T H E S P I R A L C A L E N D A RA N D IT S E F F E C T O NF I N A N C I A L M A R K E T S A N D H U M A N E V E N T SW R I T T E N & IL L U S T R A T E DB YC H R IS T O P H E R C A R O L A NN E W C L A S S IC S L I B R A R YT heS piralC alendaranditsE ffectonF inancialM arketsandH umanE vents1992byC hristopherL .C arolan.A llrightsreserved.N opartof thisbook maybeused or reproducedinanymannerwhatsoever withoutwrittenpermissionexceptin thecaseof brief quotationsembodiedincriticalarticlesor reviews.V iolators will be prosecuted to thefullest extent of thelaw.D esign,layout,dustjacketartwork,F igures3-6and3-9photographyand allillustrations except F igure 3-7byC hristopherC arolan.P rintedintheU nitedS tatesof A mericabyB raun-B rumfield,I nc.F irstE dition:O ctober1992S econdP rinting:F ebruary1993T hirdP rinting:S eptember1995F orinformationaddressthepublishers:N ewC lassicsL ibraryP ostO fficeB ox1618GainesvilleGeorgia30503U .S .A .I S B N 0-932750-21-4S piralC alendarisa trademarkof C alendarR esearch,I nc.2forMadeleineA C K N O W L E D GM E N T ST heexchangeofideaswithfellowtechnicalanalystsisavitalresourcetoanyoneattemptingtobreaknewground.S pecialthanksgotoB obP rechterwhohashadunlimitedpatienceintherolesofadvisor,editorandpublisher.T hosewhoseassistancehasbeeninvaluableincludeD aveA llman,P eteD eS ario,H ankP rudenandD anS teinberg.I gratefullyappreciatetheunderstandingandpatienceofmyfamily.I didn'tgrasptheeffectof thisprojectonthemuntilmythenthreeyearolddaughter'sschoolclasswasaskedwhattheir parentsdid."M ydaddy livesin thebasement,"washer reply.A llchartswithpre20thcenturypricesareconstructedfromdataprovidedbyTheFoundation fortheStudyof Cycles,I rvineC alifornia.T heintra-daybondchartonpage94iscourtesyof Tudor Investment Co.F igure3-7isfromScientific American,M arch1969.T hechartforL ondonstocksinF igure10-5iscourtesyof Yelton Fiscal, Inc.3T A B L E O F C O N T E N T SF O R E W O R D 6P A R T I F O U N D A T I O NC H A P T E R 1A D E T E C T IV E S T O R Y 12T H EQ U E S T I O N / 12T H E S IM IL A R IT IE S / 13O N C A L E N D A R S / 15W H Y1987?/ 18C H A P T E R 2L U N A 20M O O N ,S U N A N D E A R T H/ 20S U N A N D E A R T H/ 23GR A V I T Y / 23M O O N A N D M A N / 24T H E M O O N A N D O T H E R L IF E / 25S U M M A R Y / 26C H A P T E R 3F IB O N A C C I & 27T H E O T H E R L E O N A R D O / 27P R O P E R T IE S O F Fn1 28T H E R A T I O SO F / 30T H E GE O M E T R YO F / 31F I B O N A C C I A N D I N N A T U R E / 32I N T H E A N C I E N T W O R L D / 33A N D T H E S P IR A L C A L E N D A R / 36P A R T I I M A R K E T S I N T IM EC H A P T E R4S P I R A L C A L E N D A R R E V E A L E D 41717M O O N S :P A N I C T O P A N I C / 41P R E C I S I O N / 42GO L D E N S E C T IO N S / 42H owB I G?/ 45M A GN I T U D E / 47S P IR A L SI NT I M E / 47T H E F ocus/ 53H A R M O N Y / 58F O R W A R D vs.B A C K W A R D / 60A P P R O X I M A T I O N / 62S U M M A R Y / 634C H A P T E R 5T H E C A U S E O FP A N I C S A N D C R A S H E S 64A U T U M N F A L L S / 64A N A L O GI E S / 66U N K I N D M A Y / 71P A N I C S I GN A T U R E S / 73C H A P T E R 6F O R E C A S T IN GM E T H O D S 76T O O L S / 77T H EM E T H O D / 77C R I T E R I A / 79S E N T I M E N T & T H E W A V E P R I N C I P L E / 82T R A D I N GT E C H N I Q U E S / 83T H E E X A M P L E S / 85C H A P T E R 7N I K K E I :A F O R E C A S T I N GT U T O R I A L 1llP A R T III P A S T ,F U T U R E & B E Y O N DC H A P T E R S S P I R A L C A L E N D A RP A S T 125T H E S P IR A L C A L E N D A R I N T H E A N C I E N T W O R L D ?/ 125Y O MK I P P U R / 125C A L E N D A R S A N D H O L I D A Y S / 126T H E M E S O P O T A M I A N S / 127T H E B I B L I C A L C O N N E C T I O N / 129S T O N E H E N GE / 131C H A P T E R 9B E Y O N D M A R K E T S 133T H E C O L L A P S E O F C O M M U N I S M / 133T H EH O S T A GE S / 137C H A P T E R 10F U T U R E F O R E C A S T S 139O C T O B E R 1995/ 14027J U L Y 1998/ 1411O C T O B E R2004 / 14226J U N E 2023/ 1451D E C E M B E R2032/ 146C H A P T E R 11IsT H E F U T U R EP A S T ?148F I B O N A C C I ,T I M E A N D L IF E / 148W H Y ?/ 151I N D I V I D U A LB E H A V I O R P R O D U C E S GR O U PO R D E R/ 151A P P E N D I C E S 154B I B L I O GR A P H Y 1585F O R E W O R DO ntheeveningofW ednesday,O ctober11,1989,I metC hristopherC arolanforthefirsttime,atadinnerwithfourstockmarketprofessionals.T heD owJ onesI ndustrialA veragehadhitanewall-timeprinthighthepreviousday.M r.C arolanwasveryemphaticabouthisnewmarkettimingdiscovery,whichwasbasedonalunarcalendar."I tcallsforasharpdropinthestockmarket,andwe'reoutoftime.I tneedstohappenrightnow."T henextday,theD owsoldoff13points.T hefollowingday,F riday,itbegancollapsingrightfromtheopeningbell,finallyclosingdown189points,thethirdlargestone-daypercentagedropofthedecade.M onday'sdownopeningbroughttheD owtoalevelalmost300pointsbelowthehighithadregisteredfourtradingdaysearlier.T he juxtapositionof theseeventsgotmyattention.A taboutthesametime,from3000milesaway,aprofessionalfriendcalledtotellmewhathehadseenontelevisionthatnight.O neof themostpopularastronomersintheU .S .wasdiscussingahugemuralcenturiesold.I twasabsolutelyremarkable,hesaid,thatpeoplewaybackthen,withouttheaidofatelescope,hadsuchadetailed,preciseknowledgeof thetimingof themotionsoftheearthrelativetothesun,moonandseveralplanets.W hatkeenobserverstheywere!W hatlengthyrecordstheymusthavekept!A ndhowadvancedscientificallyandmathematicallytheyhadtohavebeen!T henhepointedtoanareaofthemuralthatappearedtocorrelatethemotionsofthesunandmoonwithdramaticnaturalandhumanevents.T heastronomerdismissedthisportionofthemuralwithasmirkandawaveofhishand.B eliefinsuchcorrelation,hesaid,isofcoursesuperstitiousnonsense.I sthatassertiontrue,orinfactcontradictorytotheevidence?L et'sexaminetheattitudeoftheastronomer.S cientists(includingtheoneontelevision)andmathematiciansprobablyconstitutethelargestblockofhumanityleast6Forewordlikelytoholdsuperstitiousideas.S incehumannature,byallotherevidence,hasremainedconstantandreliablethroughthecenturies,itseemsunlikelythatscientistsofanyprior agewerehighlysuperstitious.J ustasscientistsdotoday,theywouldhavebeenthemostlikelypeopletoask,"W hereistheevidenceforthisclaim?"T heywouldhardlybepronetomarryingtheirintellectualachievementswithdetailedsuperstitiousinvention.T hemodernastronomer,then,wasinfactpresentingtwocontradictoryideas,first,thatagroupofpeoplewereconsummateobservers,detailedrecordkeepers,andofscientificbent,andsecond,thatthesesamepeoplewereassertingsomethingthatwasobviouslyfalse,infactsilly.M oreover,thesupposedfalseassertionscouldbesupportedordisprovedpreciselybyobservationanddetailedrecordkeeping,atwhichtasktheywereprovenexperts.T heknowledgeofhumaneventsdisplayedonthemuralreflectedthesamedetailedrecordkeepingasdidtheastronomicknowledge.W ho,then,wasactuallybeingsuperstitious?T hepreviousscientists,orthemodernonewhowascertain,notmerelywithoutevidence,butcontrarytocharacter,thatpreviouslylivingscientistswereasgullibleaschildren?E videncethatearlierrecord-keeperswereontosomethingcameinthe1970s,asclimatologistI benB rowning,tothenearlyuniversalcondemnationofthegeologicprofession,publishedresearchindicatingacorrelationbetweenpeaktidalforces,ascreatedbylunarandsolar(actuallyE arth)cycles,andtheoccurrenceofearthquakesandvolcanoes.I ndeed,asI writethispage,themostpowerfulearthquakeinsouthernC aliforniainfortyyearshasjustoccurred,withinhoursofavolcanoblowingoffinA laska.A lthoughallcommentarycallstheseevents"unrelated,"theyaretakingplace36hoursfromasolareclipsenearalunarperigee,duringayearD r.B rowningdescribedoveradecadeagoasbeingnearthecyclicpeakforsuchphenomena.D r.B rowningdiscoverednotonlythiscorrelation,whichhelpsforecastoccurrencesthatarethemselvesoftenverydramaticeventsinhumanhistory,butfoundfurthermorethatmajorvolcaniceruptions,becausetheycloudtheatmosphere,aretypically followed byacoupleof yearsofcoolerseasons(thisyearisalreadyshapingupasoneofthecoolestofthepastcentury),ashrinkageinfoodgrowingacreage,aresultingeconomicdepression,socialstress,andoftenclassichumanresponsestosuchconditions,suchasthebreakupofpoliticalunionsandwar.I notherwords,heexplainedthemechanismforadirectlinkbetweenlunarandsolarcyclesandanumberofthehumansocialexperiencesthatprovidedramainhistory.M r.C arolan'sdramaticmarketcall,D r.B rowning'swork,plusthecontradictionintheideasheldbythemodernastronomerdiscussedabove,appeartosuggestaninterestingidea,thatsomeofourancestorsweremoreobservantofthehumanconditionthanweare.I ndeed,theymayhavediscoveredsomevaluableinformationaboutpredictingthefuturethatisbeingdismissedcavalierlybythosewhocomeacrossanysuchindication.I tishardlyasurprisethatmanypre-industrialculturesdevelopedakeeninterestinobservingthemotionsinthesolarsystem.T oday,theavailabilityandsophisticationofnighttimeelectronicentertainment(whileablessing,tobesure),hasreducedtonearzero7TheSpiralCalendarformostpeoplethedesiretoobservethemotionsofE arthandmooneverynight.C enturiesago,theskywasprobablythebiggestattractiongoing,onarelativebasisasbrightandbeckoningasL asV egasoravideoscreen.U ltimatelyitbecametheequivalentofagiantdetectivegame,astheparticipantstriedtodeterminetheoutcomeofitsmotions.H owinterestingthatplayingthisintriguinggamemightalsohaveledtotherevelationof anevenmorestartlingfact,thatthesolutionsboreacorrelationtoeventsinmankind'shistory.R atherthanresultingfromsuperstition,then,perhapstheultimateintensityofourforefathers'interestintheskywastheresultofacompellingnewunderstanding,availableonlybecauseofmeticulousobservationanddetailedrecordkeepingspanningcenturies.A nymysticalaurathatmayhaveattendedsuchknowledgewasprobablyduetothefactthattherewasnophysicallylinkingmechanismimaginablebyancient scientists.T heattainmentofsuchanunderstandingmayalsoaccountforthepassion,dedicationandeffortbehindthebuildingoftwoofantiquity'sgreateststructures.T hebuildingof theGreatP yramidinE gyptwasapparentlyundertakenandaccomplishednotonly,asoneauthorpostulated,toserveasanastronomicobservatory(apoolofstillwateraroundwhichastronomersapparentlysatreflectedasmall"moving"patchoftheskythatshonedownalongstraightshaft)butclearlyalsotopreserveformillenniacertainmathematicalknowledge,includingthatofthegolden,orF ibonacci,ratio.T helinkageofthesetwopurposesestablishesthepossibilitythattheE gyptiansrelatedastronomiceventstoothernaturalphenomena.T heknowledgeofthelunar-solarcalendarhaditsmonumentfortheagesaswell,theastronomicclockknownasS tonehenge,whichbrilliantlytracksthe19-yearM etoniccycle(atwhichintervallunarmonthspreciselyfitintosolaryears)andallitssmallercycliccomponents.E achof thesemonumentsisphysicallysosubstantialthatitsimportancetothebuilderswasclearlygreaterthanthatofanobservatory,amathematicalformulaoraclock.T heobservationof a connectionbetweenlunarandsolarcyclesanddramaticnaturaleventsand/or humansocialbehaviorwouldexplainthededicationandcommitment,indeed,theintellectualpassion,ofthebuilders.W hy?B ecausesuchaconnectionwouldbesomonumentallycrucialbothtounderstandingthepatternsinthehistoryofmankindandtoanticipatingfutureevents.T heremaybeanotherreasonthatthesestructuresaresounusuallymassive.A fterall,mendon'tundertakesuchprojectstoday.W ekeepourpresentknowledgeinperishableform,undertheassumptionthatitwillbearoundforever.C onsiderthatanaturalresultofstudyinglongtermsociallypredictivephenomenaisthatoneiseventuallyledtocontemplatethegreatcyclesofcivilizationandtorealizethatthepresentstateofcultureandsociety,whateveritmaybe,isephemeral.U ndoubtedlythearchitectsofthesemonumentswereintentuponpreservingtheirmostimportantknowledgeforfuturecultureseveniftheirowncivilizationweretocollapse.T heseprojectsmighthavebeendesigned,then,byscientistsforanoblescientificpurpose.F orcenturies,itwaswidelyacceptedthatthesestructureswereerectedsimplytosatisfythewhimsofaglorygreedymonarchusingslavelaborandamysticalcultofrobedanimal8Forewordsacrificers.