April2015

Atlanta,Georgia

EvidenceofModernPhysicalKnowledgefromAsiaticAntiquity

subtitled:Reintegration:NineRealmsofMiddleEarth

R.QuincyRobinson

Alumnus,GeorgiaInstituteofTechnology

TheCreativeknowsthegreatbeginnings.TheReceptivecompletesthefinishedthings.

TheCreativeknowsthroughtheeasy.TheReceptiveaccomplishesthroughthesimple.

Bymeansoftheeasyandthesimplewegraspthelawsofthewholeworld.

Whenthelawsofthewholeworldaregrasped,thereinliesperfection.

Chapter1,TheGreatTreatise

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IntroductionPrefaceHoTu,LoShu,&FivePhasetheoryQuantumofActionontheAshtapada8x8fieldThe5thPhase:PivotofEqualizationDivinationSymbolsandProbabilitiesontheAshtapada8x8fieldTetrahedralStructureofNumberDivinationSymbolsandProbabilitiesontheComplexUnitSphereHexagramSpectralValuesHexagramSpectralValuesintheContextofUnityQuantumofActionontheComplexUnitSphereSummaryAppendices

Imagesofthe4thDimensionGoogle+postBloggerpost

Errata

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IntroductionTheearlieststepsoftheauthoronthispathbeganearlyinhisundergraduateyears,whenhetookpossessionofamassmarketeditionofJamesLeggestranslationofYiJing.Aroundthatsametime,theauthorwasreadingfromanothermassmarketpaperback,TheDancingWuLiMasterswrittenbyGaryZukav.Thesebooksfascinatedandinteractedinthemindoftheauthorinsuchawaytosuggestthattherespectivesubjectsofbookswereindistinct.ThisistosaythatZukavs1979layrenderingofquantummechanicsandLegges1899translationoftheChineseclassicseemedtobediscussingtheverysamesubject. ThefirstworksthatemergedfromthisvisionofunitycamewhileinvestigatingTerenceMcKennasTimewavehypothesis.Thoseideaswerecommittedtopagearound2006,wherebytheauthorwasproposingtoexpandthesystemproposedbyMcKenna.ThateffortultimatelybetrayedtheWengroups,orwhatarenowtermedspectralnumbers.AlsoaroundthattimetheauthorviewedWhattheBleepDoWeKnow!?,apopularrenderingofquantummechanicalconceptsandtheirpossiblerelevancetothehumancondition.Thefilmwascriticizedonitsnonscientificorinaccurateattributionstoquantumtheorythataresimplynotsupportedbythetheory.Nonetheless,thedirectorscutofthevideowasinterestingandusefulbecauseitpresentedthefullinterviews,segmentsofwhicharepresentedinthefilm.ThecontentoftheseinterviewsvarywidelyfromBuddhistmonktophysicist,tochiropractor,tochannelermedium.Bythistime,theauthorwasinformallypenningideasintonotebooks.Thenextmajorpushoccurredaround2010,whentheauthorreceivedvisionsthatthecube,aPlatonicsolid,isfairrepresentationoftheTreeofLifeaspresentedinvariousmythosortraditions.Theauthorbeganproducingartwork,recordinghisresearch,andregularlyjournalingincompositionbooks.Theauthordiversifiedhisefforts,embracingthepracticeofyoga,producingjewelry,charms,mandalasandyantratogainclarityonhisvisionandtohelpshareitwithothers.Hebeganablogandpostedregularlyoverthenextseveralyears.Duringthistime,anotherpopularrenderingofquantummechanicscapturedtheauthorsattentionandimagination:QuantumRealitywrittenbyNickHerbert,memberoftheFundamentalFysiksGrouptowhichGaryZukavisadjunct.Herbertiswellgroundedinthescientificdisciplineofquantummechanics,andhisrendering,whileperhapslessaccessiblethanZukavs,helpedtofillthegapsintheauthorsunderstandingofmodernphysics.QuantumRealityandDancingWuLiMastersareregardedasexamplesofthebestwrittenlaytreatmentsofquantumphenomena.ThefinalmajorpushoccurredafterencounteringtheauthorofSumofThings,PaulMartynSmith,whowouldbecomecoauthorofDecomposition.Paulandtheauthormetonline.Aftersomedelay,theauthorcommittedtostudyPaulstreatmentofBookIIofYiJing.TheauthoradmitstohavingoncebeenadmonishedtomastertheMaterialbyDr.StephenKarcher,translatorofYiJingandTaChuaneditions(ownedbytheauthor).InSumofThings,PaulsynthesizestheHoTumapandLoShuwritingintoaconstructfromwhichemergeseveralwellknowncosmiccycles,includingtheprecessionalperiodof25,960years.PaulwaspatientinsharinghisknowledgeoftheHoTuandLoShumaps,personallyguidingtheauthorinhiseffortstolearntheMaterialwhichconstituteanancientphilosophyofmathematics.

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PrefaceThisworksetsouttoproveapriorithatYiJing,theChineseBookofChangesisavalidschemafordepictingquantumsystems.Thishypothesisisallbutassumedastruebymanylatterdayauthors,butnoexplicitprooforchainofreasoningjustifyingthisassumptionisknowntotheauthoratpresent.TheauthoracknowledgesthatYiJingisaslightlydiminishedformofIfa,adivinationsystemoriginatingontheAfricancontinent,asthelattersystemiscomprisedofeightbinarylines.TheauthoralsoacknowledgesthattheSumerianTabletofDestinies(DupShimati)hasbeenindicatedasapossiblereferencetoYiJing.Theformerpaper,"Decomposition"waspresentedinfinishedformstrictlyonthebasisofscriptureandmathematics.Manyofitsassertionsandsuggestionswereleftundefended,however,thusbluntingtheforceofitsarguments.Inthispaper,weexpandonthetopicslightlytreatedintheformerworkandbringthemtotheirnaturalconclusions.Asecondarygoalofthispaperistoinquirewhyanaccessibleexplicationhas,todate,notbeenmadepublicbytheoriginatorsoftheoriginalformofquantummechanics(Bohr,Pauli,vonNeumann,Heisenberg,Dirac,etal.)oranyoftheirfollowers.Reasonsforthismay(should)beexaminedinasociologicalcontext.ThepossiblereasonsareperhapsrelatedtothedegradationovertimeofthecomprehensionoftheunitythatrevealsthedivineAbsolute.Lessbenignreasonsarealsowithintherealmofpossibility.ThereseemstobesufficientcircumstantialevidenceminimallyestablishingthatasubsetofthefoundingfathersofquantummechanicsmadeYiJingtheirsubjectofinquiry. WolfgangPauliconsultedYiJingforinsighttothecontentofhisdreams.NielsBohr,aleadingpioneerinthescience,andforwhomascienceinstituteinCopenhagenwasnamed,tookashisfamilycrestthetaijituoryinyangsymboluponreceivinghisknighthood.TheinscriptiononthecrestreadsCONTRARIASUNTCOMPLEMENTA,orOppositesAreComplementary.Inthisway,BohrmadeclearhisunderstandingandtheoreticalrelianceontheprincipleofcomplementarityasconveyedbythesymbolwhichrepresentsChangeitself. Thequestionremainswhynobreadcrumbtrailwasleftfromvisionaryoriginstopublishedtheory.Onepossibilityisthatexpressionsofspiritualismorreligiosityarevastlydiscouragedinthescientificcommunity.Wearetemptedtoopinethatintentionalobscurationwasappliedthroughtheabstractformalization,whichmakesthetheoryintelligibleonlytoskilledmathematiciansandmathematicalphysicists.Theauthorhasspokentothisobscurationofcommonsensemeansofunderstandingunity(i.e.,theDivine)amountstotheftfromallmankind:

