15_OER

586

GEOMETRY 587

15

Geometry

The Wild SurmiseWasthereeveramoredaringexampleofextremeschol-

arshipthanJosephSmithsannouncementoftheforthcom-ing publication of his translation of the Book ofMormonfromthegoldplatestowhichhehadbeenintroducedbyanangel?Inlesstimethanittakesacollegestudenttoproducea respectable term paper and after devastating advancenotices in the press, the twenty-five-year-old dirt farmerfromupstateNewYorkhadprepareda588-pagebookcov-eringeverymajoraspect in the lifeofanancient civiliza-tionoveraperiodofonethousandyearsandwasdiligentinplacingitinthehandsofaninvinciblyhostilepublic.

The answer to our opening question, by the way, isYes,therewasanotherevenbolderventure:fiveyearslaterwhen Smith surpassed his first effort by laying out first-handaccountsbytheancientleadersofthesevenmajordis-pensations of sacred history. The separate histories rangein length fromsingle chapters to elevenpages (five chap-ters)oftheBookofAbraham.ThebooksofAdam,Enoch,Noah, Abraham, Moses, Jesus Christ, and Joseph Smith,each giving a firsthand account of his dispensation,maynowbechallengedandtestedbyalibraryofancientapoc-ryphalwritings,whichGzaVermscallsTheRewritten

Bible.1 In a time when those apocryphal writings werealmosttotallyunknowntotheworld,JosephSmithcouldbehisusualunshakablyconfidentself;buttodaywehavethelibraryof texts tosupportorrefutehim.Wehavealreadyshownhowsomeoftheaboveancientauthors,mostnota-blyEnoch,Abraham,andMoses,havebeenvindicatedbynewlypublishedtexts.InoneinstanceJosephSmithhasrunwhathiscriticshavecalledhissureannihilationnamely,theinterpretationandtranslationofcertainEgyptiantexts.When he published three facsimiles from three Egyp-tianpapyriwithhisowninterpretation,hegavehiscriticssomething theycouldreallyget their teeth into.Thefinalandhardesttest,theinfalliblecoupdegrcewouldbetheEgyptiandocumentswhich JosephSmith interpreted andtranslatedtherehewouldnothavea chanceagainst theexperts.Butasweshowinchapter1,thethreeattemptstodiscredithimin1896,1912,and1967allstruckout.Hereanotherapproachisuseful.

The lifelong trials of Abraham begin there. Here wehavetwoscenes:(1)AbrahamonanaltarinFacsimile1and(2)AbrahamonthethroneinFacsimile3.Somethingsensa-tionalmusthavehappenedbetweentheevents,andthisisindicatedbythemajesticexpanseofthecosmos,whichtheherohadtopassthroughtogetfromtheformertothelattercondition.On this subject, surprisingly,wehaveawealthofancient literature.2Thestory is somewhatgarbledas todetails,forSatanandhisagentsmademorethanoneattempttodestroythehero.Butrightatthepointofdeaththeangelarrives, asdepicted, andnotonlydeliversAbraham fromthealtarandthefire,butconductshimonaflightthrough

1. GzaVerms,Scripture and Tradition in Judaism: Haggadic Studies(Leiden:Brill,1961),67,95. 2. Seethediscussionofthe Testament of Abraham andthe Apocalypse of Abraham inchapter9,TheAscensionDramas, in thisvolume,pp.****.

588 ONEETERNALROUND GEOMETRY 589

theheavens,eventothesixthheaven,wherehebeholdsGodonhisthrone,aswellashisownthronewhichawaitshim.

On their return journey to earth, the angelpointsoutthevariousheavenlybodiesandanswersAbrahamsques-tions,instructinghimtomakeachartoramapofwhathehasseenwhentheyreturntoearth.Ancientpilgrimstoholyplaces,usuallytemples(whichwerethoughttobereplicasofheavenandearth),werewonttoleave,usuallyatthegateoftheshrine,awrittenaccount,sometimesinscribedonastele,oftheirmomentousjourney.Thepilgrimalsoreceivedanofficialtokenattheshrine,likethefamousseashellsofSantiagodeCompostela (fig. 68), bothasproofofhisdis-tinctionasanendowedpilgrim,liketheMuslimhajj,3andasacontributiontotheholinessoftheplace.WecansafelyassumethatourFacsimile2issucharecord.