*S uchshallowandquicklyadoptedexplanationsdemandananswertothequestion,howmanysuchanti-intellectualinstitutionshavebeenknowntoallythemselveswithscientistsandbuildbreathtakingtriumphsofknowledgeandengineering?I stheacceptedconclusionreallysensible?O risitanotherexampleofthecondescendingattitudetowardantiquitydisplayedbytheaforementionedtelevisionastronomer?I farelationshipbetweenhumansocialbehaviorandlunarcyclesiseverformallyestablished,perhapsamonumentthatclearlyrecordsandcelebratesthelinkwillbebuiltsome time during the next millennium.I fsuchamonumentisbuilt,muchofthecreditwillgotoC hristopherC arolan,whoappearstohavere-discoveredasubstantialportionofalostunderstandingofthecorrelationbetweenthemotionsofthesun,E arthandmoonandman'ssocialbehavior.O f course,correlationaloneisnotproof.T hemechanismconnectingphenomenamustbeidentifiedaswell.A physicalexplanationforanyimmediateanddirecteffectoflunaractivityonmasshumanemotionhasyettobeproved,althoughscientificstudieshavevariouslylinkedchangesinlunar phase tochangesintheE arth'smagneticfield,thesolarwindbombardingE arthandtheelectromagneticstateoflivingentities,suggestingthattheideaof alinkiswithintherealmof possibility.A tminimum,itisalaudablefeatthatM r.C arolanhasestablishedacorrelationintricateenoughtomakeacommittedscientificinquiryintoamechanismworthundertaking.H avingreadM r.C arolan'streatise,andhavingobservedanumberofhisforecastsoverthepastthreeyears,botherrorsandsuccessesincluded,I amconvincedthattheS piralC alendarcouldwellprovetobeasignificantdiscoveryinultimatelyestablishingacriticallinkbetweenthecyclesoftimeand humansocialbehavior asevidencedinfinancialmarkets.T heS piralC alendarandtheW aveP rinciplearecomplementarytoolsinthefieldofmarketforecasting.O ntheonehand,theW aveP rincipleisuniqueinrevealingvaluableknowledgethat canoftenbeused toforecast extent(futurepricelevel,inmarketterms)withstrikingprecisionwellinadvance.O ntheotherhand,theS piralC alendarrevealsvaluableknowledgethatcanoftenbeusedtoforecasttimingwithstrikingprecisionwellinadvance.Bothbodiesofknowledgepossessapowerfulutilityatthetimeofatrendchange.B othalsopertaintodegree,i.e.,thesizeofexpectedtrendchanges,whichisacriticalaspectof theinformation,sometimestothepointofbeingoflife-and-deathimportancetoindividualsandinstitutions,anduponoccasioneventocultures.I ncollege,it'scommonlyunderstoodthatifyouwanttomajorintheeasiestsubject,youchoosesociology.T hereasonisthat modernsociologyisnotveryscientific.I nfact,it'ssoheavilyladenwithopinionandguessworkthatonlythetrulylazycan'tpassthesubjectwithacreditabletermpaper.C urrentlypublishedmaterialonpopularculturaltrendsissimilarlyvacuousandworthless,assuchtrendscanbeunderstoodintheirfullsignificanceonlywithanunderstandingoftheW aveP rinciple,andnow,*T hatS tonehengewasbuiltbyD ruids,certainlyamorefarfetchedideathanpresentedhere,wasrecentlychallengedbyprehistorianA lbanW all,whodemonstratedquiteconclusivelythatitwas the C elts.9TheSpiralCalendarperhaps,oftheS piralC alendaraswell.I amquitecertainthatthe20thcenturywillultimatelybeknown,asthe17thcenturyistodaywithrespecttothephysicalsciences,asthecenturythatbroughtcohesionandgrandvisiontothefieldof sociology.T hetwoconceptstogether,pluswhateverundoubtedlysubstantialknowledgeremainsmissing,mayultimately,if civilizationdoesnotregresstoofaratanytimeoverthenextseveralcenturies,produceafascinatingscienceofsocialman'sintimaterelationshipwithhisuniverse.A tminimum,asthisbookreveals,thereappearstobeevidencethathumanbeingsinsocialsettingssometimesfindthemusicofthespheresirresistible,andtheyarecompelled,uponrecognizingthateveryoneelseisreadytodothesamething,toleapupoutof their chairsanddance.R obertR .P rechter,J r.,C M TGainesville, GeorgiaJ une28,199210P A R T IF O U N D A T I O NC H A P T E R 1A D E T E C T IV E S T O R YT H E QU E S T IO NI nwhatwaydidthe1987stockmarketcrashmostcloselyresemblethe1929crash?W iththeanswertothisquestionI haveformulatedamethodforforecastingperiodsof extremeemotionalhumanbehavior.T heseperiodsoftenoccuratatimewhenmarkets,suchasthoseforstocksorcommodities,willchangedirection.T hismethodisalsoapplicabletomasshumanbehaviorinotherareassuchaspoliticalmovementsandtrends.T hereareadditionalimplicationsofmyresearch,namely,thecauseandstructureof thelarger forcesthat createthepatternsof lifeand growthintheaffairsof men.B eforeI examinethequestionthatledtoalloftheseideas,I willexplainwhyIaskedthatquestionatall.I n1987,I wasastockoptiontraderontheP acificE xchangeinS anF rancisco.T hereI stoodina"pit,"aconglomerationof humanswhooftenbehavedmorelikebeasts,hungryforadollar,competitive,greedy.P eopleyelledandscreamedforanyadvantageavailable.S urvivalofthefittestwastheacceptedrule.I nthisenvironmentmanydecisionsareinstinctive,intuitiveoremotional.T heconventionalwisdomexplainsthemovementsofthemarketsaslogicalreactionsorprogressionsofthelatest economicinsightor data bit,but theviewfromthepitisdifferent.T he"crowd"dividesintobuyersandsellersandthepitiswheretheymeet.T heirenergyislikeelectricpotentialwithtwopolarities.T hetradesarethesparksthat jumpthegap.T radersreactby"feel."S ometimesthisisanadvantage,andyetwhenemotionsprevail,itisahandicap.W ithtimespentinapitcomestheabilitytosensethepulseof thecrowd.T hemostdramaticeventforatrader towitnessisthesuddenandcompleteconversionof the12ADetectiveStorycrowdfrombuyersintosellersorviceversa,aneventthatoccursatimportantmarketturns.I frequentlywitnessedpricesmovingquicklyinonedirectionwhilethecommentaryofthecrowdcheeredon.P ricesaccelerateandthecommentarybecomesevenmoredefinitive."T hey'llnevergodown!"tradersshout.T hen,pricesreverseandmovejustasdeterminedlyintheotherdirection.T hemoodofthecrowdhaschanged.T heyareconvincedtheyhaveseenanimportant junctureinthemarket.T heirchangeofheartisnotareactiontothechangeinthemarket,butratherasimultaneousoccurrence.T hesearetraderswho,individually,willtellyoutheyarebehavingwithlogicandreason,yetwillhaveaconsensusopinionandthenallchangethatopiniontogether,asifoncue.I tislikewatchingaflockofbirdsflyingintightformationsimultaneouslyreversedirectiontogether.T hisisnothowallmarketsbehaveatalltimes,butitismostevidentintimesof extremeemotion.I wasasixyearveteranonthetradingfloorinlate1987whenthecrashoccurred.M yowninterestinobservingtheemotionsoftraderswassharpenedbythesinglemostextraordinaryeventinthelivesof mostcurrentmarketparticipants.H erewasemotionsogreatitwascalleda"panic,"thefirstseenbymygeneration.I thappenednotjustinonepit,butinmanypits;notjustononeexchange,butonmanyexchanges;notjustintheU .S .,butworldwide.F ear,panicanddespairgrippedmillionssimultaneously.W hy?T hiswasnotauniqueevent.I t had happened before,most recentlyin1929.B eforethecrashin1987,somehadseenparallelsto1929andhadprofitedhandsomely.A fterthecrash,newspaperswerefullof chartscomparing1987and1929.F igure1-1comparestheD owJ onesI ndustrialA verageforthesetwoyears.T hischartshowsallof1929,J anuarythroughD ecember,comparedtothesameperiodin1987.T hecomparisonisamazing.M anyanalystswerecaptivatedbytherelationshipbetweenthosetwoyears,untilthemarketsstoppedfollowing thesame path.T hesubsequentyears,1988and1930,werenotsimilar.Y etthefactremainsthatthepatterntracedbypricesin1987followedstepforstepwhathadoccurredfifty-eightyearsearlier.W henyoulookatthesetwocharts,considerthattheyreflectapatterncreatedbythebehaviorofagroupofindividuals,and that eachindividualhasthefreewill tobuyorsellonanygivenday,andthatthesegroupsof individualsareseparated bymorethana generation.Y et,thepatternsare similar. W hy? H ow?T H E S IM IL A R IT IE ST herearefourmarkedsimilaritiesinthechartsof thetwoyears.T hesefourpointsgivethecomparisonof thetwochartstheirstriking resemblance.1.B othmarketshadasmallcorrection that endedinlatespring;M ay31in1929andM ay20in1987.2.S harpralliesinbothmarketspeakedinlatesummer;S eptember3in1929andA ugust25,in1987.3.B othmarketsattemptedralliesintheearlyfall,whichfailedtoexceedthesummerpeaks.I n1929,thelowerpeakwasonO ctober11.I n1987,itwasO ctober2.13TheSpiralCalendarFigure1-1PatternorCoincidence?4.T hecrashesoccurredinO ctober.T he1929crashwasO ctober29.T he1987crashwasO ctober19.E achofthesefourpointsonthe1987chartfallsninetoelevendaysearlierthanthesamepointonthe1929chart.W hiletheseanalysespointoutthesimilaritiesofthetwocharts,itisalsoevident that theyarenotstrictlythesame.N owitistimeforsomedetectivework.W hatarethestrengthsandweaknessesofthisanalysissofar?F igure1-1isachartof thetwoyearswearecomparing.T hechartisinteresting,buthowisitmade?T herearethreepiecesofdatathatarecriticaltotheappearanceofF igure1-1.T woofthepiecesareobvious,thestockpricedatafor192914ADetectiveStoryand1987.T hethirdisnotreadilyapparent,butiscrucialtotheappearanceof F igure1-1.I tisthecalendarthatdetermineshowthetwodatasetsalign.T hechartshowsthefirstdayoftradingin1929alignedabovethefirstdayoftradingin1987,etc.B yaligningbothchartsatthebeginningoftheyear,weareusingthecalendarastheyardstickofcomparison.T hefactthatthesimilaritiesinthechartsoccurredatthesametimeofyeariswhatmakesitmostinteresting.I fonecrashhadoccurredinM archandtheotherinO ctober,theshapeof thechartsmaystillbesimilar,butthecomparisonwouldnolongerbeasstriking.The similarity in the two charts is greatly a function of the calendar.T heexaminationoftheexactdatesofthesimilaritiesalsousestheyardstickofthecalendarastheabsoluteagainstwhichconclusionsaredrawn.T hequestionsI wantedansweredwere,"Isthiscalendartheproperyardstick?"and"W hatwouldhappenifthetwoyearsof tradingdataarecomparedwithsomeothermethodof countingtime?"O N C A L E N D A R ST hecalendarweuseistheGregoriancalendar.I tisanimprovedversionoftheJ uliancalendarnamedforJ uliusC aesar.O urcalendar,withitsR omanorigin,doesaverygood jobofkeepingtime.T hetimeitkeepsissolartime,therotationoftheeartharoundthesunonceevery365.25days.T hereareotherunitsoftimeacalendarcoulduse.A ncientcalendarsusuallymeasuredthetimeittakesforthemoontocircletheearthrelativetothesun.M ostcalendarsmeasureonetypeoftimeaccuratelyandthenapproximatetheothertypeoftime.O ursolarcalendarapproximateslunartimewithitsmonth,butthisisaninaccuratemeasurementoflunartime.T odistinguishbetweenthemodernmonth,whichisl/12thofasolaryear,andtheancientmonth,whichisonemoon,I willusetheterm"moon"fortheancientmonthand"month"initspresentcontextasafractionof asolaryear.T able1-115TheSpiralCalendarFigure1-2Thetwomarketsalignedwithalunarcalendar.T heJ ewishcalendarisalunarcalendar,whichisstillinusetomarkreligiousholydays.O neoftheholydays,Y omK ippur,fellexactlyonthethirdpointofthefourmarketsimilaritiesin1987,thefallmarkettop.T hereisanoldstockmarketadage,"buyonR oshH ashanah(anotherholyday),sellonY omK ippur."S ellingstocksonY omK