Naturalscienceisnotinherentlydifficulttounderstand.Arcaneformulationsofscientificprinciplesbysubjectmatterexpertsandacademicsmakethoseprinciplesdifficulttocomprehendbylaypeople,thuscreatinganartificialdeficitofscientificunderstandingamongthecommonpeoplewhomaythenbemanipulatedthroughtheirinducedignorance.

Modernscienceviewsdivinationasthepracticeofseekingknowledgeofthefutureortheunknownbysupernaturalmeans.Herein,weshalldemonstratethatdivinationasdevelopedbyChineseandAfricansfromantiquityoperatespurelybymathematicalmeansthatarefairlyequivalenttomodernphysics.Earlierversionsoftheideasofferedinthispaperappearinthebloglinkedhere andtheGoogle+postlinkedhere andpresentedhereinasan1 2appendix.

1http://sharqtank.blogspot.com/2012/07/ichingandblochpoincaresphere.html2https://plus.google.com/+QuincyRobinson_prime/posts/29kJ8tK5Dvw

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TetrahedralStructureofNumberWecasuallyalludedtothistopicintheintroduction,belyingitsimportancetotheoverallwork.Apartfromthe64hexagramfigures,thebestknownpartsofYiJingarelikelytobetheHoTumapandtheLoShuwriting(figuresfromWilhelmBaynesedition).Muchhasbeenwrittenontheseearliestofwrittenrecords.Together,theserelicsarerequiredreadingforunderstandingthemathematicsunderpinningYiJingspictureofwholeness. AtleftistheLoShuwriting,itdiscussesthecorrespondencesbetweentrigrams,cardinaldirections,WuXing(FivePhase)theoryanddecimaldigits.Observe

thatthearrangementofdigitsinthediagrampermitsthemtobeinterpretedasa3x3magicsquare.LoShuisactuallyatemplatefor3x3magicsquares.LoShu,magicsquareoforder3,magicnumber15,hascuriouslysubtlefeatures.Itmaybefracturedintofourpairofdigitssuchthateverypairofdigitsbearsaparticularrelationshiptoaninth,centralnumber.Notethesymbolindicating5atthecenterofbothdiagrams.ThepairsontheHoTushowthismoreclearly:Water(16),Metal(49),Fire(27),Wood(38).Thesepairsalldifferbyexactlyfive.HereweseeahintofthequantizationofactiondisplayedinMartynSmithsNineStackmodel.AnotherlookattheLoShudiagramshowsthesepairspositionedtogetheratthemiddleofeachsideandtheadjacentCCWcorner.Thesearetheobviousfeatures.Observehoweverthatwemaybreakoutthedigitsontoseparategridswhereweseethatopposingdigitsreveala10scomplementrelation,indicatingthatthepairssumtoten.Themeanofeachpairisfive,nowremovedfromitscentralpositiononthegrids.

Notealsothatthedigitsarenaturallyseparatedintooddandevenontheirrespectivegrids.Observethatthecolored3x3gridfairlyreproducesthegroupingsontheHoTumap(fouroddevenpairsaroundacentralmeanvalue)whilepreservingthe10scomplementrelationrevealedbytheLoShuwriting.Nowconsiderthefour10scomplementpairs:[(46),(37),(28),(19)],asthecentralvalue(x=5)andavariance[x:1,2,3,4].Thisexpressionexplicitlyreproducesthefour10scomplementpairsintermsofthecentralandaveragevalue,five.TheHoTualsorevealsthisfeatureitsdigitsarepresentedintwogroups:aninnergroup[1,2,3,4]andanoutergroup[6,7,8,9]withfiveatthemean(central)position. Thefigureatright,thefoursimplexor5cellisageometricrepresentationofthestructurehintedbyHoTuandLoShuitisaregulartetrahedronwithacentralnode.Observethatthecornernodescorrespondtoinnernumbers.Projectionofaninnernumberthroughthecentralvalue(5)ontoafaceproduceseachcorrespondingouternumberassumoftheremainingthreevertices.Theseouternumbersaresameasthesymbolicvaluescorrespondingtothefour(xiang)symbolsproducedbytheyarrowstalkoracle.

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Divination,SymbolsandProbabilitiesontheAshtapada8x8field 3

Xiangmeanssymbolfoursymbolsareusedtoperformthedivinationritual.Chinesefolkloregivesusthefollowingassociations[ofthesymbols]tothefoursacredguardiananimals:

AzureDragon,East,Spring VermilionBird,BirdSouth,Summer WhiteTiger,West,Autumn, BlackTurtle,North,Winter

Projectedontothe8x8field,thexiangfieldcomprises4concentricbands(4,12,20,28)eachbandbearingaproportiontothewholeidenticaltotheprobabilityofitscorrespondingsymbolbeingselectedwiththeyarrowstalkoracle.Thesymbolicdivinationritualvaluesandtheirprobabilityfrequenciesaregivenas:

stableyin[8],showninred:28/64=0.4375stableyang[7],ingreen:20/64=0.3125dynamicyang[9],inblue:12/64=0.1875dynamicyin[6],showningold:4/64=0.0625*notetheprobabilitiesarereduciblebyafactorof4TaChuangivesusthe"numberofallthings":11520,commonlyreferencedasthemyriad(tenthousand)thingsasfollows:

Changeencompasses64hexagrams,eachcomprising6lines. 11520dividedinto64partsyieldstheaverageof180perhexagram.180dividedbythesixplacesgivesanaverageof30perline. Alternatively,thetotalnumberoflinesinChangeis384,equalto64hexagrams. Eachhexagramhas6lines. 11520dividedinto384partsalsogives30perline. Wededuce,reasonably,thattheaveragevalueofalineis30.Torecap,eachbandofprobabilityfrequencycorrespondstoaspecificxiangsymbol.Eachsymbolcorrespondstoaparticulardivinationritualvalue[6,7,8,9].Thesesymbolsvaluessumto30(=6+7+8+9).Eachlineproducedthroughtheoraclecanbeconsideredascontainingallfoursymbolsinpotentia.