Theplacingof thehypocephalus(Fac.2)betweenearthand heaven (Facs. 1 and 3) points to its function as a linkbetween the two. In this it performs the same function asthe temple,which is, as itwere, adownwardprojectionofheaventoearth(seep.**,fig.42B).Themessageofthehypo-cephalusisbroughtoutinasubtleandsecretway,farbeyondthecapacityofJosephSmithoranyofhiscontemporariestoconceivenamely,inawaysimilartotheancientmysteriesofPythagoras,theKabbala,Templars,Masons,Rosicrucians,etc.,wherethemembersidentifiedthemselvesbysecretsignsand tokens. Themost remarkable of these are two starssingular geometrical emblemsthe pentagramor pentacleand thehexagramor Solomons seal, sometimes called theshieldorStarofDavid.Thesesealsbindheaventoearthandthepilgrimtoboth.Theyaretwostars,thetwomostsacredemblemsintheancientworld,andtheirfunctionistoblendtogetherallthingsholyinbothtimeandplace.

3. SeeHughNibley,SacredVestments,inTemple and Cosmos: Be-yond This Ignorant Present, CWHN 12(SaltLakeCity:DeseretBookandFARMS,1992),127,fig.31.

Figure68.StephenIIIPraun(154491),aGermanpilgrimtoSantiagodeCompostela,preservedhisbeautifullymadewalkingoutfit,completewithstaff,sandals,andraincoat.Havingreachedtheshrine,hewasentitledtowearthecockleshellsymbolonhishatandcoat.Inv.-Nr.T550556,StephanPraun,pilgrimcostume,1571.GermanischesNation-almuseum,Nrnberg.


Far from being rooted in magic or superstition theseobjectsaresupremelyutilitarian;thesolidstructureofreal-ityinourownworlddependsonthem.Theyaretheprac-tical, nay indispensable, tools of the builder. By them thepillarsofSolomonstempleweresetuptoestablishbeforethe world the meeting place of heaven and earth.4 Theyembodytheworkofcompassandsquare,foundcarvedonthetombofanoblemanin1484(fig.69).5Tothisdaynoonecanexplainhowtheycanoffertheirparadoxicalservices.

4. SeeHughNibley,TheCircleandtheSquare,inTemple and Cos-mos, CWHN 12:15051,fig.37. 5. J.-P.Laurant,Lepentagrammeetltoileflamboyante,Revue de lhistoire des religions180(1971):12729.

Figure69.ThefuneraryplaqueofGuillaumeLetelliercommemorateshisdesignofaGothicchurch,asshowninthescrollattheupperright.Hisskeleton,areminderofthecertaintyofourowndeath,holdsalargecompasswithaset-squareinside,representingjudgmentandthesymbolicuseofhisdraftingtools.Onthelowerright,theplumb-boblevel,settingmaul,andtrowelrefertotheactualconstructionofthechurch,whichhedidnotlivetoseecompleted.17th-centurydraw-ingof1484marbleoriginal(nolongerextant)inChapeloftheVirgin,ChurchofNotreDame,Caudebec-en-Caux,France.

The Measure of All ThingsHypocephali as suchare singulardocuments; to reca-

pitulatebriefly:1. There are a littlemore than 100 surviving from a

periodof500years.2. Theywereprivateandintimatedocumentsbelong-

ing to a very limited social class, families of priests andpriestessesofAmon-Re.