ippurin1987wasaverygoodthingtodo.M ydesiretoknowwhenY omK ippurfellin1929wastheoriginalimpetusformycalendarinvestigations.T heJ ewsborrowedtheircalendarfromtheB abylonians.A partialhistoryofancientcalendarsisexploredinC hapter8.I ntheJ ewishcalendar,thenewmoonisthefirstdayofthemooncycle(month).T henewmoonnearthespringequinoxmarksthefirstdayof thefirstmoonoftheyear.W iththiscalendarI realignedthecomparisonofthe1929and1987stockcharts.T heresultisF igure1-2.S hownalsoarethefourpointsofsimilarity.T able1-116ADetectiveStoryshowsthelunarandsolardatesforthesamefourpoints.Thecrashof 1 987 occurred onthe samelunar dateasthecrashof 1 929!T heotherthreepointseachfallonedaylateronthelunarcalendarthantheir1929counterparts.W ithinoneday,the1929autumnalmarkettopoccurredontheJ ewishholydayof Y omK ippur.C ouldthisbeacoincidence?T hereare354daysinalunaryear,29.5306dayspermoontimes12moons.T he1987crashcouldhaveoccurredonanyoneofthose354datesintheyear,butitoccurredonthesamedateasthe1929crash.T hat'sa1in354chance!A ddtothattheoddsoftheotherthreepointsfallingonthesamedateplusorminusoneday.T hey'reveryremote.I t'smorelikelythat youandI bothwinthelottery.Iundertookthisinvestigationonthebelief,bornintheexperienceoftradingduringthecrash,thatitwasnotarandomevent.T heeerieparallelsto1929spurredmycuriosity.N ow,herewasalinkbetweenthetwomarketsthatwasabsoluteanddramatic.I nsolartermsthe1929-1987analogywasanintriguingcoincidence.N owseeninlunarterms,thelikenessisanempiricalfact.W heretherewassimilarity,thereisnowsameness.W hatappearedasaresemblanceisrevealedtobea replication.C onsiderthatthestockmarketchartofeachyearisapatternordesigncreatedbyhundredsof thousandsormillionsof peoplefromtheireconomicbehaviorthroughtime.I thaslongbeenawonderofnaturetowatchbirdsmigrateorsalmonreturntospawn.T heymiraculouslyrecreatetheirancestors'patternsinminutedetailwithoutconsciousknowledgeoftheiractions.H ereisevidenceofthesameactivityinhumans.I n1987,tradersandinvestorsdancedtothesametuneofspeculationandpanicthathadplayedfifty-eightyearsearlier.H erealsoistheevidenceof whatwasplayingthetunethatledthosedancers.T hephasesof themoonprovidedtherhythmsof greedandfearbywhichtheyboughtandsold.I twas1988whenI madethediscoveryofthelunarconnectionbetweenthecrashes.I understoodimmediatelythatthiswasanimportantadvanceintheunderstandingofbothmarketsandman.H owever,itdoesnotprovideamethodforforecastingmarketbehaviorbyitself.I twasacluetothemarket'smechanism,anditpointedthedirectionI shouldlooktoforadditionaldiscoveries.I twassomemonthsbeforeI madefurtherprogress.M ostofthosemonthswerespentlookingatthatsamepairofchartsinF igure1-2,oneyearpositionedabovetheother,likeaphotographofsiblings;andI knewwhotheparentwasontheothersideof thecamera: Luna.I tisnotanewideathatthepositionof themoonandsunintheskyhaveinfluenceonthebehaviorof man.I ntheW est,itistracedtotheM esopotamiansof 5000yearsago.T omodernman,theideaisonthefringeofacceptedknowledge,supposedlylacking"proof."I submittheextraordinaryrelationshipinlunartimebetweenthestockmarketsof1929to1987asthefirstbrickinawallof proof I willconstruct.N oticealso,thatitiswhenseenthroughthecalendarofthosesameM esopotamians,theevidenceismadeplain.T heM esopotamianswillreceivetheir fulldueinC hapter8.17TheSpiralCalendarW H Y1987?T hequestionI neededtoanswerwas,"W hydidthespeculationandcrashof1987happenthenandnotinsomeotheryear?"M yknowledgeofthelunarrelationshipgavemenocluetothatanswer.I slowlycametounderstandthatthechartI waslookingatwasnotpointingmetowardsthenextpieceofthepuzzle.I waslookingat1929and1987asisolatedyears.I neededtolookattheentiretimespanofstockpricemovements.It wasthe interval of timein between that wasthekey.B efore,I focusedoneachofthefourpointsbeingattheexactsamelunarcalendarpoint.N ow,I sawthesimilarityinadifferentway.T hedistanceintimefromthespringlowof1929tothespringlowof1987wasthesameasthedistanceintimefromthe1929toptothe1987top.T hiswasalsothesameasthedistancefromthe1929fallpeaktothe1987fallpeak,andagainthesameasthedistancefromcrashtocrash.T hetwoyearshadtobeseeninadifferentwaytobeunderstood.T histimespanalsoneededafreshperspective.I twasn'timportantthatfifty-eightyearsseparatedthesecharts.I twasn'timportanthowmanydaysseparatedthesecharts.W eeksandmonthsalsoweren'trelevant.T hequestionwas,howmanymoonsseparated theseevents?S eeF igure1-3andT able1-2.Figure1-3E acheventin1987occurredapproximately717moonsfromtheanalogouspointin1929.T heaverageofthefourdistancesis717.03.I feltthatiftherewerealargerpatterntobefound,thisnumbershouldturnouttobesignificant.M yfirstinclinationwastocheckifitwasaF ibonaccinumber.I firstlearnedofF ibonaccinumbersinhighschool.I nextencounteredthemandtherelatedgoldensectioninmycollegiatestudiesofancientartandarchitecture.M ythirdexperiencewithF ibonaccinumberscamethroughtheworkofR .N .E lliottandR obertP rechterandtheiranalysisofthestockmarket.Y et717isnotaF ibonaccinumber.I wonderedif717mightbesomepermutationofaF ibonaccinumber,maybeamultipleorasquareroot.W ithcalculatorandpencilinhand,I proceededthroughthesequenceofF ibonaccinumbers.W henI enteredthe29thF ibonaccinumber,514,229,intothecalculatorandpressedthesquarerootbutton--there18A Detective StoryT able1-2was717.0976.M yfirstthoughtthenisstillveryclearinmymemory:"T heworldisaverybeautifulplace."T hiswastheseconddiscoverymydetectiveworkhadproduced.T hesediscoveriesareempiricalfactslinkingthetwogreatfinancialpanicsofthiscenturywithconcreteexternalphenomena,moonphasesandF ibonaccinumbers.T henexttwochapterswilltakeusawayfromthemaintopic,forecastingemotionalmarketbehavior,inordertoobtainafoundationof knowledgeaboutthemoonandF ibonaccinumbers.T hisisnecessaryfor theideasthatwillfollow.19C H A P T E R 2L U N AM O O N ,S U N A N D E A R T H .W hatarethephasesof themoon?W hydoesthemoonsometimesappearlargerinthesky?W hyareeclipsesoccasionalandnotonceamonth,excuseme,onceamoon?W hatfollowsisanoverviewof themovementsofthemoon,sunandearthandthetimeperiodstheyrepresent.T heunitoftimeofprimaryinteresttousistheonemarkedbythephasesofthemoon,calledalunation.I tisthetimeelapsedfromfullmoontofullmoonornewmoontonewmoon.T hemoonisfullwhenitisontheoppositesideof theearthfromthesun.T hesideof themoonthatfacesthesunisthereforethesamesidethatisfacingtheearth,givingusafullyilluminatedviewofthemoon.A sthemoonorbitstheearth,ittravelsaroundtobeonthesamesideoftheearthasthesun.A tthistime,thesideofthemoonthatisfacingtheearthisfacingawayfromthesun.T hissideisinshadow,andwecannotseeit.T heeffectiscalledthe"newmoon."SeeF igure2-1.T hetermnewmooncomesfromtheancients,whowouldwatchthemoonshrinktoasliverbeforedisappearingor"dying."A fterthreedays,thereemergingmoonwascallednew.T henewmoonisalignedtoocloselywiththesuntoview,notonlyonthedayofthenewmoon,butusuallythedaybeforeand thedayafteraswell,hencethreedays.T heperiodof timefromnewmoontonewmoonis29days,12hours,44minutesand2.8secondsor29.5306days.T hisisthesynodicmonth,whichI willrefertosimply20LunaFigure2-1asa"moon."I tisnotthetimeittakesforthemoontoorbittheearth.T hatis27.3216days,calledthe siderealmonth.T ounderstandthedifference,startatafullmoon.H ere,themoonisoppositethesun,withtheearthinbetween.I n27daysthemoonwillhavecompletedonerevolutionaroundtheearth,buttheearthwillalsohave journeyedfartherinitscoursearoundthesun.T hemoonneedstoorbitanadditionaltwodays,thedifferencebetweenthesynodicandsiderealmonth,torealignwiththeearthandsunforthenextfullmoon.T hepointtoemphasizeisthatweusuallyconsiderthephasesofthemoontobetheproductof themotionof themoon.P rimarilytheyare,buttherearethreecontributorstothephases,thesun,earthandmoon.T hemovementsof thethreecombineto produceone unit of time,a lunation.E clipsesoccurwhenthesun,earthandmoonalignsothattheearthcastsitsshadowonthemoonorviceversa.I nalunareclipse,theearthcastsitsshadowonthemoon.T hiscanonlyoccuratthefullmoon.A solar eclipseisthemoonpassingbetweentheearthandsunatthenewmoonandcastingitsshadowontheearth.E clipsesdonothappenat everynewandfullmoonbecausethemoonorbitstheearthatanangle.T heplaneonwhichtheearthorbitsthesunistheelliptic.T hemoonorbitstheearthonaplanethattiltsfromtheellipticatanangleof58'.T hemooncrossestheelliptictwiceinitsorbit.T hepointsofcrossingarethenodes.T hemoonmustbenearanodeandatneworfullmoonforaneclipsetooccur.I nF igure2-2,thenewmoonisbelowitsnode,sothereisnoeclipse.T hetimeittakesforthemoontotravelfromanodethenbacktothesamenodeaverages27.2122days.T hatisthe nodicalor draconicmonth.T hepaththemoontakesaroundtheearthisanirregularellipse.T heearthisnotinthecenter of theellipse,makingthemoonmoveperiodicallycloserto,thenfartherfrom,theearth.T hepointofthemoon'sclosestapproachtoearthistheperigee,andthe21TheSpiralCalendarFigure2-2farthestistheapogee.T hetimeittakesthemoontogofromperigeetoperigeeorapogeetoapogeeaverages27.5545days,theanomalisticmonth.A tperigee,themoonappears14%largerintheskythanatapogee.A llofthesetimeperiods,synodic,nodical,siderealandanomalisticareaverages.A nysingletimeunitwillvaryfromtheaverage.T heorbitofthemoonissubjecttogravitationalpullsfromotherplanets,aswellaschangesinvelocitythateffectthesetimeunits.T heprimarytimeunitofourconcernisthesynodicmonth.O verthecenturiesithasgraduallylengthenedasthemoonveryslowlymovesawayfromtheearth.T heothermooncyclethatwillconcernussomewhatbesidesthesynodicisthesaroscycle.T hisisthe6585.32dayor18year,11dayeclipsecycle.A sarosmarksthetimewhenthemoonhastraveledawholenumberofsynodic,nodicalandanomalisticmonthsandsolaryears.A fter6585days,themoon,earthandsunwillreturntonearlytheexactpositionsrelativetoeachother,usuallycausinganothereclipse.A notherinterestingtimeunitisthelunarday.T hisisthetimeittakesthemoontoappearatthesamepointabovetheearth,24hoursand50minutes.T hedifferencebetweenthisandthe24hourrotationoftheearthissimilartothedifferenceofthesynodicandsiderealmonths.I fthemoonisaboveacertainpointontheearth,after24hourstheearthwillrotatebacktothatsamepoint.T hemoon,however,willhavemovedfurtheronitsownorbit,whichmeanstheearthmustrevolveanadditional50minutestopositionthemoonabovethatsamepointontheearth.T helunardayisthesourceof themaintimeperiodof thetides.H ighandlowtidesoccur twiceeverylunarday,onceevery12 hours and 25minutes.22LunaFigure2-3S U N A N D E A R T HT heearthcompletesonerevolutionaroundthesunevery365.249days,thesolaryear.T hepolaraxis,onwhichtheearthspins,tiltsfromtheperpendicularoftheearth-sunellipticatanangleof2326'.F igure2-3showsthetiltintheaxischangesrelativetothesunthroughouttheearth'sorbit.A tthewintersolstice,thefkstdayofwinter(aroundD ecember22),theN orthP oleisatitsfarthestpointfromthesun.T hesummersolstice(aroundJ une21intheN orthernhemisphere)iswhentheN orthP oleisatitsnearestinclinationtothesun.T heequinoxesmarkthefirstdayofspringandfall(aboutM arch21andS eptember22).A tthesepoints,neitheraxisisclosertothesun.T helengthsofdayandnightareequal.N otethatthetiltoftheearthdoesnotchangewith theseasons, just the orientation of the tilt toward or away from thesun.L ikethemoon'sorbit,theearth'sorbitof thesunisanellipsewiththesunnotatthecenter.T heearthisattimesclosertothesun.T hepointofclosestapproachistheperihelion,whichoccursinJ anuary.T he aphelion,thepointwhentheearthisfarthestfrom thesun,occurs in J une.GR A V I T YI fI amtoconsiderwhichforcesrelatedtothemovementsofthemoon,sunandearthcouldpossiblyeffecttheemotionalbehaviorofmarketparticipants,gravityisa23TheSpiralCalendarlikelystartingpoint.I