3ThissectionpreviouslypublishedtoGoogle+

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Sinceanyproperquadrantofthe8x8fieldisfractalofthewhole,wecanarraythesymbolsona4x4field(onequarterthesizeofthe8x8field)tosimplifyandreducethenumberofmanipulableelementsThen,tomaintainproportionalityrelativetothe8x8field,wescalethesymbolicvaluesby4.

dynamicyin[32],denotedinred:dynamicyang[28],denotedingreen:stableyang[36],denotedinblue:stableyin[24],denotedingold:

Computingtheweightofthefield,bymultiplyingthecorrespondingsymbolicvaluebyitsprobabilityfrequencyonthefield,wecalculate:(24*1)+(36*3)+(28*5)+(32*7)=496. Recallthattheaveragevalueofanabstract,unmeasuredlineis30thencewemightexpectatotalweightof480(=30*16)forthefield.Aswehaveshown,however,theweightedvalueofthe4x4fieldcomputesto496whichfactorsinto31*16,implyinganaveragevalueof31overthe4x4field.Thedifference,(496480=16)isexactly1/30th(onethirtiethpart)ofthemeanvalueof480.Theeffectofthemultiplyingthefieldsexpressionofyarrowstalkoracleprobabilityfrequencieswiththexiangsymbolicvaluesistoraisethemeanvalueofalinebyunity(from30to31).Bearinmindthatthearrangementofthesymbolsonthebandsofthesquarepreciselyreplicatestheprobabilityfrequenciesusedtoderivehexagramlines.Inthisway,wewitnesstheunityofformandfunction.TheformallanguageofChangeembodiesitssacredcharacter.ArthistorianTitusBurckhardtcitesthisqualityasatraitessentialtosacredart:

Anartcannotbeproperlycalledsacredsolelyonthegroundsthatitssubjectsoriginateinaspiritualtruthitsformallanguagemustalsobearwitnesstoasimilarorigin.[...]Noartmeritsthatepithetunlessitsformsthemselvesreflectthespiritualvisioncharacteristicofaparticularreligion.Everyformisthevehicleofagivenqualityofbeing.[...]thereexistsnosacredworkofartwhichisprofaneinform,forthereisarigorousanalogybetweenformandspirit.Aspiritualvisionnecessarilyfindsitsexpressioninaparticularformallanguage.Everysacredartisthereforefoundedonascienceofforms,orinotherwords,onthesymbolisminherentinforms.Itmustbeborneinmindthatasymbolisnotmerelyaconventionalsign.Itmanifestsitsarchetypebyvirtueofadefiniteontologicallaw...asymbolis,inacertainsense,thattowhichitgivesexpression.FoundationsofOrientalArt&Symbolism,ch.1:IntroductiontoHindu,Buddhist,andTaoistArt

Thesignificanceofthefoursymbols(xiang)inthedivinationiseasilyoverlookedinthetraditionalpresentationoftheoracleritual,whichservestoobscuretherelationshipbetweenthesymbols,lines,figures,andthedivinationritual.Consequently,Yijingasitiscommonlypresented,appearsmoreasanarbitrarycollectionoffeatures:yarrowstalks,lines,trigrams,andhexagrams,ratherthananholism.Whenviewedproperly,itssacredcharacterofwholenessismorereadilyobserved.

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EvaluatingtheFieldElementsTheYellowRiverbroughtforthamapandtheLoRiverbroughtforthawritingtheholymentooktheseasmodels. TaChuan,pt.I,ch.11,sec.8

TheHoTumap(fig.4)demonstrateshowthedecimalcountingsystemcanbeexpressedwithintheWuXingsystemoffivephases.ThesePhasesappearasnumberpairssuchthateachpairdiffersfromitsmatebyexactly5(five).ThisestablishesabasisforquantifyingandquantizingthesystemofChange.TheWuXing,(wxng)alsoknownastheFiveElements,FivePhases,theFiveAgents,theFiveMovements,FiveProcesses,andtheFiveSteps/Stages,isafivefoldconceptualschemethatmanytraditionalChinesefieldsusedtoexplainawidearrayofphenomena(Wikipedia)TwobasicconditionsaredefinedinChange,theDARKandtheLIGHT,or

YINandYANG.TheyarecalledLiangYiortheGreatExtremes.Wearegiventhattheseconditionshavelimits,beyondwhichtheconditionabruptlytransformsintoitscomplementaryopposite.ThistendencyforabrupttransformationcapturestheessenceofchangewithinthesystemofChange.Thesystemcomprises384fundamentalLINEentitiesthatmaketakeeitheroftwodiscreteconditions:DARKorLIGHT.Alinesconditionmayalsobestatic(fixedacrossaduration)ordynamic(varyingacrossaduration)thisaccountsforfourallowablestatesforeachlineinthesystem.LinesappearinorderedtriadscalledTRIGRAMS,historicallyknownasBaGua,or8diagrams.ConsideringTRIGRAMS,wearegiventhatthebottomlineisfirstandeldestposition,toplineisthirdandyoungestposition.ThePrimalsequenceofBaGuafollows:

Wearegiventhatthefundamentalactionoccurringwithinthesystem(ofChange)involvesoneormoreLINESalternatingstatesbetweenDARKandLIGHTandthatthisactionoccursacrossalltheFivePhasesofChange.HoTushowsustheFivePhasesaregivenhereaspairsofDARKandLIGHT:HoTudefinestheintegralmagnitudecorrespondingtothealternationbetweenthevaluesoftheGreatExtremesofDARKandLIGHT(LiangYi)as5Changeunits,irrespectiveofPhase.ThisfactestablishesLINESasthequanta(smallestparts)ofthesystemofChange,andtheircorrespondingChange(Effect)as5.Wilhelm,inhiscommentaryonTaChuan,pt.I,ch.1,sec.8,sharesthat'Yi'hasafourthmeaning:achangeofthesmallestparts.ThisconstantdifferencebetweenDARKandLIGHTplaysoutuponthefieldofactionsuggestingaquantumofeffectonthefield,asthisdifferenceconstitutesthesmallesteventthatmayoccurwithinthesystemofChangenamely,thealternationbetweenDARKandLIGHT.