3. Theyweredesignedtothepersonaltasteandoftenbythehandoftheowner,whopassedthepracticedownfromfathertoson.

4. They are found in fewplaces, notablyThebes andTelel-Yahudiah,theJewishsettlement.6

Since themorecompletehypocephalialldisplaysimi-lar symbols in a similar set order,wemust conclude thattheyare far frombeing thecasual jumbleof superstitioussignsthatmostEgyptologistshavechosentoignore.Moreimportant, thereare certain featureswhich set the JosephSmithhypocephalusapartfromalltherest,includinginter-esting trivia suchas that (1) ithasbeenuntil recently theonlyhypocephalusintheWesternHemisphere; infact, itsimplausibleappearanceon theWestern frontier in1835 isquitesurprisingandhasbeennotedbysomewritersonthesubject.7Thenagain,(2)itisthefirsthypocephalusevertobepublishedandinterpreted.ThenextpublicationwastheFlorencehypocephalus,which,although themummywas

6. AllofthisinEdithVarga,Lestravauxprliminairesdelamonog-raphiesurleshypocphales,Acta Orientalia Academiae Scientiarum Hun-garicae12(1961):23547. 7. ElmarEdel,Hypocephalus,inLexikon der gyptologie, 7vols.,ed.WolfgangHelckandEberhardOtto (Wiesbaden:Harrassowitz,197592),3:693;HansBonnet,Reallexikon der gyptischen Religionsge schichte,2nded.(Berlin:deGruyter,1971),390;ConradLeemans,Hypocphale gyptien du Muse Royal Nerlandais dAntiquits Leide,ActesdusiximeCongrsIn-ternationaldesOrientalistes,tenuen1883Leide.Pt.4,sect.3(Leiden:Brill,1885),95.


unrolledinSeptember1827,wasnotpublisheduntil1855.8Samuel Birch takes due notice at the time of the JosephSmithhypocephalusbutproteststhattheinscriptionissobadlyengravedthatitisimpossibletomakeoutitsmean-ing, and Smiths interpretation throws no light upon it.9(3)Inthemid-nineteenthcentury,somescholarswerequiteexcited about hypocephali but suddenly lost interest, andbetween 1896 and 1942 therewere only four publicationsdealingwithhypocephali.10Hence,ofall thehypocephali,theJosephSmithalonehasbeentheobjectofseriouscon-templationthroughtheyears.

TheauthoritiesoftheChurchhavewiselyabstainedfromissuinganystatementsonthe interpretationof thefacsimi-les.TheBookofAbrahamstillawaitsfurtherrevelationwithstrict limitations for the present: writings that cannot berevealeduntotheworld(Fac.2,fig.8,explanation), thingsthatoughtnottoberevealedatthepresenttime(Fac.2,fig.9,explanation),andthingswhichwillbegivenintheownduetimeoftheLord(Fac.2,figs.1221,explanation).OthersaretobehadintheHolyTempleofGod(Fac.2,fig.8,expla-nation).Meantime,Iftheworldcanfindoutthesenumbers,soletitbe.Amen(Fac.2,fig.11,explanation).(4)TheJosephSmith hypocephalus is the only onewhose owner bears apharaonicname,Sheshonq,Solomonsfather-in-law,theonlypharaohnamedintheBibleandtheonemostcloselyassoci-atedwithIsrael.(5)ThefrequentreferencetoHeliopolisonthe Joseph Smith hypocephalus ties it to the headquartersof the Jews inEgypt. (6)Aspecialoddityabout the JosephSmithhypocephalus is that forall thehundredsofcurious

8. ArcangeloM.Migliarini,AccountoftheUnrollingofaMummyatFlorence,belongingtotheGrandDukeofTuscanybyProfessorMi-gliarini:TranslatedfromtheItalianMs.ofProfessorMigliarinibyC.H.Cottrell,withSomeNotesandObservationsbyS.Birch,Archaeologia36(1855):161. 9. Ibid.,173. 10. Varga,Lestravauxprliminaires,23637.

viewerswho examined it inKirtland andNauvoo, no onehaseverdescribeditasaseparatedisk,cushion,oranythingbutaregulardrawingonpapyrus.Whatshouldhavebeenitsmost striking feature isnevermentioned. It containsnomentionofthestandardfunctionofthehypocephalus,asacushionundertheheadorheadrest tosupplywarmthbs,flameforthehead.ThustheFlorencepapyruswasfoundover thehead, between theprotecting cowl and thebodyofthemummy,andiscomparedwithanotherhypocepha-luswhichwasabovetheocciput.11ThereisnohintofthisfeatureintheJosephSmithhypocephalus.Itisinparticularacosmographicalchart.Fromallthis,wemayinferthatitspurposewasthatofmanyotherobjectssimilartothehypo-cephalus,whichistoaidthestudentinthewell-knownpre-occupationofAbraham,thestudyofthecosmos,thecreation,andtheplaceofmaninit.