amnotgoingtodetailthephysics,justrelatehowthepreviouslydescribed movements fit the gravitational picture.Gravitationalattractionvariesinverselywiththesquareofthedistancebetweentwoobjects.T hestrongestattractionwefeel,otherthantheearth's,isfromthemoon.T hesunisthenextstrongestpull,thentheplanets.T hisattractionisevidentinthetides.T hefactorsaffectingthepullof thetidesareproximityandalignment.T healignmentatnewandfullmoonscauseshighertides.T hesun,moonandearthalignonaplanewhenthemoonisatanode.T heresultishighertides.H ighertidesalsooccurwhenthemoonis closer at perigee and when thesun iscloser totheearthat perihelion.T hehighest tidesoccurwhenallthesebodiesareatmaximum proximityandalignment.M O O NA N DM A NT heconnectionbetweenhumanlifeandthemoonhasbeenrecognizedforthousandsofyears.A ccordingtomythologistJ osephC ampbell,theoldestevidenceforanunderstandingbymenandwomenofthesignificanceofthemoonisastoneagefigurinefromtheS outhofF rance,whichisabout17,000yearsold.1I nthefigure,awomanholdsacrescentshapedhorninherrighthand.T heotherhandrestsonherbelly.T hecrescentshapehasthirteenmarksonit.I tissuggestedthatthehornreferstothemoon.T hehandonthecrescentandthehandonthebellyshowanunderstandingof thelinkbetweenthemoonandthefertilitycycle.T hethirteenmarkingsrepresentthenumberoflunationsinayearandthenumberofdaysfromthefirstappearanceofthenewmoontothefullmoon.T hemoonhasbeenlookedtosinceasadeviceforcountingandasthesourceof thereproductivecycle.W ecantracehowtheseideashavecomedowntouswithlinguistics,thestudyoflanguage.O urword,month,derivesfromtheA nglo-S axonmono,meaningmoon.T hewordsmenses,menstruationandmenopauseallhavetheirrootinmeno,Greekformonth.T hemenstruationcycleinwomenapproximatesthelunarcycle.D onotunderestimatehowimportantthisis.T hereproductivecycleisthestartingpointofhumanlife.I ftherewerenoreproductivecycle,therewouldbenohumanlife.O urlanguagereflectsthisunderstanding.T hebaseunitoftimethatdefinesthehumanlifecycleisthemoon.T hestoneagefigurinealsomakesreferencetothemoonasadeviceforcounting.O urwordmeasureisderivedfrom mas,anancientwordformoon.T hemoonwastheancientmeasureof time.L unaticandlunacyarewordsthatindicatealinkbetweenhumanbehaviorandthemoon.T heyderivefromtheL atinwordformoon,luna.W ethinkoflunacyasasynonymforinsanity.T hewordsoriginallyhaddifferentmeanings.S omeonewhowasinsanewasalwayscrazy,whilealunaticwascrazyonlyduringcertainlunarphases,usuallythe fullmoon or new moon.1 Mystery of the Full M oon-P ublicM ediaV ideo-1986.LunaT hischapterisnottheplacetoexaminetheaccumulatedevidencethatlunaractivityaffectshumanbehavior.H owever,afewexamplesdeservemention.I directthereader tothebibliographyfor further information,particularlyK atzeff s Moon Madness.A rnoldL eiberstudiedthetimingofmurdersandaggressivebehaviorinD adeC ountyF lorida.2H efoundthatmurdersandaggressivebehaviorpeakedatthefullmoonandagainimmediatelyafterthenewmoon.H isworkwascriticizedbyotherscientistswhocouldnotreplicatetheresults.L eiberpointedoutthathiscriticshadusedthetimeofdeath,whichcouldbedayslater,andnotthetimeofinjuryasthebasisfortheirconclusions.I twasthetimeofinjurythatcorrelatedwiththelunarphases.I njuries,presumably,aremorelikelytobesustainedwhenapersonisthinkingmoreemotionallythan"normal."Icehockeyisaviolentsportknownforitsemotionalaggressionandfighting.K atzeffreportsthatastudyofthe1976-1977seasonoftheW orldH ockeyA ssociationshowedthatpenaltyminutesservedbyplayersonthenightofthefullmoonwere33%higherthanthepreviousnight.3S irM atthewH alearguedinthe1600sthatlunacywascausedbychangeinthemoon'sphases, particularly at the solstices andequinoxes.4A linkbetweentheeffectofa particularlunarphaseanditsproximitytoasolsticeorequinoxwillplayapartinmylaterinvestigationsof economicbehavior.C hronobiologistsstudythetimingofbiologicalrhythms.T heyhavefoundthatthenaturaltimeunitofthehumanclockisthelunardayof24hours50minutes.5W henhumansaredeprivedofseeingthelightanddarkcycleofthesolarday,theirsleepcyclelengthenstoapproximatethelunarday.T hisexplainswhyjetlagisnoticeablyworsefortravelersflyingE astthanW est.I fyoucrossthreetimezonesgoingE ast,yourdaywillbetwenty-onehourslong,threehoursandfiftyminutesshorterthanthenaturallunarday.I f youtravelthreehoursW est,yourdaywillbetwenty-sevenhourslong,twohoursand tenminuteslonger thanthelunarday.T headjustmentforyourinternalclockwillbelessonwesterlytravelbyonehour and forty minutes.T H E M O O N A N D O T H E R L IF EL unarrhythmsarepresentinnon-humanlifeforms.A quaticlife,withitscloseconnectiontothetides,isoneexample.F eedingtimeformostshellfishisaccordingtotidalcycles.T hequestionis,aretheshellfishrespondingdirectlytothelunarpullortothetide,whichisa responsetothelunar force?2Leiber.pp23-25.3Katzeff.p184.4Ibid.p9.5Wall Street Journal.10A pril1990.p1.25TheSpiralCalendarI nthe1950s,F rankB rown,aprofessoratN orthwesternU niversity,setouttoanswerthisquestion.H etookatrayofoystersfromN ewH aven,C onnecticutandbroughtthemtoE vanston,I llinois.T heywerefittedwithpressuresensitivedevicestorecordtheopeningof theshellatfeedingtime.F ortendays,theycontinuedtofeedatthetimeof thehightidesinN ewH aven.T hentheystopped.T heoysters,inunison,begantofeedatthetimesthatwouldhavebeenhightideinI llinois,iftherehadbeenanoceanthere.6H econcludedthattheoystermustbetakingitsfeedingcuesdirectlyfromthetidalpullof themoon,andnotfromtheflowof thetide.M anymarinecreaturesfeedand/orspawnaccordingtolunarrhythms.T hepaloloworm,whichlivesintheoceannearF iji,spawnsonagivenlunardate,sevendaysafterthefullmoonofO ctoberorN ovember.T hisisanexampleoflifebeingregulatedbyboth thelunarandsolar calendars.T hemoonmust beintherightphase andthesunmustbeattherighttimeof yearfor thespawningtotakeplace.S U M M A R Y1.T heprimarytimeunit,thesynodicmonth,lunationormoonis29.5306days.2.A ncientmanbelievedintheconnectionbetweenlifeandthemoon.T hisobservationsurvivesinour language.3.M odernsciencehasdocumentedthelongperceivedconnectionsbetweenreproductionandthemoon,andbehaviorandthemoon.6P almer,B rown,E dmunds. Introduction to Biological Rhythms.A cademicP ress.N ewY ork1976 pp. 215-216.26C H A P T E R 3F I B O N A C C I &T H E O T H E R L E O N A R D OR enaissancemeansrebirth.I nI taly,the"rebirth"wastherediscoveryoftheknowledgeof theancientsthatwaslostintheD arkA ges.I nthefieldsof artandscience,wethinkofL eonardodaV inciasthesupremefigureoftheR enaissance.H isage,theQ uartrocento,orfifteenthcentury,epitomizestheR enaissanceimage.I nmathematics,theR enaissancehadoccurredtwo-hundredfiftyyearsearlier,andthecentralfigurewastheotherL eonardo,L eonardodaP isa,knownalsoasF iliusB onacci,meaning"sonofB onaccio,"shortenedtoF ibonacci.I nthearts,theR enaissancewaspropelledbytheexcavationofGreekandR omanartifacts,whichgavecontemporaryartiststheirinspiration.T herewasnogreatmathematicaltraditionoftheR omanstorediscover.T hemathematicsoftheGreeks,primarilythegeometryofP ythagoras,hadoriginatedwiththeE gyptians.T hroughtheD arkA ges,mathematicalknowledgewaspreservedandexpandedbytheA rabsandtheI ndians.F ibonaccireceivedpartofhiseducationinE gypt,wherehisfatherconductedbusiness.T herehewasexposedtothemathematicaltraditionsof theA rabworld,whichhe introduced to E urope.F ibonacci'sgreatestcontributionwastheintroductionofA rabicnumeralsandtheplacevaluesystem.I nthedecimalplacevaluesystem,eachnumeral'svalueisdeterminedbyitsplacerelativetothedecimalpoint.T heseconddigittotheleftofthedecimalpointisworthtentimesthesamedigitinthefirstpositionandonetenththevalueofthatdigitinthethirdposition,etc.T heplacevaluesystemgreatlyfacilitates27TheSpiralCalendarmanipulationofnumbers.I fyouhavetriedtomultiplyR omannumerals,youunderstandtheadvantageof theplacevaluesystem.T heintroductionofthissystemtoE uropeproducedtheR enaissanceinmathematics.F ibonacciwrotethreemajormathematicalworks,theLiberAbacci,PracticaGeometraeandtheLiberQuadtratorwn.O neofthese,LiberAbacci,containstheproblemwhoseansweristhenumbersequencewhichbearsF ibonacci'sname.H ereistheproblem:A ssumethatapairofrabbitsstarttobreedattwomonthsofageandtheyproduceonepair,onemaleandonefemale,permonth.S tartingwithonepairinthefirstmonth,howmanyrabbitswillyouhaveinoneyearassumingnorabbitsdie?Inthesecondmonth,youwillstillhaveonepair,astherabbitsareonlyonemonthold.I nthethirdmonth,theyproduceonepairforatotaloftwopairs.T hefourthmonthaddsonepairtothetotal,makingthreepairs.I nthefifthmonth,thesecondpairbreedsalso,addingtwopairs,foratotaloffivepairs.A ftertwelvemonths,therewillbe144pairsofrabbits.T able3-1showstheanswertotheproblemfor thefirst thirty-fivemonths.T heleft column,labeled,isthenumberof monthsandtherightcolumnisthenumberofpairsofrabbits.F ibonaccihimselfdidnotinvestigatethepropertiesof thesequenceinanyof hismajorworks,thoughtheyhaveattractedtheinterestofmathematiciansthroughthecenturies.I nthenineteenthcentury,mathematicianE douard L ucasnamed thesequenceafter F ibonacci.P R O P E R T IE S O F FnT heF ibonaccisequenceisdenotedasFn,wherenisthesequencenumberandFnisthecorrespondingF ibonaccinumber.I nT able3-1,wheren=12,Fn=\44,etc.T hetwomostnoteworthypropertiesoftheF ibonaccisequenceare:first,eachF ibonaccinumberisthesumofthetwonumbersprecedingit,thusitisanadditivesequence;andsecond,theratioofeachF ibonaccinumbertoitsprecedingnumberisalternatelygreaterorsmallerthanthegoldenratio.A sthesequencecontinues,theratioapproachesthegoldenratio,T able3-128Fibonacci &1.6180339..,knownalsoas(phi).T hisistrueof allsuchadditivesequences.A nadditivesequenceiscreatedbyaddinganytwonumberstogetherthenaddingthesumtothesecondnumber,etc.T heF ibonaccisequenceisthemostperfectadditivesequence,becauseitsratiowillapproachwith fewer iterations.T hegoldenratio,,istheirrational numberI tcanalsobeexpressedasT hisnumberanditsinverse(.618...)arethefoundationforthedesignof ancientartandarchitecture.T heuseof theGreeklettertodenote1.618isareferencetotheGreeksculptorP hidias,whobelievedinthenumber'sroleindeterminingproportioninartandnature.P hidiasconstructedthechryselephantinestatuesofA thenaintheP arthenonandZeusatO lympia.T helatterwasoneof thesevenwondersof theancientworld.H ereisapartiallistof theintriguingpropertiesof F ibonaccinumbers.T hesumof anytenconsecutiveF ibonaccinumbersisdivisibleby11.E verythirdF ibonaccinumberisdivisibleby2,everyfourthisdivisibleby3,everyfifthisdivisibleby5,etc.,thedivisorsbeingF ibonaccinumbersinsequence.C onsecutiveF ibonaccinumbershavenocommondivisorotherthan1.1and144aretheonlyF ibonaccinumbersthataresquares,andtheyareboththesquaresoftheirsequencenumber.T hefactthattherearenosquareF ibonaccinumbershigher than144wasprovenin1963byJ ohnC ohn.1addedtothesumof allF ibonaccinumbersto Fnequals Fn+2-F oreveryintegerm,thereisaninfinitenumberofF ibonaccinumbersevenlydivisibleby mandatleastoneexistsinthefirst m2F ibonaccinumbers.T hesquareofanyF ibonaccinumberwilldifferby1fromtheproductoftheF ibonaccinumbersoneitherside.T hedifferencealternatesasplusoneandminusoneastheseriesprogresses.A nyadditivesequencewillhavesuchaconstantdifference,butonlytheF ibonaccisequencehasthenumberoneasthatdifference.T hisisexpressedmathematicallybyFn2= (Fn+l )(Fn_1) +1.F or anyfourconsecutiveF ibonaccinumbersA ,B ,C ,D :C2 B2 A x D.E veryprimeF ibonaccinumber except threehasa primesequencenumber.T