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EnumeratingtheTrigramSequences

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Asdescribed,aTRIGRAMisanorderedsetofthreeLINES.Theorderingdefinesapositionalnotationsimilartothatabase10numbere.g.(4*100)+(2*10)+(9*1)=429wheretherightmostdecimaldigitistheleastsignificant.IntheNineStackmodelwhichusestheinvertedPrimalsequence,theleastsignificantpositionofaTRIGRAMisthebottomLINEmostsignificantisthetopLINE.Thus,whenaLINEvaluechanges,itsrankintheTRIGRAMmustbeconsideredwhencalculatingaTRIGRAMSvalueordelta.Thesignificanceofpositionistreatedbythefollowingtable.InthefirstWaterRealmoftheNineStack,DARKhasvalue6whileLIGHThasvalue1theirdifferenceis5.So,whenthe1stLINEinagivenTRIGRAMchangesfromDARKtoLIGHTintheWaterPhase,itsvaluechangesfrom6to1,anetloss/expenditureof5unitsofpotential.IfthatlinechangesbacktoDARK,itgains5unitsofpotential.However,ifthe2ndLINEofaTRIGRAMundergoesthesametransformation,thenetloss/gainis(2*5=)10actionpoints,owingtotheseniorityofthatLINE.Likewise,iftheeldestlineundergoesthesamealternation,itschangeinpotentialis(3*5=)15asitistheeldestLINE.Thisruleholdsacrossall5Phasesthevaluesmaydiffer,butthechangeinpotentialiscalculatedinthesameway.WhenthevariousconfigurationsofTRIGRAMSarecharacterizedandassignedvalues,sevendiscretenumericvaluesareobtained.AlthoughthereareeightTRIGRAMS,ZHENandXUNarefoundtohaveidenticalvaluesinanotherwisediscreteset.

Ifinstead,onechoosesthetoporyoungestlinetobemostsignificant,thealternativePrimalsequenceensues,whichissimplythefigurateinversionoftheprioreachguanowstandsonitshead.Here,TRIGRAMSGENandDUIexhibitequality.Neitherthevaluesnortheirorderarealteredinanywaybygloballyinvertingthesymbolsusedtodepictchange.Hereweareremindedthatorientation(aglobalfactor)hasnoobservableeffectontheobject.

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QuantumofActionontheAshtapada8x8fieldTRIGRAMSarenormallyencounteredinasuperposition(gua,stack)calledanHEXAGRAMitstwocomponentTRIGRAMSarereferencedasinnerorlowerTRIGRAM,andouterorupperTRIGRAM.AnHEXAGRAM,thereforecomprisestwoTRIGRAMSandsixLINES. TRIGRAMSaretreatedasnumbersexpressedintermsofthreebinarydigits.TRIGRAMsumsarewrittenasapairofstackedorsuperposedTRIGRAMS,justasintegeradditioncanbewrittenasastackofdecimaldigitsaccompaniedbythe(+)plussymbol.Below,fieldcellspresentsumsofthevaluesofthestackedTRIGRAMSforeachofthe64combinations:

FigureTheFirstRealmofWaterfromtheNineStackMountainmodel

1)WaterRealmsdefineLIGHTLINESas1andDARKLINESas6,showninsetabovebytheredborderedtable2)EachlinehasacoefficientcorrespondingtoitspositionintheTRIGRAM3)EachlineofaTRIGRAMmayberegardedasabinarydigitTherowofsymbolsabovethefieldcorrespondstotheouterTRIGRAM. ThecolumnofsymbolsatleftofthefieldcorrespondstotheinnerTRIGRAM. TofillthetableofHEXAGRAMS,eachTRIGRAMiscomputedavalueaccordingtotheREALMandPHASE.Tocalculateacell,simplyselecttheouterandinnerTRIGRAMSfromtherespectivecolumnandrow,thenaddtheirrespectivevaluesasshown.ThecompletedfieldabovepresentsallthepermutationsofsummedTRIGRAMpairsfromthefirstRealmofWater.CellscontainsvaluescorrespondingtoTRIGRAMsuperpositionsthataretreatedassums.CellsinthesamecolumnshareacommonouterTRIGRAM.CellsinthesamerowshareacommoninnerTRIGRAM.Thusalladjacentcellssharearoworcolumn,indicatingtheyshareouterorinnerTRIGRAM.Adjacentcellsmaydifferatmostby1,2,or3LINESperTRIGRAM.Thecoefficientofthetoppositionaboveis3,ofthemiddlepositionis2,andthebottompositionis1.Thisordering(impliedbytheredborderedtableinsetabove)producesaninversionofthePrimalsequence,describedearlier.Coefficientspreservetherelativesignificanceofthethreepositions.Consequently,DARKat3rdisgreaterthanDARKat2ndorDARKat1stlikewiseforLIGHTacrossthethreedigits.Movingortransitioningbetweenadjacentcellsinvolvesproperlyaccountingthetransitionsinthosepositionswherechangeisobservedtooccur.

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Figure:FirstRealmofWaterfromtheNineStackMountainmodelForexample,movingfrom12atupperleft,tothe17toitsimmediaterightentailsadifferenceofasingleLINEatthebottompositionoftheouterTRIGRAM.TheinnerTRIGRAMSoccupythesamerow,hencethesame,nochange.Pertheredborderedtableinthefigureabove,alternationinthebottomplacefromLIGHT(1)toDARK(6)entailsadifferenceof+5units.

Asubsequenttransitiononthefieldinthesamefashion,from17to22,entailstwolinechanges:bottomandmiddle.ThealternationfromDARKtoLIGHTinthebottomplaceaccounts5units,whilethealternationfromLIGHTtoDARKinthemiddleplaceaccounts+10.Thesetwoalternationsareoppositeinnature,thesignsareopposite,thenetdifferenceis+5units.

Athirdexampledescribesatransitionfrom27to27onthetoprowofthefield,wheretheouterTRIGRAMdiffersatallthreepositions:+5atthebottomposition,+10atthemiddleposition,and15atthetoppositionforanetdifferenceof0units,astobeexpectedsince2727=0.Aswefollowvaluesacrossadjacentcellsofthefield,weagainobserveaconstantdifferenceof5betweenvaluesinadjacentcells.Theexceptionstothisrulearerestrictedtothecentralcolumnsandrowswherethedifferenceis0becausethemiddleTRIGRAMSofthesequencehavethesamevalueasdiscussed.IfweinterprettheeightTRIGRAMvaluesassimplescalarsandorderthem,weproduceakindofbasisvector,sincewehavebothmagnitudeanddirection.Ifwethenarrayourbasisvectorsatrightangles,theresultisanorderedfieldcomprisingsquaredunits.ThissuggeststousthatourTRIGRAMsuperpositionsmayinvolvemultiplicationalso/insteadofsummationofTRIGRAMS.TheWheel,presentedbelow,isanalternativedepictionofthefieldwehavejustimagined,allowsforthisinterpretation.

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The5thPhase:PivotofEqualizationInseemingdeparturefromthefieldsoftheNineStackMountain,thePrimalandTemporalTRIGRAMsequencesaredepictedhereonthecircleasacompositionofperiodiccircularandstepwiselinearcycles.Theprominentfeaturesarethecircularfield,thesequences,thefourquadrants,andthesilvercross.