Reliable Measurements?Butwithall this therearestillmorespecificqualifica-

tionswhichsetourFacsimile2apartinamostremarkablefashion. When we discussed figure 3 we referred to thedrawingintheChurchHistoriansOffice(seeappendix3),whichwasdoneaftertheoriginalhadsufferedaddeddam-agefollowingthe1842engravingofReubenHedlock(figs.7and8wereapparentlystill therewhenHedlockdidhisengraving). Ifwe startmeasuring the lines that form thevarioussectionsofFacsimile2,itatoncebecomesapparentthat theyarealldrawnwithcare,and the specialdimen-sionsofFacsimile2arenotduplicatedinanyofthehundredorsootherhypocephali.

Some Cousins of the HypocephalusThe recent publication of a magical text describing

a horoscope board, such as that used byNectanebos (see

11. Migliarini,AccountoftheUnrollingofaMummy,165.


p.**,fig.65),showshowvariouselementsofastronomyandalchemycanbebroughtintoasinglescheme:Muchismadehere and in the romance [of Alexander] of the materialsusedforthepieceswhichrepresenttheheavenlybodies.12Thetablet (,pinax),accordingtoPseudo-Callisthenes,combinesthefourelementsofthealchemist,beingmadeofivory,ebony,gold,andsilver.13Itisdividedintothreezones:thefirstisthecycleofthethirty-sixdecans;uponthesecondarethetwelvefiguresoftheZodiac;andinthemiddlearethe sunand themoonplacedona chariot.14Thediviningpieceswhich can bemoved around represent seven starsandtheplanets;thesunisanicycrystal,themoonanama-ranthine,Ares/Marsahematite,Hermes/Mercuryanemer-ald,Aphrodite/Venusasapphire,Kronos/Saturnaserpen-tine,andthehoroscopeisofwhitemarble.15Othersuchlistshavedifferentstones:Letthestarsbesetupontheboard(astheyare)bynature,exceptfortheSunandMoon,andlettheSunbegolden,theMoonsilver,Saturnofobsidian,Mars of reddish onyx,Venus lapislazuliwith gold flecks,Mercury turquoise; let Jupiter be crystalline.16Hermes isalwaysemeraldorsmaragd.

The alchemists disk of Cleopatra (see p. **, fig. 58),theTablet of Cebes (seep. **,fig. 59), and the JosephSmith

12. IregardasitsnearestparallelthedescriptionofthekitusedbytheEgyptianNectanebos to cast thehoroscopeofOlympias (Pseudo-Callisthenes,Historia Alexandri,I.4).KlausMareschandZolaM.Pack-man,eds.,Papyri from the Washington University Collection, St. Louis, Mis-souri. Part II,AbhandlungenderRheinisch-WestflischenAkademiederWissenschaften(Opladen:WestdeutscherVerlag,1990),4142. 13. Pseudo-Callisthenes, Historia Alexandri 1.4.5, in Wilhem Kroll,Historia Alexandri Magni (Berlin:Weidmann,1958),4. 14. Pseudo-Callisthenes,Historia Alexandri1.4.5,in ibid. 15. Pseudo-Callisthenes(Armenianversion)8,inAlbertM.Woloho-jian,The Romance of Alexander the Great by Pseudo-Callisthenes(NewYork:ColumbiaUniversityPress,1969),26. 16. Pseudo-Callisthenes,Historia Alexandri1.4,in MareschandPack-man,Papyri from the Washington University Collection, 46.

hypocephalus illustrate three different ways an ancientartist could present a three-dimensional object on a two-dimensionalsurface.Ineachcasetheproblemwastorep-resent three circular zones, one above another. Cleopa-tras tablet lets us knowwe aremounting towardheavenby occupying the center circle with the heavenly bodies,includingacomet,andbytellingusinthetwoouterringinscriptionsthattheuniverseisallone.IntheCebestablet,theartistischallengedbyaverbaldescriptiontodepictthethree circular zones,which is done bydrawing things inperspectiveastheywouldappeartoaviewerfromtheside,crowningtheinnermoststructurewithathrone,apalace,andanembracethesuccessfulascenttoheaven.