hefinaldigitsoftheF ibonaccisequencerepeatincyclesof60.T hefinaltwodigitsrepeatincyclesof300.T hefinalthreedigitsrepeatincyclesof1,500,fourdigitsin15,000and150,000forfivedigits,etc.T herearetwomethodsforcalculatingagivenF ibonaccinumberwithoutaddingthesequence.29TheSpiralCalendarR ounded to the nearest integer,will provide the exact F ibonaccinumber for n.T H ER A T I O S O FT able3-2T heratiosofanditsinverseare theproportionsofaF ibonaccinumberwithitsadjacentF ibonaccinumbers.T hisistherateatwhichtheF ibonaccisequencegrows.A lternatenumbersintheF ibonaccisequencearerelatedby. T he ratiooftwoF ibonaccinumberswillapproximateoritsinverseraisedtothepowerofthedifferenceoftheirsequencenumbers,n.T able3-2showsthefirstfourratiosofT heseratiosarethemselvesanadditivesequence.S irT .A .C ookinTheCurvesof Lifecallsthisthegoldenseries:.236,.382,.618,1,1.618,2.618,4.236...T heratiosof thisseriesarethesameasthesequenceitself,eachnumberbeing1.618timestheprevious.T hefollowingequationsconcerningtheratioshold true.T helastequationisthemostinteresting,added toitssquareequalsmultipliedbyitssquare.P hi'suniqueplaceamongnumbersasthebridgebetweenadditionandmultiplicationiswhyitandtheF ibonaccisequenceareubiquitousinlifeandnature.P hi'ssourceintheadditiveprocessof F ibonaccibecomesthevehicleforthemultiplyingnatureof life.T helifeprocess, reproduction,is another termfor multiplication.I nnature,likereproduceslike,multiplyingitsformsthroughout.S tockpricesin1987wereapatterncreatedbylivingthingsthatreplicatedthepatternof1929.T hroughthegeometryofwecanunderstand howformscanbegrownor reproduced.30Fibonacci &T H E GE O M E T R Y O FT hegoldensectioncanbedescribedasthepointonalinethatwillcutthelinewherethelargersectionwillbeinproportiontothesmallerasthewholeistothelarger,asshowninF igure3-1.T heproportionthuswillbeT hegoldenrectangle,F igure3-2,isconstructedbytakingasquareA B C D withsidesequalto2.F romcornerB ,lineB EbisectsC D .B ytheP ythagoreantheorem,B EequalsW ithacompass,createE F .E F +C E =.GF =2.T heratioofT hesmallerrectangleB GD F mustalsobeagoldenrectangle.A smallersquareD H I F canbedrawnwithinthissmallergoldenrectangle,F igure3-3.T hiswillproduceanotherrectangle,B GI H ,whichwillalsobegolden.E achrectanglecreatedis.618thesizeofthenextlarger.T heprocessalsoworksintheotherdirectionbyaddingsuccessivelylargersquarestoformgoldenrectangles,each1.618timesthepreviousinsize.T headditivenatureofthisgeometricalprocessisre-creatingthesameshapeinlogarithmicallylargerorsmallersizes.I tcancontinueineitherdirectiontoinfinity.T hesquaresthatareaddedto,orsubtractedfrom,thegoldenrectangleare gnomons.A gnomonisashapethat,whenaddedtoasecondshape,reproducesthesecondshape.T hisprinciplein nature is called gnomonic growth.F igure3-1F igure3-2F igure3-3F igure3-431TheSpiralCalendarD rawing arcs within the squares ofthenestinggoldenrectanglesproducesagoldenspiral,F igure3-4.T hegoldenspiralisalogarithmicorequiangularspiral.T helogarithmicspiralhasaconstantratioorproportionbetweenradiiseparatedbythesameangle.F igure3-5shows that thegoldenspiralhas radiithatareinproportionatrightangles.T hefigureshowstheradiiinF ibonacciunits.T hegoldenspiralmaintainsitsproportionasitspiralstoinfinityineachdirection.T hereisnodifferentiationinshapebetweenthesmallestandlargestpartsofthewhole.T hespiralisacommonmanifestationofF ibonacciandin nature.F I B O N A C C I A N D I N N A T U R ES piralsappearinseashells,pinecones,animalhornsand patternsof plantgrowth(phylotaxis).T heshelloftheNautilus Pompilius,showninF igure3-6,isnature'smostperfectapproximationofalogarithmicspiral.A mongothershells,the Haliotis ParvusradiiprogressatrightanglesaccordingtoT he Dolium Perdix hasas the ratio governing itsspiralprogression.S piralsappearinnon-livingnaturalobjectssuchasgalaxies,andinnonliving natural processessuchashurricanes.T hegrowthpatternof theseedsof thesunflowerformtwospirals,oneclockwiseandonecounterclockwiseasshowninF igure3-7.T henumberofspiralsineachdirectionareadjacentF ibonaccinumbers,usually55and89or89and144.V erylargesunflowershavebeenfound that have144and233spirals.T hisisastrikingexample ofthe F ibonaccisequencein nature.T hedaisyformsasimilarspiralpatterninthecenter ofitsflower.F ibonaccinumbersinthepetalsforsomeflowertypesareshowninT able3-3.V itruvius,theR omanarchitectandauthorof De Architecture,said,"N aturehasdesignedthehumanbodysothatitsmembersaredulyproportionedtotheframeasawhole."S tudiesshowtheproportionsofarefoundinman.T heaverageheightfor thenavel of a manis.618thetotalbodyheight.1T hehumanbody,includingthehead, has aF igure3-5'F ibonacciQ uarterly.174. Golden Mean of the Human Body.D avis,A ltevogt.pp.340-344.32Fibonacci &T able3-3F ibonaccifiveappendagesattachedtothetorso.T hehandsandfeeteachhavefivefingersortoes.O ursensesalsonumberfive,sight,smell,taste,touchandhearing.T heF ibonaccisequencehasbeenfoundinthearrangementofthesolarsystem.P lanetswithmorethanonemoonhaveaF ibonaccicorrelationinthedistancesfromthemoonstotheplanet.A similarF ibonaccirelationshipholdstruefor thedistanceof theplanetstothesun.2I N T H E A N C I E N T W O R L DT heproportionwasknownandreveredbytheancientGreeks.E uclidsolvedtheproblemoffindingthegoldensectionofaline.P ythagorasincorporatedthepentagramasthesymbolofhisP ythagoreansociety.T hepentagramisconstructedfromthediagonalsofapentagon.E achdiagonalcutstheothersatthegoldensection.S eeF igure3-8.P ythagoraswasaphilosopheraswellasamathematician.F ortheGreeks,thephiproportionrepresented theharmoniousorder throughout nature.T heknowledgeofinancientGreeceextendedbeyondthemathematiciansandphilosopherstotheartistsandarchitects.T hetemples,themostsacredandenduringstructures,werebasedonT heP arthenon'sheightandwidthformagoldenrectangle.T hetempleofP oseidonatP aestum,oneofthebestpreservedGreektemples,alsohasthegoldensectionrectangleforitsheightandwidth.T helengthofthetempleisexactly2.618timesthewidth.T henumberofspacesbetweencolumnsareF ibonaccinumbers,fivebythirteen.S eeF igure3-9.P ythagoras,likeL eonardodaP isa,traveledtoE gypt.I tisbelievedhelearnedhisgeometryfromtheE gyptians.T hegreatpyramidinE gyptisbuiltonthedimensionsofthesquarerootofphi.T heratiooftheheightof thepyramidtohalfitsbaseisasistheratioof theapothemtotheheight.T heapothemisthelengthfromthetoptothemidpointofasideofthebase.T heratiooftheapothemtohalfthebaseisT hesethreeproportionalrelationshipsareshowninF igure3-10.T hedimensionsofthegreatpyramidareoftendescribedintermsof n.F ibonacciQ uarterly.O ct.1970. Fibonacci Series in the Solar System.R ead,pp428-438.33TheSpiralCalendarF igure3-6F igure3-7F igure3-934Fibonacci &F igure3-8AF igure3-11AC =355.5cubitsAB =279.5cubitsBC =219.7cubitsCDimensionsoftheGreatPyramidFigure3-1035TheSpiralCalendarT hisispossiblebecause nisrelated toso thatD idtheE gyptiansusenortodesignthepyramid?A ccordingtothemeasurementsofL ivioS tecchini,thepyramidwasconstructedusingwiththenrelationshipabyproduct,andnottheotherwayaround.3T hevalueofimpliedinthepyramid'sdimensionsis1.61812.T heimpliedvalueof nis3.14418.T heapproximationtcis closer to the mathematical ideal.T heE gyptianuseofpyramidsasmonumentsderivesfromtheM esopotamianziggurats.T heearliestversionsof E gyptianpyramidsresembledthesesteppedstructures.I nterestingly,thezigguratatU rinsouthernM esopotamiawasbuiltasatempletothemoon.A studyofproportionsinarthasfoundexamplesfromM esopotamia.T he"D yingL ioness"relieffromN inevahca.650B C ,nowintheB ritishM useum,containsdiagonalsthatcutthegoldenrectangleatthegoldensections.4T hephiproportionwasnotjustamathematicalentitytotheancients.T heyincorporateditintotheirphilosophy,artandreligions.T heysawtheconnectionbetweenphiandtheharmonicorder of theuniverse.A N D T H E S P I R A L C A L E N D A RT heset of numbersthat formsthebasisof thetheoryof thisbookisthesquarerootoftheF ibonaccisequence,T able3-4.T heratiothatgovernstheproportionofthissequenceis1.2720201...A sthesequenceprogresses,theratioofeachnumbertoitsprecedingnumberapproaches1.272...T able3-4depictsthesquarerootsequenceintwocolumns,labeledoddandeven,toillustrateitsmostimportant facet.Thesquarerootfunctionapplied tothe Fibonacci sequenceresultsintwoseparateadditivesequences.I neachcolumnoftheT able,asthesequenceprogresses,thesquarerootnumberbecomesthesumofthetwoprecedingnumbers.T hegoverningratiooftheseseparatesequencesisthusO nesequencecomprisestheevensequencenumbers,theother,theoddsequencenumbers.I f weconsider thesquarerootsequenceasasinglesequence,T able3-5outlinestheratiosthatgovernthisset.B ecausealternatesquarerootsofF ibonaccinumbersareanadditivesequence,thealternatingratiosarethesameasthesetofratios.T heratioofanytwosquarerootsof F ibonaccinumbersisraisedtothepowerof thedifferenceofthesequencenumbers,or itsinverse.3T ompkins. Secrets of the Great Pyramid, pp 367-368.4F ibonacci Q uarterly. The Golden Section and the Artist. H edian.19 5. p. 407.36Fibonacci &F igure3-11showsthesquarerootrelationshipsinthegoldenspiral.T heradiiseparatedby45degreeswillhaveproportionsof1.272...I nthefigure,theyareshowninunitsofsquarerootsofF ibonaccinumbers.A lternateradiiareat90degreesandhaveproportions.T hetwosetsofradii,diagonal(dottedlines)andhorizontal-vertical(solidlines),areavisualrepresentationofthetwoadditivesequencesof thesquareroot ofO urpurposeinacquiringtheknowledgeoftheproportionsoftheT able3-4T able3-5geometryofthegoldensectionandgoldenspiral,andthepropertiesofthesquarerootoftheF ibonaccisequenceistoapplythesetothedimensionoftime.T heoriginalobservationwasthattheconnectionbetweenthemarketcrashesof1929and1987isaunitoftimebasedontheF ibonaccisequence.T heS piralC alendarisasetoftimeunitswherethenumbersofmoonsaremeasuredinsquarerootsofF ibonaccinumbers.T able3-6showsthefirst35timeunitsintheS piralC alendarcalculatedforeachofthethreenaturaltimeunits:moons,daysandyears.37TheSpiralCalendarT heS piralC alendarT able3-638Fibonacci &T hefirstunitofthecalendaris1moon,thetimeunitofthehumanreproductivecycle.T hetwelfthtimeunitisexactly12moons.A t354days,itiselevendaysshortofasolaryear,365.25days.T hesearetheonlyS piralC alendarunitsthatareanevennumberofmoons.T he15thunitis1.1daysshortofanexacttwosolaryears.T he29thtimeunit,theoneobservedtoconnect1929-1987,is8.5daysshortof 58solaryears.I ndays,the10th,llthand12thS piralC alendarperiodsare219days,278.6daysand354.4.T hesenumbersareclosetothedimensionsofthegreatpyramidinE gyptinroyalcubits,theoriginalmeasuringunitoftheE gyptians.S eeF igure3-10.N otonlydoesthepyramidcontaintheproportionsofbutthemeasurementscloselyapproximateS piralC alendar unitsindays.T heS piralC alendarisasystemofmeasuringtimequitedifferentfromconventionalclocksandcalendars.M ostnumericalsystems,havingdefinedabaseunit,proceedinarithmeticincrementsof thatunit.The Spiral Calendar movesinlogarithmicincrements.T hesecondnumberof thesequenceisnottwicethefirstandthethirdisnotthreetimesthefirst,asinanarithmeticprogression.T hehumanmindnaturallyprefersarithmeticmethodsofcounting.T heconventionalanalyticapproachoftimecycles(regularlyrepeatingintervals)reflectsthisfact.H owever,itisthelogarithmicfunctionthathaslongbeenunderstoodasthegrowthfunction,intimatelyconnectedwiththelifeprocess.T oapplythisideatothedimensionof timerevealsthepatternsoflifeproducedbythelogarithmicbuildingblocksof theS piralC alendar.39P A