ThesilvercrossitselfrepresentsMiddleEarth,acolloquialismforthe"fifthstageofChange"astermedinWilhelm(1950),or5thTransformativeMomentperRitsemaandKarcher(1994).Coincidentally,theastrologicalsigilforEarth()isacrosswithinacircle. TheangularspanofthissilvercrossontheWheelisidenticaltothatofthecoloredquarters.Thiscrossrepresentingthe5thPhaseisitselfdividedintoquarters,whichsimultaneouslypartitiontheWheelintofourthsandfifths.

[OntheFiveTransformativeMoments(wuxing)]:"Theseprocessesarenotsubstancesbutstagesoftransformation." Ritsema&Karcher(1994),p.72[The5thPhaseisdenoted]t'uortheEarthymoment:soil,dust,clay,ashes.Itisthepointaroundwhichtheseasons,thecardinalpoints,andthealternationofdayandnight[areheldto]revolve.ItrepresentsthePivotofEqualizationwhereDARKandLIGHT(structionandaction)areinbalance,neutralizingeachotherorcreativelyinteracting.Itsactionis*bringingforth*.Itmanifestsassowingandhoardingacrop.Theearthymomentconnectsthehemicycles.Itisthemomentofbalanceandtransition."(ibid),p.73

InthePrimalsequence,the5thPhaseissetoffbyTRIGRAMS4&5. IntheTemporalsequence,the5thPhaseissetoffby1&9.Theangulardifferencebetweenadjacentpositionsuponthe5thphaseisequalforbothsequences.ThisreflectsthePivotofEqualizationmentionedabove.Suchapivotsuggestspointsofinterchange,orintersectionswherethetwocyclesareequipotent,suchasrootsorzerocrossings.ObservethatthispivotoccursatthemidpointofthePrimalcycleandbetweensuccessiveTemporalcycles.ThisobservationrelatestheWheeltothesolsticesandequinoxesofthesolarcycle.

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DecomposingtheWheelintoquadrantsfourPrimalsequenceseachof =2 90producesthehemicycles.WesawaboveintheNineStackmodel,thedifferenceorpotentialbetweenthe4thand5thpositionsofthePrimalsequenceis0(zero)aswasseenintheFirstRealmofWater.ThisconditionobtainsontheWheelat =45and4

itsreflections.4thand5thpositionsofthePrimalsequenceareeffectivelyidentical,justasare1stand9thintheTemporalsequence.Eulersidentityinformsusthatspiral(helical)motioncanbemodeledbytwocycleswithphaseoffset =2

90.ItfollowsthatwemayuseTRIGRAMStomodelcosineandsinetodescribehelicalmotionintermsofcomplexnumbers.decomposethePrimalsequenceintotwosubcycleseachwithperiod =90.Oneis2

describedastheDARK(real,cosine)hemicycle,theotherastheLIGHT(imaginary,sine)hemicycle.

Orthogonal(90)outofplanerotationoftheCOSINEphaseeffectivelysynchronizesitwiththeSINEphasebyreplacingtheoffsetintimewithanangularoffsetinspace.ThisdecompositionofthePrimalcycleisimaginedastwocongruentverticallyorientedrings(cycles)intersectingat90,imitatingthesineandcosineexceptthatsineandcosine,normallythoughtasoffsetintimeby90arebelowoffsetinspaceby =90angularmeasure.Theredringoccupiesthexzplane,2

blueringintheyzplane.Thesetwointersectingringsarethenintersectedatfourpointswithagreen,equatorialringcharacterizedbystepwiselinearmotion,namelytheTemporalsequence,havingperiod =90.TheTemporalsequenceis2

effectivelyprojectedontoacircumferenceinthexy(real)plane,.Systemicevolutioncannowbedescribedinintermsofangulardisplacementoverarotationalcycle.WenotethisdepictionbeginstoresemblethequbitmodelsofBlochandPoincar.Torecap,theTRIGRAMSwerefirstmappedontothecomplexplane(Wheel)insuchawaytoresembleapairofcoordinateaxes.Weidentifiedfourrootsattheintersectionoftheunitcirclewithy=x.ThecomplexdomainpermitsustorepresentTRIGRAMSasvectorsemanatingfromtheorigin.Suchvectorsaredefinedinpolarnotationbytheirmagnitude(=unity)andtheirCCWangulardisplacementfromthepositivexaxis.Productsofcomplexnumbersarecomputedbyaddingtheangulardisplacementsandmultiplyingthemagnitudes(=unity).

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TheComplexUnitSphere

Poincarssphere(left)definesapointPbytwoparameters,anglesand.ThevectordenotesthedirectionanddisplacementfromtheorigintothepointPonthesphericalshell.Thissphericalsurfaceisatwodimensionalcomplexspace.ItisconceptuallysimilartotheBlochspherewhichisageometricalrepresentationofthepurestatespaceofatwolevelquantummechanicalsystem(qubit).Thus,definedpointPdenotesaparticularstate(psi)ofthequantumsystemorqubit.

Ingeneral,atangentmeasuresarateofchange.Itiscommonlydefinedasrisedividedbyruninplanetrigonometry.Inthisway,itisroughlyequaltoadefinedslope.SincethePoincarsphereisaunitsphere(radius1),acentralanglemeasuredinradiansfromareferenceaxismaybedefinedasaslope,whichisfairlyequivalenttoatangent.Bothangleandtangentareequaltothemagnitudeofthearclengthsubtendedbytheangle.Thatis,=TAN=s.Thisequivalenceallowsustoredefinetheparametersofthequbitstateintermsofslopesratherthanangles.Inaslightlydifferentcontext,atangentismeasureofthederivativeofafunctionatthemeasuredpointPwheretheslopeintersectsthefunction.TheslopeasmeasuredfromtheoriginisalwaysreciprocaltothetangenttothefunctionatthemeasuredpointP.Whyslopes?Slopesarefundamentaltonaturalgeometry.Apersonmaynotknowtomeasurearbitrarydegrees,orradians,oranyotherangularunit.Thepicture(atright)ofChinesefolkheroesFuxiandNuwabearingthecompassandsquaredatestheuseofthoseimplementsto~3000yearsBCE.Everymancandeterminethemagnitudeoftheresultingarclengthorangleorslopefromtheconstructedtangent,whenitisdefinedinradiansupontheunitcircle/sphere.

