeitisfeasibletocommunicatemathematicsalongaesthetic

channelstothegeneralaudience."1

Tocreatehisexquisiteforms,Helamanutilizesmethodsfromtraditionalsculpting,thecomputer,andmathematicalequations.

HisworksbearsuchnamesasWildSphere;KleinBottleWith

Cross-capAndVectorField,Torus,UmbilicalTorusWithVector

Field.,WhaledreamU(hornedsphere).

1IvarsPeterson.EquationsinStone,ScienceNewsVol.138September8, 1990.


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MATHEMATICS&ART73

LAYINGANECCMATHEMATICALLY

WhenRonaldDaleReschacceptedthecom

missiontodesignagiganticEasieregg

sculptureforVegreville,Alberta,he

soondiscoveredhewouldhaveto

developthemathematicsforthetask

virtuallyfromscratch.

OvertheyearsReschhasrefined

theartofmanipulating2-Dobjects

into3-Dforms.Hiswork andthe

problemshehassolvedpointto

mathematics,yethehashadlittle

formalmathematicaltraining.

Workingwithsheetsofvariousmaterials

suchaspaperoraluminum,he

transformsthemintoworksofartbyfolding

techniqueshehasdeveloped.Hesolvesgeometricproblemsusing

intuition,ingenuity,mathematics,thecomputerandhishands.

Hisinitialinstinctsaboutthedesignoftheeggwerethathecould

maketwoellipsoidsfortheendsandabulgingcylinderforthe

center.Hequicklyrealizedthiswouldnotwork.Discoveringthatavailablemathematicsfortheeggwaslimited1,herealizedhe

wouldhavetogoitalone.HisresultingdesignforthemagnificentEastereggrequired2,208identicalequilateraltriangulartilesand

524three-pointedstarstil

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