R T I IM A R K E T S I N T IM EC H A P T E R 4S P IR A LC A L E N D A RR E V E A L E DT heevidenceofS piralC alendaroperationinmarketsisintheempiricalmeasurementsofpastdata.T hequantityandqualityoftheevidencepresentsadramaticcasefortheS piralC alendarasthetimingmechanismof marketbehavior.717M O O N S :P A N I C T O P A N I CI tisafactthatthedistancefrommarkettop1929tomarkettop1987isthesquarerootof aF ibonaccinumberof moons,asisthedistancefrom1929crashto1987crash.W ithoutfurtherexamplesoftheserelationships,thelinkbetweenF ibonacci,themoonandemotionalmarketbehaviorwouldbeacoincidence.I n1799,afinancialpanicoccurredontheE uropeancontinent.H amburgwasthecenterofthepanic,whichC harlesK indleberger,authorof Manias,PanicsandCrashes,datesfromA ugustthroughN ovemberof thatyear.T he29thS piralC alendartimeunitof717moonsis58yearsand10days.F ifty-eightyearsafterthepanicof1799,thelargestpanicyetof the19thcenturysweptbothsidesoftheA tlantic,beginninginA ugust1857intheU .S .andreachingH amburginN ovember.T hisisasecondexampleoftwopanicsseparatedby58years.I n1942,theD owJ onesI ndustrialA veragemadealowonA pril28whichrankswith1932and1974asoneofthemostsignificantbottomsofthe20thcentury.T houghpricesdidnotfallaslowasin1932,tradingvolumeandthepriceofexchangemembershipsreachedlowerlevelsthanin1932.I fwesubtract717moons,or21176days,fromA pril28,1942,theresultisM ay5.1884.D ailypricesoftheD owJ ones41TheSpiralCalendarA veragesarenotavailablefor1884.H owever,S obel,inhisbook, Panic onWall Street,devotesanentirechaptertotheeventsleadingupto,andincluding,thefirsttwoweeksofM ay1884.T hepanicof1884wascalled"Grant'spanic."T heformerGeneralandU .S .P residentlenthisprestigetoabrokeragehousethatcontainedhisson'sname,GrantandW ard.T heGrantnameattractedlargeamountsofpubliccapital.T hefirmalsoborrowedheavilyfromtheM arineN ationalB ankformarketspeculations.W henthemarketcracked,thebankfailedandpanicensued.T hemarketcrackedonM ay3.thebankfailedonM ay6andS obelreported thepanicaswidespreadonM ay10.P R E C I S I O NN owwehavethreeexamplesof panicsorimportantlowsseparatedby717moons:1929-1987,1799-1857and1884-1942.T heseexampleshaveonecharacteristicthatismostextraordinary.T hemeasurementineachcaseisremarkablyprecise.P revioustreatisesoncyclesinmarketsorbusinessinvariablyrefertotheaveragelengthofaparticularcycle.E xamplesoftheK ondratieff,orlongwave,ofapproximately55yearswillincludeoccurrencesof53yearsand57years,withtheproofofthewaves'existencerelyingonanaverage.W hatI amdemonstratinghereisentirelydifferent,aunitoftimethat covers generations,whoseprecision isaccuratewithin days.P recisionseparatestheS piralC alendarfromallprevioustheoriesoftimerelationshipsinmarketsandisitsmost potent forecastingaspect.GO L D E N S E C T IO N SA retheseexamplesportionsoflargergoldensection,orspiral,formations?T he27thS piralC alendartimeunit,443.19moons,or13,088days,is.618timesthelengthofthe29th.A ddthistimelengthtothe1857panicofA ugustthroughN ovemberandyouhaveJ unethroughS eptemberof1893.T hepanicof1893iswelldocumented;ittooreceivedachapter'sconsiderationfromS obel.W eknowfromtheD owJ onesaverages,theninexistence,thatthelowpointof thepanicof1893occurredonT hursday,J uly27,withintheJ unethroughS eptembertimewindowcalculatedfromthepanicof1857.T hetotaldistanceintimefromtheH amburgpanicof1799tothepanicof1893isthe31stS piralC alendartimeunit.T hepanicof1857occursexactlyatthegoldensectionofthisdistance.I fweadd13,088daystothelowofA pril28,1942,theresultisF ebruary26,1978.M arch1,1978wasthedateofthelowestD owJ onesI ndustrialspriceinthefouryears,1976to1980.H ereisaS piralC alendartimeunit35.8yearslongwhichmeasuresfrommarketlowtomarketlow,andthatisprecisetothreecalendardays.S eeF igure4-1.T hisdatecanalsobecalculatedbyaddingthe31stS piralC alendarnumbertothepanicof1884.I fweuseM ay10,1884,whenthepanicwasreportedlywidespread,andadd34,264days,theresultisM arch3,1978,twodaysafter thelow.42Spiral Calendar RevealedF igure4-1T he1987topandcrash,plus13,088daysyieldsJ uneandA ugustof2023.W edon'tyetknowtheoutcomeofthiscalculation.I nsteadofaddingthe.618timeunitto1987,subtractthe1.618timeunit,the31st,whichis1160.29moonsor93.8yearsor24,364daysfromthe1929top.T heresultputsusin1835,theyearofamajormarkettop.S tockpricesdidnotreturntotheir1835levelsforover 25years.F artherbackintime,therecordofsecuritiespricesislesscomplete.T hedataforthepre-20thcenturychartsinthisbookarefromtheF oundationfortheS tudyofC ycles,whichhasreconstructedamonthlyindexofstockprices.T hisindexaccuratelydepictsthetrend and majorswings,but ifamarketisconcentratedinoneortwoissuesandthoseissuesarenotincludedinthereconstructedindex,thedataN omenclatureS piralC alendarperiodsaredenotedbythesquarerootsymbolfollowedbytheletterF andthentheappropriateF ibonaccisequencenumber.T hemoonsymbolindicatesthattheunitofmeasurementismoons.T henumberfollowingthemoonsymbolisthenumberofdaysthedepicteddistancevaries,plusorminus,fromthepreciseS piralC alendarperiod.T helabeleddistanceisthe15thS piralC alendarperiodminustwodays.729-2equals727days.43TheSpiralCalendarF igure4-2maynot bereliable.I havecombinedthisdatawithanecdotalandnewsaccountstoformamorecompletepicture.T he31stS piralC alendarperiodsubtractedfromS eptember3,1929,thedayoftheD owJ oneshigh,yieldsN ovember11,1835.T hisdateisnotconfirmedasatopbythereconstructedindex,whichpeakedinJ une.T heanecdotalevidencefromS obeldiffers1.H ereportsthatthecornerinthestockoftheH arlemR ailroadbeganinS eptember1835andlasted"amatterofweeks,"forcingthestockfrom60to195.T hescandalthataccompaniedtheendofthesqueeze"couldnotendthebullmarketof1835."S tockswere"driftinglowerinlate1835,"hesays,afterthegreatfireofD ecemberconsumedmostof W allS treet.T hisaccountplacesthetopnoearlierthanthebeginningofO ctober,andmorelikelylater,withinonemoonofthecalculateddateof11N ovember.I tdoesnotruleoutthatthetopcouldhavebeenprecisewithindays.F igure4-2showsthegreatsignificance,or magnitude,of both the1929and1835tops.T hethreeexampleseachcomprisethreedateswhich,inturn,formthreeS piralC alendartimeunits.T hetwosmallertimeunitsaddtogethertoformthelargerunit,whichisthereforepartitionedat thegoldensectionbythesmallerunits.S eeF igure4-3.1callthesecontractinggoldensectionsbecausethepatternsappeartocontractinlengthastheymoveforward in time.^obel.pp.58-59.44SpiralCalendarRevealedF igure4-3T heimplicationsofF igure4-3growwhenyoucontemplatethegoldensectionsrelativetothesizeoftheirtimespan.I n1893,tradersinN ewY orkwereactinginamannerthatformsalinkwiththebehaviorof tradersinH amburgnearly94yearsearlier.T heparticipantsofthetopin1987werepartofamuchlargerpatternthatcanbetracedto1835.A 152yearlinkageisamazing.D olargeronesexist?H owB I G?T hereisaS piralC alendarrelationshipcoveringovertwocenturieswithprecision,whichisaweinspiring.T heM ississippischemeandS outhS eabubbleof1720wereamongthegreatestspeculativemaniasandcollapsesof moderntimes.I willfocusontheS outhSeabubbleinE ngland.ItinflatedandburstinJ une-J ulyof1720.T heM ississippischemewastheF renchcounterpart that toppedin J anuary.W hite,inCrashesandPanics,relatesthattherearetwohistoricalaccounts,F recke'sandC astaing's,ofthetimeandpriceofthebubble'speak.2F reckenotesthehighesttradeinS outhS eastockwasat1050onJ une24,1720.C astaingrecordsthe2White. pp. 42-44.45TheSpiralCalendarF igure4-4highestpriceas950onJ uly1.C astaing'spriceislessa10%stockdividend,sobothpricesare roughlyequal.B eforewecanaddS piralC alendartimeunitstodatesofthisperiodwemustadjustforchangesinthecalendar.T heE uropeancontinenthadconvertedtotheGregoriancalendarfromtheolderJ uliancalendarinthe16thcentury.T heE nglishkingsresistedthechange,notwantingtogiveaC atholicpopecreditforagoodidea.E nglanddidnotchangetotheGregoriancalendaruntil1752,sothedatesfortheS outhS eabubbletopmustbeadjustedbyelevendays,thedifferencebetweenthetwocalendars.E levendaysaddedtoF recke'sJ une24dateisJ uly5,1720.T othisdate,addingthe35thS piralC alendarperiodgivesF ebruary10,1966.T heU .S .stockmarketmadeanimportanthighonF ebruary9,1966.T heintra-dayhighof1001.1ontheD owJ onesI ndustrialaveragewasnotexceededuntilN ovember1972,oversixandahalfyearslater.B yanothermeasure,thestockhighofF ebruary9,1966hasnever beenexceeded.I nflationmaskstherealvalueof stock prices.A naccuratepictureoftherealvalueofstocksisobtainedbyadjustingstocksfortheeffectsofinflation.T hisisdonebydividingstockpricesbyanindexofcommoditypricesoranotherinflationgauge.T hemarketin1966reacheditspeakinaclimateoflowinflation.W henstocksrecoveredtothoselevelsinthemid-1970s,thepriceswereexaggeratedbyrapidlyincreasinginflation.F igure4-4showsnearly300yearsofstockpricesadjustedforinflation.T his"real,"or"constantdollar,"indexmarksthe1966topasthealltimehighforstocksinmoderntimes.T hechartshowswhatweallknow,today'sinvestorbuyingtheD ownear3000haslesspurchasingpowerandalowerstandardof livingthanthe1966investorbuyingtheD owat1000.46SpiralCalendarRevealedI fF recke'sdateforthetopiscorrect,theobserveddistancediffersfromtheidealmathematicalS piralC alendarnumberby22.2thousandthsofonepercent!T hehighestrealstockpricesofthe18thand20thcenturyarelinkedbyaS piralC alendartimeunitthatis precisetooneday.C astaingandF recke,takentogether,corroborateatimeframefor the1720bubbletopthatvariesfromthepreciseS piralC alendarunitbynomorethantendaysinaperiodlastingover245years!M A GN I T U D EA tendencyemergesamongtheS piralC alendarmarketrelationshipsdetailedthusfar.T hemarkettopsof1720and1966,separatedbyweretherealvaluehighsoftheirrespectivecenturies.T hetopsof1835and1929,separatedbywerenotasimportantoraslargeinmagnitudeas1720-1966.T helowsof1978and1942,separatedbyonachartdonotappeartobeassignificantcomparedtotheotherexampleswherewaslarger.T hereisaproportionalrelationshipbetweenthesizeofaS piralC alendarsequence number, the degree,and the resulting magnitudeof themarket turn.T hisobservationenablesaforecastertoapplyarelativescaletomarketexpectations;avery powerful tool, even moresowhen combined with precision.S P I R A L SI NT IM EI havedemonstratedtheexistenceofgoldensectionsconstructedofS piralC alendartimeunitsinthestockmarket.W hatothergeometricpatternsbasedonF ibonacciandexist?I ftherearelogarithmicspirals,whatdotheylooklikeinthesingledimensionof time?F igure4-5projectsthetwodimensionalspiralshapeintoaonedimensionalconstructiononahorizontaltimeaxis.T hecenterofthespiralbecomesthepointintimelabeled focus.T hetimeunits,labeledinF ibonaccinumbers,spiraloutfromthisfocalpoint.T hisspiralturnsoutwardclockwise,producingwhatC ookcallsaright-handedspiral.T hepatternmovesbackwardintimefromthefocus.A left-handedspiralwillproducetheopposite,timeunitsspiralingforward,F igure4-6.T heillustrationinF igure4-5isabeautifulrepresentationofthegrowthpatternofF ibonaccinumbers.T hepatterncreatesthreesequencesofF ibonaccinumbers.T henumber34ispartoftheprimarysequenceinwhicheverynumberinthesequenceisanchoredtothefocus.T hesecondsequenceisthedifferenceofadjacentnumbersof theprimarysequence,i.e.34=89-55.T hethirdsequenceisproducedbythedifferenceof thealternatenumbersoftheprimarysequence,34=55-21.T hisisimportantbecause,ifagivenS piralC alendartimeunitliesonalargerspiral,therearethreepossiblepositionsforitonaright-handedspiral.I tmayalsolieonaleft-handedspiral.T hismakessixpossiblelocationsfor thefocus.47TheSpiralCalendarF igure4-5F