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Regardingtheunitsphere,wehaveassertedthatCCWangulardisplacementfromafixedreferenceaxis(positivexaxis)servesasoneoftheslopesrequiredtodefinepointPonthesphericalsurface.Theotherslopeisgivenasinclinationordeclinationfromthexyplane(realplane),andissignednegativetoindicatedeclination.Underacertaininterpretation,isarunaroundthecircumferenceoftheunitsphere,whileisthecorrespondingriseabovethecircumferentialplane.Posingthisasaratio offersakindof complextangent.Withthispairofslopes(orthepointswithwhichtheyintersectatthesurface,weareequippedtomodelaqubitstate. TheBlochsphereisdefinedfor0

DivinationSymbolsandProbabilitiesontheComplexUnitSphereTheyarrowstalkcountsproducedduringdivinationare[24,28,32,36]thesestalkcountsarescaledbytoproducetheritualnumbersareusedtoconstructtheresultinghexagramfigure.Theritualnumbersthatcorrespondtothesestalkcountsaregivenrespectivelyas[6,9,7,8].Aswehavejustdemonstrated,theprobabilityfieldoftheyarroworacleisaunitycomprisedofthefollowing4

fourratios[ , , , ].Theseratioscompriseasetoffrequenciesthat116 316 516 716 depicttherespectiveprobabilitiesofthefourpossibledivinationoutcomes.Thefull8x8fieldmaybeinterpretedascomprisingfourdiscretebandsofprobabilityfrequencies.Thereducedfield,termedtheFieldofFourSymbols(FoFS,atright),expressesthesameyarrowprobabilityfrequenciesasexceptatscale.

Weobservethatthefourcharacterizedpoints[ , , , ]bearresemblancetotheprobabilities41 43 45 47 frequenciesoftheyarroworacleonFoFS.Wedefendourobservationasfollows:

Thefourpointsnormalizedbymultiplyingby =1torepresentthemincomparabletermstothe44 FoFS(scalefield,atupperright).Weobtain[ , , , ]fromthisoperation.The164 1612 1620 1628 numeratorsofthepointsnowcorrespondtotheratiosontheFoFS.However,thesumofthepointsisnow4wehavelosttheunitypreviouslyexhibitedonthescalefield.

Weassertthatthepointsofthesurfaceoftheunitsphererequirescalingbyonequarter justasis5requiredbytheyarrowstalkoracletomaintaintheoriginalunityemblemizedbythe8x8field.Thisdivisionbyfourgives[ , , , ]which,whensummed,gives =.644 6412 6420 6428 6464

Providedthisproposedequivalenceormappingofpoints/tangentsontheunitspheretooracleprobabilityfrequenciesismathematicallysound,weareledtoinferthatinthecontextofthecomplexunitsphere,atleastthegroundoftheyarroworacleprobabilityfieldisrelatedto,thehalfcircle.Thatis,itselfisrelatedtothefundamentalqualityfromwhichemergethesymbolicquantities[6,9,7,8]thatareproducedwhenaquerentdivinesanhexagramfigurefromtheyarrowstalkoracle.

Theoriginaltwotrigramseachaccount onaverage.Infact,thereexactlyexisttwentytrigramsthatfully2 accountthehalfcircle(shownbelow).Theyarroworaclegivesusthat,onaverage,ofthelinesitreturnswillhaveachangingcharacter.Theremainingofdivinedlineswillexhibitastaticcharacter.Weknowthisbecausetheyarroworacleratioscorrespondingtochanginglinessumtoofthefield.Whenthequerentdivinesoneormorechanginglinesusingtheoracle,additionalquadrantsarecharacterizeduponthecircle.Thus,yarrowdivinationproducesaresultthat:

minimallycharacterizesthehalfcirclewhennochanginglinesarethrown mediallycharacterizesthreeofthefourquadrantswhenchanginglinesappearinexactlyonetrigram maximallycharacterizesfourquadrants,whenthequerentreceiveschanginglinesinbothtrigrams.

4https://plus.google.com/u/0/+QuincyRobinson_prime/posts/1TMTZyFjYwm5Instructionstotheyarroworaclespecifycountingtheyarrowstalksintobundlesoffour.Thisrequirementemergesthroughtheinteractionoftheshapeorformortheyarroworaclei.e.501=49=7x7stalksandthesetofrequisiteoutcomes[y:24,28,32,36].Yarrowstalkoraclerequiresthatwedividethenumberofcountedstalksbyfourtoreceivethecorrespondingsymbolicvalue[x:6,9,7,8].Dividingpointsbyfourisassertedasequivalenttocountingstalksbyfours.q.v.:http://sharqtank.blogspot.com/2011/07/ichingritualdivinationnumbers.html

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QuantumofActionontheComplexUnitSphereThediagramabovecontextualizesthehexagramspectralvaluesintermsofunity(theAbsolute),depictedaboveasapolarviewoftheupperhemisphereofthecomplexunitsphere,wherethefourpointsarelocated.Weobservedthatthehexagramspectralvaluesdefinearangeof or72andthatthespectrum52 valuesofhexagrams#1and#2,calledtheGatesofChangeare216and144whichequal and3 * [ 52]

respectively.Theirsumis2or360,asrequiredbythetext.2 * [ 52] 72 78 84 90 90 96 102 108

72 78 84 90 90 96 102 108

144 156 168 180 180 192 204 216

Havingdeducedtheangularidentityoftheeighthexagramsfigurescomposedofdoubledtrigrams,wemaynowdeducetheangularidentitiesoftheeightTRIGRAMSorBaGua.Weknowthattheaveragevalueofatrigramis andthattwotrigramsdefineanhexagram.WeknowfromtheGreatTreatisethat2 theaveragehexagrammeasures,andthatapairofhexagramscharacterizeuptofourquadrantsontheunitcircle.Bearinginmindthatlinepositionissignificant,thepresumedlinevaluesarescaledbytheirlinepositionseniorityinthetrigram,indicatedintheleftmostcolumnofthetablefollowing:

1 12 18 12 18 12 18 12 18

2 24 24 36 36 24 24 36 36

3 36 36 36 36 54 54 54 54

72 78 84 90 90 96 102 108

FinallyweareabletorevealtheangularvaluesoftheLiangYi,thequantaofactionofthesystemofChangeontheunitsphere.TheLIGHTandDARKrepresentthesmallestobservablepartsofChange.Inthepriorpaperonthistopic,welikenedthe8x8fieldsoftheNineStackmodeltofieldsofaction,whereLINESalternatebetweenLIGHTandDARKthroughtheexchangeofactionquanta.Theseactonquantaareshowntoaccountfiveunitsonthe8x8fields.Wealsosuggestedthatthesystemcouldbeextendedtothecomplexdomain. Here,weconcludethatanactionquantaisproportionalto or6ofangulardisplacementscaledbythe30 traditionalvaluesoftheLiangYi(LIGHT=3,DARK=2).Inotherwords, or6,multipliedbysome30 constant,isthecorrespondingquantumofactiononthecomplexunitsphere.Hereweshouldmentionthattheunitsofactionareidenticaltotheunitsofangularmomentum,whichisequaltotheproductofmomentofinertiawithangularvelocity.ThissuggeststhatLiangYispecifythegivenangularmomentum,dividedbytheirrespectiveangularmassesgivenabove