igure4-648SpiralCalendarRevealedF igure4-7F igure4-3showsthreegoldensectionformations.I neach,thetimeunitclosertothefutureisshorterthantheother.T hatis,thegoldensectioncontractsasitmovesintime.Y oucanfindthreeinstancesof particulargoldensectionformationsinF igure4-5.F or55-34,thereareA B C ,B D F andB C F .T hefirsttwoarecontracting;thethird,B C F ,isexpandingintime.O naleft-handedspiraltherearetwoexpandingandonecontractingexamplesof eachgoldensection.T herefore,anygivengoldensectionhasatotalofthreepossiblespiralsandfocalpoints.F oracontractinggoldensection,twolieonright-handedspiralsandoneliesona left-handedspiral.T hespiralof F igure4-5showsasingleadditivesequence.F igure4-6isaspiralintimeconstructedfromtheS piralC alendarwithbothalternatingadditivespiralscreatedbythesquarerootofF ibonaccisequence.T heunitsoftheoddsequencenumbersareshownwithdashedlines,theevenwithsolidlines.L ikethesingleadditivespiral,therearethreeseparatesequencescreatedinthestructure.O neisanchoredtothefocus,i.e.thesecondiscomposedofalternatesequencenumbers,andthethird by fourth alternate numbers,49TheSpiralCalendarT hereproductivenatureoftheS piralC alendarcreatesreplicasofshapesandsequences.W ithonepieceof thepuzzle,findingthelargerspiral,if anyexist,requiresanunderstandingoftheformitwilltake.T hesetwochartsaretheroadmapsforthepatternswewillencounter.H owtofindthespiralsandfociwillbecoveredlaterinC hapter6.I ntherealworld,someS piralC alendarunitsexistassinglephenomena,othersgroupintogoldensections,stillothersformlargerconstructionsthatdonothavetheregularityoftheidealspiralofF igure4-5.F inally,thereexistperfectspiralsofincrediblebeautyandprecision.I distinguishspiralsfromgoldensectionsbythenumberofturnsinthestructure.Goldensectionshavethreeturns,whilespiralshavefourormore.W hatwasagoldensectioncanbecomeaspiralasmoreturnsoccuronthesequence.O thertimes,thepatternwillstopafterthreeturns,leavingonlythegoldensectionstructure.F igure4-7isaperfectspiraland,likeaflawlessgem,ararity.H erethebeautyofthelogarithmicspiralmeetsthetruthofamarketcreatedbymillionsofinvestorsandtradersmatchingwitsandemotions,profitandloss,fearandgreed,hopeanddespair.I tisincrediblethatwecanconstructanidealmathematicalstructureinF igure4-5andthenfindthatexactstructureinastrikingvisualpatternofstockprices.T hechartshows13monthsoftradingintheD owJ onesI ndustrialaverage.T henumbersonthespiralarethedistanceincalendardaysfromlowtolowandfromlowtofocus.T helegendatrightrevealstheoddsequenceS piralC alendarlengthsindays.N otethe precisionwithwhichthestructureadherestotheidealS piralC alendarnumbers.T hefocus,A pril6,1990,isnota market turningpoint,butitisthedefiningpointforthefivelowsinthestructure.S omepointsareconnectedtoeachotherbyS piralC alendarunits,butallpointsareconnectedbyS piralC alendarunitstothefocus.I usethedateofthefocustonamethespiral,makingitthe April 6,1 990 spiral.T heA pril6,1990spiralis"perfect"becauseof itsregularfeatures.T herearenomissedturns.F romthefirstturninthespiral,everyalternateS piralC alendar numberresultedinaturn.A llturnsarethesametype.H eretheyaremarketbottoms.T helowof J une1989,thefirstinthesequence,isnotaslargeinmagnitudeasthesubsequentlowofO ctober1989.T he"minicrash"isofsmallerdegreeonthespiral.T heconceptofmagnitudewouldsuggestitshouldhavebeenlesssignificantamarketturnthantheJ unelow.H owever,the"mini-crash"isthelargermagnitudelowbecauseitliesontwospirals.S piralscannesttogether,sharingacommonpointbetweenthem.F igure4-8isalargerspiral,withoneofthelowstheO ctober"minicrash."T hedegree,orsequencenumber,ofthe"minicrash"onthelargerspiral,showninF igures4-8and4-9,isgreater thananylowsonthesmallerspiral,showninF igure4-7.T hisexplainswhyitsmagnitudeisgreaterthantheotherlowsonthesmallerspiral.T hislargerspiralisalsoperfect.I tconsistsonlyof lows,andeveryalternateS piralC alendarpointhasproducedaturnasofthiswriting.T heturnsofthisspiraldefinethemajorpatternoflowsofthe50SpiralCalendarRevealedF igure4-8F igure4-951TheSpiralCalendarF igure4-10mid-1980sandearly1990s.T hefocusisJ anuary8,1993.O nceagain,theprecisionisamazing.T hesmallerspiralsharesmorethanthecommon"mini-crash"pointwiththelargerspiral.T hefocusofthesmallerspiral,A pril6,1990,cutsthedistancefromthemini-crashlowandtheJ anuary1991lowatthegoldensection.S eeF igure4-9.T hosedistancesarethemselvesS piralC alendarunits.T hesmallerspiralnestsinthelargerspiral.T hepatternsarethesame,themagnitudeanddegreearedifferent.T helargerspiralof F igure4-8sharesapointwithastillgreaterspiral,illustratedinF igure4-10.T hegreatspiralinF igure4-10hasitsfocusinthe21stcentury,D ecember25,2015.I tsfirstlowoccurredinthe19thcentury.I tisperfectinthateveryalternateS piralC alendarunitsincethefirstoccurrencehasproducedaturnandeveryturnhasbeenthesame,alow.T hecrashof1987isthepointcommontothetwospirals.I fthecrashlowhadfallenpreciselytothedayontheJ anuary1993spiral,itwouldhavebeenS aturday,O ctober17.T heprecisemeasurementfortheD ecember2015spiralisO ctober24.T heprintlowontheD owJ onesin1987wasO ctober20,effectivelysplittingthedifferenceof thetwospiralmeasurements.T herearesomecharacteristicsthatsetthisgreaterspiralapartfromthefirsttwoexamples.T heseturnsfallontheevensequencenumbersoftheS piralC alendar,whiletheotherspiralsareontheoddsequence.B ecauseofthisfact,thefocusoftheJ anuary1993spiraldoesnotlieonagoldensectionofthegreatspiralofD ecember2015.T he52Spiral Calendar RevealedF igure4-11lowof1942inthisspiralisnotprecise.I tvariesfromtheidealby32days(lunation+3days).T urnsthatlackprecisionI callapproximate.Iwilldemonstratelaterhowturnswillclusteratpointsthatareprecisetooneormorestructuresandapproximatetoothers.R emember,the1942lowfallspreciselyonadifferentstructure,agoldensectionthatincludesthe1884panicandthe1978low.T H EF ocusT hefocusofaspiralorpartialspiralwilltendtofallonamoonphasenearasolarseasonalchange,and/oranemotionalmarketturningpoint,oragoldensectiondivisionofalargerspiral.E achofthespiralsinF igures4-8and4-10hasafocusthatisthefirstfullmoonafterthewintersolstice.T hiscombinationoffullmoonandwintersolsticeseemstoproducethemostperfectofspiralsthatareassociatedexclusivelywithmarketlows.Ihaveonlyfoundthemtomovebackwardintime,notforward.T hesecondfrequentfocal53TheSpiralCalendarF igure4-12pointisthenewmoonnear(usuallyafter)thespringequinox.T hismoonwillproducespiralsthatmoveforwardorbackward.T heforwardspiralsareofsmallerdegree,producingturnsofsmallermagnitude.T hesespiralsproducetopsorbottomsandcanproduceturnseitheronanalternatingS piralC alendarsequence(oddoreven)oronconsecutivesequencenumbers.T henewmoon,springequinoxspiralswillmorefrequentlyhavemissingturns,i.e.thepatternskipsa beat.F igures4-11through4-14shownewmoon,springequinoxspirals.T hespiralwiththefocusofS pring1982,showninF igure4-11,hasturnsspiralingbackwardsontheoddS piralC alendarnumbers.T hespiralcontainsfourhighsandthreelows,foratotalofseventurns.T wooftheturnsvaryfromtheidealbytwelveandeightdays.T hesedifferencesaretoogreattobeprecise.F igure4-12isanewmoon,springequinoxspiralmovingbackwardsfrom1963.T heturnsreflecttheoddsequence,skippingO neturnalsofallsontheevensequenceV F 20.T hisspiraldefinessomeof themajorturnsofthelate1950sandearly1960s.F igures4-13and4-14areforwardspirals,using54SpiralCalendarRevealedF igure4-13F igure4-14TheSpiralCalendarF igure4-15consecutivesequencenumbers,fromthenewmoon,springequinoxesof1976and1987.B othgeneratedturns,mostlytops,onconsecutiveS piralC alendarpoints.T hespiralsterminatedattheO ctoberhighsof1978and1989,whichprecededO ctobermini-panicsinbothyears.T hefocusof F igure4-14isanexampleof afocusfallingonalunar-solaralignmentandnearanemotionalmarketturn.M arch23,1987wasthehighfortheadvancedeclinelineinthebullmarket from1974.A nextraordinaryforwardspiralinmagnitudeandprecisionisshowninF igure4-15.H erethefocusisthefullmoonfollowingthefallequinox.T hisistheonlyexampleIhavefoundofthislunar-solaralignmentproducingaspiral.M oreimportant,thefocuswasapointofextremeemotionalmarketbehavior.T hefullmoonofO ctober1946wasonthe10th.T hemarket'sclosinglowfor1946wasO ctober9.O nthemorningofthe10th,stockstradedlowerstillbeforerecoveringandclosinghigher.F igure4-16showsthedailypricesforthesummer andfallof1946.E motionalmarketsareone-sidedmarkets.T hesearetimeswhenalmosteveryoneisonthesamesideofthemarket.P anicsareone-sidedsellingwavesandmaniasaretheirbuyingcounterpart.T headvance/declineratioisanexcellentmeasureofone-sidedextremesinmarkets.T hisistheratioof thenumberof stocksthatclosehigheronthedaydividedbythenumberofstockslowerontheday.F igure4-16containsadvance/declineratiosforsevendaysin1946.T hesixnegativeratiosonthechartareamongthe10056SpiralCalendarRevealedF igure4-16mostnegativeadvance/declineratiosfrom1934tothepresent.T hetwoworst,1:31and1:28,arerankedeighthandeleventhinarankingofthe100lowestratiosfrom1934.T hisshowstheextremeone-sidedsellingleadingintothelowofO ctober1946.O nO ctober15,theratioreachedtheoppositeextremeof28advancestoeachdecline,thesecondbestratiofrom1934.T hemarketlowofO ctober1946sawadramaticchangeintheemotionalsentimentoftradersandinvestors.T heone-sidedurgetosellwasreplacedwithanequallyone-sidedurgetobuy.T hepointofextremeemotionalinflectionwasthefullmoonafterthefallequinox.L ookattheconsequencesofthispointintimeasthefocusofaS piralC alendarsequenceintothefuture.T hespiralforwardfrom1946producedthetopofD ecember1968andthelowsofD ecember1974andA ugust1982onconsecutivesequencepoints.T heprecisionisuncanny.N opointvariesfromtheidealbymorethantwodays,eventhough35yearsseparatethe1982lowfromthefocus.T hethreeturnsproducedarearguablythemostimportantturnsofthe14yearsfromthelate1960sthroughtheearly1980s.T hemagnitudeoftheturnsiscommensuratewiththedegreeoftheS piralC alendarsequences.H erethebeautyof themathematicalideal joinswiththetruthof man'spatternof marketbehavioroverdecadesof time.T hespiralforwardfrom1946posesthelogicalquestion:whyisthe1968topthefirstpointonthesequence?W hyaretherenopreciserelationshipsofsmallerdegreebefore1968?I ngeneral,thisisanissueofmagnitude.L argemagnitudeeventsareconnectedwithotherlargemagnitudeeventsbylargedegreeS piralC alendarrelationships.T helargeemotionallowof1946isnotexpectedtorelatetosmallturnsclosein time.Icanoffer nospecific reason whymoonsisthefirstsequencenumberandnotmoonsormoons.A notherexampleisF igure4-4.T heemotional1720mania wasmuchgreaterthan1946.A contemporarydescribedthetradingscenein1720,"A sifallthelunaticshadescapedoutofthemadhouseatonce."T hisisconsistentwithmagnitudeinthattheverylargedegree,moons,isproportionaltothesizeofthemaniaitmeasuresfrom.T he1966topisthefirstsequencetomeasurepreciselyfrom1720.1expectthatthisturnwillbepartof alarger pattern,butthereisnoexplanationfor57The Spiral Calendarwhythe1966topisthefirstrelationship.T he1720high,ifF recke'sJ uly5(Gregorian)topdateiscorrect,occurrednomorethanonedayfromanewmoon.T hismeans,thatasafocusintothefuture,1720wasthenewmoonfollowingthesummersolsticeandaveryemotionalturningpoint.O neof thegoldensectionexamplesshownpreviouslyispartofalargerspiral.T hegoldensectionformationoflows,1884-1942-1978,hasthreepotentialfoci.T he1884paniccouldbethefocusofaforwardspiral;the1978lowcouldbethefocusofabackwardspiral;or,thespiralmovesbackwardandthefocusis18-21F ebruary,2036.IwillexplaininC hapter6howthislaterdateisfigured,althoughitcanbededucedfromF igure4-6.B ycalculatingbackfrom2036,aspiralthathasasplitfocusappears.T heoddsequencedatesmeasurebackto1978,1942and1884.A secondfocusabout32dayslatercalculatesusingtheevenS piralC alendarsequencetothehighofN ovember1916,thepaniclowof J une1962andtheK uwaitiwarslideof A ugust1990.S eeF igure4-17.W hythesplitfoci?T heevensequencefocusbestcalculatesasM arch21-25.T hisisbetweenthespringequinox,M arch20,andthenewmoon,M arch27.T hefocusoftheoddsequencefallsintheeclipseseasonbetweenthetotalsolareclipseofF ebruary11andthelunareclipseofF ebruary26.T hechartshowstheprecisemeasurementscalculated from thedatesshown.T hespiralexhibitsgoodprecisionfromeachofthesplitfoci.M ovingforwardintime,thereisamarkeddroppingoffinthemagnitudeoftheresultingturns.T he1978lowislesssignificantthanallofthepreviousturns.T heA ugust1990lowdidnotholdformostaverages,althoughitwasthelowfortheyearinbondpricesandtheutilityaverage,thehighpriceforgoldandthepointofmaximumdownsidemomentuminstocks.Iexpect thisspiral'sinfluencein thefuturewillcontinueto lessen.H A R M O N YA llS piralC alendarunitsarenotequal.S omeperiodsseemtooccurmorefrequentlythanothers.T hisfactmaybeafunctionof howacertainS piralC alendarunitrelatestothenaturalunitsof time,moons,solaryearsandsaroscycles.I faS piralC alendardistanceapproximatesawholenumberofnaturaltimeunits,itisin harmonywith that timeunit.T he29thS piralC alendarunitisalmostexactly717moons(717.0976).T hisislunarharmony.I tisalsoalmostexactly58years(57.9776).T hisissolarharmony.T he12thS piralC alendarunitisexactly12moons,producinglunar harmony.T he15thS piralC alendarunitiswithinonedayof beingexactly2years(1.99685years),producingsolarharmony.T heeffectsoflunarandsolarharmonyarediscussedinC hapter5.T he27thS piralC alendarunitisalmost2saroscyclesinlength(1.98740).T hisis saros harmony.F igure4-18demonstratessarosharmonyatwork.I nM ayof1901,arisingmarketfueledbytakeoverspeculationintransportationstocksledtothepanicof1901(shadesof1989!).The New York Timesheadlineof M ay10read:58SpiralCalendarRevealed59TheSpiralCalendar"D IS A S T E R A N D R U I N I N F A L L I N GM A R K E TP anicW ithoutaP arallelinW allS treetT hegreatestgeneralpanicthatW allS treethaseverknown..."O n that day, a short squeeze in N orthern P acific drove the stock from its previousday'scloseof160upto1000dollarspershare.T hestockthencollapsedbacktoclosethedayat325.T hesqueezeforcedmassiveliquidationinotherissuesasspeculators,whowereshortN orthernP acific,raisedneededcash.T hismarkedtheendofthebullmarketof1901.T hedateofthepanic,M ay9,wasafterthelunareclipseofM ay3andbeforethesolareclipseofM ay18.T hissolareclipsebelongstothesarosseriesthatproducedthelongesttotalsolareclipsesofthiscentury.T hreehadadurationofoversevenminutes.T heeclipsesinthissarosduringthiscenturywere1901,1919,1937,1955,1973and1991.F igure4-18showshowthetwomarkettops,1937and1973,measureforwardfromthepanicof1901.T hetopsmeasurepreciselyusingmoons.T hetopsdonotmeasurefrom theeclipseof1901,butfromthedateof thepanicitself.T hemeasurementisnotthatoftwosaroscycles,butratherthatoftheS piralC alendarunitthatisinharmonywiththesaros.W ecansee theeffect of theeclipsecycleonthemarket here,butitisveryimportanttounderstandthat,ultimately,itistheS piralC alendarthatdeterminesthepattern,not thesaros.T hestructureinF igure4-18,inwhichthesamesequencenumberrepeats,isconsistentwiththespiralpatternswehaveseen.I nF igure4-7,youcanseehoweachsequencenumber repeats.I t is108daysfrom the J une'89lowtotheO ctober'89lowandthen106daysfromtheO ctober'89lowtotheJ anuary'90low.I f thesaroschartispartofalargerspiral,itsfocuswouldbemoonssubtractedfrom1901ifitisforward,andmoonsaddedto1973if itisbackward.P lacingthefocusatthosepointsyieldsnootnersignificantrelationships.T hestructure,then,isapartialspiralcreatedfromtheoccurrenceofapanicinproximitytoaneclipseandtheS piralC alendarnumberinharmonywith thesaros.F O R W A R D vs.B A C K W A R DI tisarbitrarythatI callspiralsmovingbackintime,"righthanded"andspiralsmovingforward,"lefthanded."F igure4-5couldjustaseasilyhavealefthandspiralattachedtothetopofthetimelinetoproduceabackwardspiral.I havefoundthebackwardspiraltobemorecommonthantheforwardspiral.T hisisconsistentwithotherformsinnature.C ookreportsthatinspiralshells,onetype,whichhecallsrighthanded,isbyfar themorecommon.B yapplying thisconcept totime,weareraisinga fascinatingissue.T heimplicationoftheforwardspiral,withitsfocusprecedingthespiral,isofapasteventgrowingintoandaffectingthefuture.T hisisnotaneasyconceptforthosewhocannotbelievethatmarketeventsof58or245yearsagocouldbedirectingthepatternofcurrenttrading.T heimplicationofthebackwardspiralisevenmorecontroversial.I nthebackwardspiral,thefuturefocusseemstodeterminethecourseofSpiralCalendarRevealed61TheSpiralCalendarF igure4-19eventsinthepresent.I tmaybejustanillusionthatthefutureappearstoaffectthepresent.T headditivenatureof thespiralin thepast isprobablytheultimatesourceof thespiral's direction, not its focus.N evertheless,it remainsunexplained whythefuturefocustendstobealunarphasenearasolsticeorequinoxora pointonanotherspiral.A P P R O X I M A T I O NT hereareS piralC alendarrelationshipsthatarenotprecise,yetformstrikingrepetitivepatterns.F igure4-19showsthreedistancesthatapproximatemoons.T hevariationsfromtheidealrangefrom 60to114days,or 2 to4moons.E achof theturnsisprecisetoothersequencesinthischapter.T hechartalsoshowstheselengthsasratiosofthelarger1929to1987distance.T heyarecloseto.786....S piralC alendarrelationshipscanseemconfusing,asseparateprecisestructuresappeartobearnorelationshiptoeachother.T heconnectionofthesedisparateelementsthroughapproximationfurther revealsthe structure and beautyof theS piralC alendar'spatterns.T hereisanother explanationforwhyaturnmaylackprecision.A marketturncantakemorethanoneform.I nadditiontodramatic"spike"turns,marketsmayform62Spiral Calendar Revealedgradualturnsthattakea"rolling"shape.T heexactextremeof thegradualturnwilllikelyoccurondifferentdatesinvariousindices.T heturnmaybepreciseinoneindex,butnotanother.Gradualturnsaremorelikelytobemarkettopsthanbottoms.S U M M A R YE motionalmarketturnsareconnectedbydistancesinwhichthenumberofmoonsequalsthesquarerootof F ibonaccinumbers,formingtheS piralC alendar.T heselengthsoftimesgrouptogethertoformlargerspiralsandsectionsofspirals.A spiralistheshapecreatedbyprojectingthesquarerootofF ibonaccisequenceontothedimensionof time.F igures4-5,4-6.S piralC alendarrelationshipsseparatingmarketeventscontainstriking precisionintheirmeasurements.T hedegree,orsequencenumber,oftheS piralC alendarunitinamarkettimespanisgenerallyproportionatetothe magnitudeof themarketevent.T hepointatwhichaspiralanchorsisthe focus.T hefocusofaspiralorpartialspiralwilltendtofallonamoonphasenearaS olarseasonalchangeand/oranemotionalmarketturningpoint oragoldensectiondivisionofalargerspiral.S piralC alendarunitsthatareevenlydivisiblebyothernaturaltimeunitsareinharmonywiththoseunitsandmayhavemorepronouncedorfrequenteffectsonmarkets.M arketturnsthatexhibitnearS piralC alendarmeasurementarein approximation.63C H A P T E R 5T H E C A U S EO FP A N I C S A N D C R A S H E SA U T U M N F A L L ST hepredilectionformarketpanicsinthefallseasonisnotarecentphenomenon.T hepanicof1873occurredonF riday,S eptember19.T heopeningparagraphinThe NewYork Timesthefollowingdayreported,"Itisasingularcoincidencethatthegoldcornerof1869,whensomanywereruined,andtheculminationofthepanicof1873,happened...inthesamemonth."I nthiscentury,twomajorcrashes,1929and1987,aswellasthemini-crashof1989,occurredinO ctober.T herewerealsothe"O ctoberM assacres"of1978and1979.A nexaminationofthesefiveyearsinS piralC alendartermsrevealsa setof seasonalpreconditionsfor O ctober panics.I amonlyexaminingtheyearsthatcontainamarkettopinO ctober,whichisfollowedbyextremeweakness.O theryearswithsharpdeclines,including1937,1957and1990,werecontinuationsofdeclinesthatbeganbeforeO ctober.T hefallseasonmarkstheendofthegrowingseasonsofspringandsummerandtheonsetofdecayaslifepreparesforthehibernationofwinter.P anicsareasharptransitionfromrisingmarketstodecliningmarkets,justasanearlyfrostinthefallcanbeasharptransitionfromsummertowinter.M arketssheddingpricesandtreessheddingleavesarenaturalexpressionsoftheseason.T heseasonsplayaroleinlife,withspringandsummeralliedwithgrowthandlife,andfallandwinterconnectedwithdecayanddeath.T heseasonsarenotaloneinaligningthemselvesononesideortheotherofthegrowth-decayequation.I believethenewmoonbelongswithspringandsummerasapositivelifeforce.T hefullmoonalignswithfallandwinterasamanifestationofdecay.64The Cause of Panics and CrashesT able 5-1T hemarketpeaksof1929and1987werewithin24hoursofnewmoons,aswasthe1720maniapeak.T he1946emotionallowandspiralfocuswasonthefullmoon.I amnotsuggestingthatnewmoonsarealwayspricetops,orfullmoonspricelows,justthatthesemoonphasesareeachassociatedwithonesideof thegrowth-decayprocess.T hefiveyearsinthe20thcenturythatsawO ctobertopsfollowedbyextremeweaknessbelongintwocategories.T hepanicyears,1929,1987and1989,haddistincttopsandbottomssandwichingaperiodofpricefreefall.T hemassacreyears,1978and1979,hadpricetopsprecedingsharpdeclines.T hemassacreyearsdidnotcontainthedramaticonedaydeclinesof thepanicyears.A llfiveyearshaveO ctobertopsmademoonsfromthespringequinox,newmoontwoandahalf yearsearlier,withavariationof +2to-4days.T heaveragedistancefromnewmoon,springequinoxtomarkettopwas926.2days,versustheS piralC alendar927.6days.T hethreepanicyearsmadelowseitherorbackwardsfromfullmoonsfollowingthewintersolstice,withavariationofnomorethanthreedays.T able5-1containsthedatesofthenewmoonstwoandahalf yearspriortotheO ctobertops,thedistancefrommoontotopandthedateofthetop.F orthepanicyears,thetableshowsthedatesof thelows,therelatedfullmoon,wintersolsticeandtheinterveningS piralC alendarperiod.T hesethreeS piralC alendarperiodsofyearscontainseasonalharmony,thatis,thetimelengthsarewholenumbersofseasonsaswellasS piralC alendarperiods.The half year increments forward fromthe springand thequarter yearincrements backward from the winter collidein the fall.SeeF igure5-1.TheSpiralCalendaranO ctoberslidetooccur.T hisobservationisimportantforforecasting.A ftertheexperiencesofthelate1 980s,everyonelooksovertheirshoulderforthenextO ctoberpanic.M anyforecasterspredictedcalamityforO ctober1991.T woandahalfyearsearlier,thefullmoonfelltwodaysafterthespringequinox.T helunar-solaralignmentprerequisitedidnotexistin1991.T heresult,nopanicthat year.F igures4-13and4-14depictnewmoon,springequinoxspiralsmovingforwardandcreatingO ctobertops.F igures4-8and4-10showthefullmoon,wintersolsticespiralsmovingbackwardtothelows.T hesearebroughttogetherinF igure5-2,whereweseethespiralgeometryofthe1987crash.T hesearetheprecisemeasurements"andalignmentsthatcausedtheextraordinarymasspsychologicalpanic.I nanytypeofsystemordynamic,itisduringtheextremesthatanunderlyingstructurebecomesmostapparent.M eteorologistscravetheopportunitytohaveafullsetof instrumentsinplaceatthevortexofatornado,sincesomuchwouldbelearnedaboutalltypesofweather.T hecrashisthemarket'sversionof thetornado.H ere,theinstrumentsof time,measuredinlunarunitsofthesquarerootsoftheF ibonaccisequence,revealtheunderlyingstructurewithallthebeauty,truthand precisionof nature'sother creations.T hediagramof the1987crash,F igure5-2,ill