LIGHT 18=3*6= 10

DARK 12=2*6= 15

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HexagramSpectralValuesRecallthatsurfaceoftheunitsphere isatwodimensionalcomplexspace,andthatpointsonthissurface6arerepresentationallyequivalenttoqubitstates.Thus,qubitsstatesarepointsonthesurfaceoftheunitspherethatarerepresentablewithcomplexnumbersandmanipulablewithcomplexoperations.Herewehavedemonstratedameansofdepictingtrigrampairs(hexagramfigures)onthesurfaceofaunitsphere,thusfunctionallyequatingtheresultofdivinationthatis,anhexagramtoaqubitstate.Inourschema,thetwoparametersaretrigramsrepresentedbytheinplaneslopefromthereferenceaxismeasuringbetween0and2,andaslopeoutofplanemeasuredfromthethereferenceplane,between0and .2 TheGreatTreatisegivesusthattheabsolutelimitofchangeis11,520thisisexpressedas32complementarypairsofhexagrams,eachpairhavingamaximalsumof360(=11520/32).Hex#1(Heaven)hasvalue216,andhex#2(Earth)hasvalue144.ThesetwofiguresarecalledtheGatesofChange.Theyexhibitthemaximaldifferencebetweenhexagrams(qubitstates)anddefinethemeasurementofChange.Whenthemathsforproducingthegivenvaluesforhexagrams#1and#2areappliedoverthefullsetofhexagramfigures,weobservethatthe64hexagramfiguresoccupytheheptadofvaluesbelowleft:

Inthediagramaboveatleft,weseethevalues216and144,givenintheGreatTreatiseforhexagram#1Heavenand#2Earth.Theotherfivevaluesweredeterminedfromtheremaining62hexagramfigures.Thesmallinsetatupperrightillustrateshowthesefoursymbolicprobabilitiesmanifestoverthesetof64discretestates,alsohowtheyaffinetotheashtapada8x8fieldfromwhicharedeterminedtheirrespectiveprobabilityfrequencies.Likewise,wehaveidentifiedfourpointsontheupperhalfoftheunitspherewithcoordinatesthatresemblethefourprobabilitiesofthexiangdivinationsymbols.

6http://en.wikipedia.org/wiki/Complex_plane#Stereographic_projections

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HexagramSpectralValuesintheContextofUnityObservebelowhowthefoursymbolsemergeprobabilisticallyfromthe8x8fieldthroughtheactofdivination,howsixsymbolsformatrigrampair(hexagram)withaparticularspectralvalue,andhowthesespectralvaluesfindangularcontextwithinFivePhase(WuXing)theoryaspresentedonthecircle.Thisdiagramalsoillustratesapolarviewofthecomplexunitsphere.

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SummaryWehaveidentifiedeight(nominallyfour)characteristicpointsontheunitspherecorrespondingtotheoctantsofEuclideanspace.Weassertedthatthefourpointsresembletheprobabilityfrequenciesgivenbytheyarroworacle.Todefendourassertionwenormalizedthepointstosixteenthsbymultiplyingby1= 4

4 sothatthepointsarecomparabletothescalefield(FoFS).Wetreatedthepointsonthesphereinthesamewayastheyarrowstalkcountsaretreatedinthedivinationritualtowit,byscalingtheyarrowstalkcountsorpointsonthesphereby.Onthisbasis,welikenedoureightpointstotheBaGua.Wedemonstratedhowthedivinationritualsymbolicvaluesemergedirectlyfromthetetrahedralstructureofnumber,towhichthenumber5iscentral. Wepresentedthespectralvaluesofthe64hexagramfigures.WedeterminedvaluesfortheeightBaGua,whichhavearangeof andaveragevalueof .5 2 WedeterminedthevaluesoftheLiangYi,thesmallestpartsofthesystemsofChange.Weconfirmedthattheunityexhibitedbytheyarrowprobabilityfrequenciesisidenticallydemonstratedbythesummationoffourpointscharacterizedonthehalfsphere.Theunitydemonstratedbythefourpointsinterpretedasasumofprobabilityfrequenciesisthusbetrayedasrelatedto,theratioofcircumferencetodiameter.Inthiscontext,appearstoberelatedtothesourceorgroundoftheprobabilityfrequenciesgivenbytheyarroworacle. Wedemonstratehowthe8x8fieldinterpretedasafieldofaction(referencedinthepriorpaper)isextensibletothefrequencydomainofthecomplexunitspherewhereprobabilityfindsitsnaturalexpression. Weofferavalueforactionquantaonthecomplexunitsphere,andaninterpretationofthisactionasangularmomentum,ortheproductofangularvelocitywithmomentofinertia.Theashtapada8x8fieldoritscomplement,thecomplexunitsphere,arethereforeofferedasanalternatepresentationordepictionof.Thesixbinarylinesofanhexagramprovideresolutiononthesquareorspheretothe64thpart.Thehalfcirclerepresentsthetwotrigramsofthethrownhexagramfiguretheotherhalfcirclerepresentstrigramsresultingfromanychanginglinesthewerethrownduringtheritual.Whenhexagramsexhibitchanginglinesinallsixplaces,asinthecaseofcomplementarypairsHeavenandEarth,theGatesofChange,thisaccountsonefullrevolutionofthecircle.Thecomplexunitsphereconvenientlydepictsthefrequencydomain.EachYiJingdivinationresulthasasmanyas4100(=64*64)possibleoutcomes.Aparticularoutcomecannotbeknowninadvancebutwillappearwithaknownandmeasurablestatisticalfrequency,asdefinedbytheashtapada8x8field.Repeated,significant,orprolongeddeviationsfromtheknownmeanofindicatethatthedivinationritualiseffectivelymeasuringastatisticallyrarestateofthesubject,andsuggestthatyarroworacledivinationactsasaquantummeasuringdevice.BydemonstratingthattheritualvaluesandprobabilityfrequenciesarefoundwithinthesystemofChange,thispapercontinuesontheheelsofthepriorinitsassertionthatthesystemofChangeiscompleteandselfcontained.

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Appendices

Imagesthatconjurethe4thDimensionfromavarietyoftraditions.

SphericalDepictionof64Hexagrams

NavagrahaYantra

Rabiflopping

CubicModelofSubspace

NordicTreeofLife

FoursimplexPolychoron

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Google+postQuincyRobinson()SharedprivatelyOct16,2014ThesisThesphericalmodelofChangedepictsthehexagram(vector)ineachofits64purestates.Thevectorarisesfromthe8x8field(ofaction),implyingitsgroundis2Dspace. Thissquare2Dspacemaybemodeledasasphericalshell(Bloch/Poincaresphere)whichbindsthegroundandmakesallpointsequidistantfromcenterremainstwodimensional,implyingNOtranslationalmovement,noexpansion/contraction.Thisrepresentationdoeserectthe"frame"ofavolumetricspace,butallowsnomovementalonganyoftheaxes,thusnomeasurement. Imagineaneyepositionedattheoriginofthexyzgraph.Eachofthe32axesrepresentsaspatialorientationordirection.Theeyecanlookalongalltheaxes.ThissuggestsanAwarenesswhosefieldisthesphericalsurface.HypothesisReformingthe8x8intoa4x4x43DspaceisaccomplishedthroughtheYarrowOracle,whichcollapsesthe8x8into4identicalquadrantsoffsetbypi/2.The[probability]metricofthisspaceisidenticaltothe[probability]metricofthe8x8,sincetheOracleproducesthesamevalues[6789]onthe4x4asitdoesonthe8x8,hencethistransformationseemspermissible.Magicsquaresare2Dregionshavingmetricsgoverningtheirexistence(dimension,meanvalue,magicnumber,etc).ApplicationoftheHoTuRiverdiagramtotheLordSun(LoShu)MagicSquareeffectivelytransportsourhexagramanditsassociatedfieldontothe2DspacedescribedbytheMagicsquare,a3x32Dmetricspace.TheRivermap[conserves]dimensionalspacebyprojectingthe3x3spaceontothe8x8space.Ifwegrantthatthe8x8fieldofactioncanbeconstructedas4x4x4volumetricspace,thuspermittingtranslationalmotion,theRivermapbringsustwoadditionaldimensiona4thforexpandingorcontractingobjects(localtemporal),andperhapsaccountingforthereference,a5thElementorPhaseof[undefinedcapability].Onthefaceofit,thisproposed5thPhaseresemblesthecomplexdomain,whichtreatsthe"nonphysical"electromagnetismforexample,thoughelectrodynamicsdohavephysicaleffects.Thecomplexdomainalsosupportsquantummechanicsaswehaveallquietlyintuited.

20141016

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BloggerentryThursday,July19,2012IChingandBlochSphere

TheBlochorPoincaresphereseemsparticularlyinstrumentaltounderstandinghowIChingmayberelatedtoquantummechanicsperhapsevenanancientdepictionofit.IChingisfoundedontwoprinciples,Ch'ienandK'un,ortheCreativeandtheReceptive,commonlyknownasLIGHTandDARK,anddiscussedatlengthinvariousappendicestoIChing(namely,appendicesIIIandIV).AppendixIII,theGreatTreatise,isquiteclearonthedistinctionbetweenhexagrams#1and#2andtheremainderofIChing. Statedclearlyinseveralverses,Ch'ienandK'unarethe"pole,""axis,or"gates"ofChange,withoutwhichChangecouldnotbeseenandwouldceasetoexist.(*providereferences*).Thesepassages

providesupporttotheintuitionthathexagrams#1and#2areintendedtoactasbasisvectorsanddefinethesphericalsystemthatrepresentstheIChing.WenowsetasideCh'ienandK'unforconsiderationoftheremainingthirtyonepairsofhexagrams.Someeffortwasmadeearliertofindmeaningfulcorrespondencebetweenthehumanbody's31pairsofspinalnervesandthe31(+1)hexagrampairs,buttheseeffortshavenotbornefruit.ApresentationfromtheViennaCenterforParallelComputingshowsusthatourcomplementaryorxiantianhexagrampairshavingscalarvalueswhichsumto65areregardedasorthogonalandidenticalstates(exceptforphase/sign)ontheBlochsphere.ThemodelofIChingpresentedintheseblogsaccountsnotonlyorthogonalhexagrampairs,butalsothepairsgivenbyWen,DukeofChou.IChingtraditiondescribesDuke'simprisonmentatthehandsoftheeviltyrantofShang,andofthepairingschemehedevisedduringhisperiodofcaptivity. Ingeneral,Wen'spairsaredistinctfigurateinversions(read:180rotation)exceptintheeightcaseswhereinversiondoesnotproduceanhexagramdistinctfromtheoriginal:[(1,2)(51,57)(29,30)(52,58)].Inthosecases,thecomplementarypairistakenastheWenpair.ThegraphicoftheBlochsphere(top)highlightsitsimportantfeatures:basisvectors|0and|1,arbitraryquantumstatepsi(),twocoordinatesystemspolar:denotedbyanglestheta()andphi()andrectangular:denotedbyorthogonalaxesx,y,z.

Selectingoneoftheremaining56'nondistinct'hexagramstoapproachthisquestionspatially,weseethatcomplementarypairsareoppositeandorthogonalpointsontheBlochsphere,whileWenpairslayonthesame(oroppositelatitude)totheselectedhexagramandmayalsobecomplementarytotheselectedhexagram.Weseeinthegraphicabovethenorthandsouthpoles(denotedby|0and|1)alongwithpsi()forminganimaginarytriangle.

Arbitrarilyselectinghexagram#41,wegetitsWenpair#42,and#31asitsorthogonalpoint.Thesethreehexagramsformacorrespondingimaginarytriangle:#41and#42analogoustothepoles,and#31asthesurfacepoint.Thehexagram,itsWeninverse,anditsoppositeformatriplet:twohexagramssharealatitudewhiletwoshareadiameter.Selectinganyhexagramfromtheeightpairs[mentioned]aboveproducesjustadiametricpairinsteadofatripletsincetheWenpairisalsotheoppositehexagram.

IfweacceptthatIChingisanalogouswithBloch'ssphereasaworkingmodelordepictionofquanta,thequestionbegs,whatistobemadeofWen'spairs?WeunderstandthatWenpairsarefigurateinverses,butwhatdoesthismeanintermsoftheBlochsphereandhowmightthisinformustoquanta?

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ErrataReviewwithPaulMartynSmith,thecoauthor,highlightedancorrectiontothedepictionoftheTemporalorLaterHeaventrigramsequence.Inourdiscussionheindicatedtomethatthearmsoftheswastika,uponwhichwemodeledtheTemporal,reversedirectionacrossthehorizontalaxis,andthishasimplicationsfortheorderingofthenumbersonthearms. PaulsoriginalconceptionoftheHoTumathematicsconsistentlypresentsthenumbersascomplementarypairs.Thus,1pairswith9,2pairswith8,3pairswith7,and4pairswith6,and5pairswithX(cipher)atthecenter.ThepresentationoftheTemporalcycleinDecompositionneglectedthatdetail. HereweofferacorrectedpresentationoftheTemporalcyclethatmatchesPaulsvision.Weseeatrightthatthenumberingofthearticulationsoftheswastikafigureiscomplementaryasdescribed.Itscorrespondingdepictiononthecirclejustbeneathnowcorrectlypairsthedigits,sothat1pairs9,2pairs8andsoforth.

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