The JosephSmithhypocephalus is a circledividedbystraight lines into three horizontal zones. This is how itwouldappearonaglobeoftheearth,viewedfromaposi-tionperpendiculartothepolaraxis.ThatFacsimile2repre-sentsaglobeisapparentfromthefactthatthefiguresinthelowerpartofthesketchareupsidedowntothefiguresintheupper,justasontheearth,andthoseabovearegoingtotherightandthosebelowtotheleft,makingclearthatthething is revolving.The limitationsonperspectivearecor-rectedby(1)endowingthecentralfigurewithfourheads,facingthefourdirections,likeaHindutempleorBuddhiststupa;17also,(2)theperceptiveviewerwouldrecognizethefour canopic figures in figure 6 as representing the fourcardinalpoints of the earth, the four elements and colorsderivedfromthem,thefourracesoftheearth,andlater,themedicalalchemistsfourhumorsofman.(3)Theupperpartofthehypocephalusrepresentstheearthandsky,whilethelowerpart,whichisreversed,representsthenetherworldorrealmofthedead,whichtogetherdepicttheentireuniverse.

17. Foramapofstupadevelopment,seeJohnM.Lundquist,WhatIsReality?inTemples of the Ancient World, ed.DonaldA.Parry(SaltLakeCity:DeseretBookandFARMS,1994),62829,fig.50.


TheEgyptianssoughttoshowarepresentationofthearrangementofpartsoftheuniverserelativetoeachother,exactly in the manner of a cartographic image.18ThisrecallshowtheLordmustconstantlyremindAbrahamthatheisseeingthingsonlyfromonepointofview,thathisuniverseisonlythesegmentheseesfromtheearthuponwhichthoustand-est,orsimplythatuponwhichthoustandest(Abraham3:5,3).Ninetimesinonechapterheisremindedofthisrela-tivity.Thisalsoappliestotime,measureddifferentlyfromdifferentplacesandatthesametimeinseparablylinkedtospaceandmatter:whichcelestialtimesignifiesonedaytoacubit(Fac.2,fig.1,explanation).KurtSetheremindsusthattherearethingswhichareopposedtoourknowledgeofrealityasbeingincorrectandimpossible.19Hewarnstheexpertsagainstnitpickingandmicromanagement.20

Almost all hypocephali, while divided into distincthorizontalzones,arealsomarkedbyastrongverticaldivi-sionrightdownthemiddle.ThisisestablishedbythetwodominantJanus-figures(Fac.2,figs.1and2),whosefaceslookbothwestandeastas theBookof theDeadputs it,to yesterday and tomorrow respectively.21 The center lineneatly divides these majestic figures, as it does the twoAmun-feathersworn by figure 2,which usually anchorthecenterlinebetweenthem.OnanumberofhypocephalitheWepwawetstaffoffigure2liesexactlyonthecenterline,suggestinganintentionalsymbolism.Itmayalsobesignifi-cantthatintheHedlockengravingthetallAmun-feathersworn byfigure 2 break through the circle into the outerspaceoftherimbeyond.Theanomalyisfoundconsistentlydisplayed in other hypocephali (see appendixes 47A), so

18. KurtSethe,Altgyptische Vorstellungen vom Lauf der Sonne (Berlin:AkademiederWissenschaften,1928),11(emphasisadded). 19. Ibid.,28. 20. Ibid.,1213. 21. BD17,line5,inRichardLepsius, Das Todtenbuch der gypter nach dem hieroglyphischen Papyrus in Turin (Leipzig:Wigand,1842).

that itmightwell bedeliberate.22 This cosmicbisecting isprominentinEgyptiantemples.ThusinthetempleofOpeteverythingontherightsideoftheworshipperinthetem-plewason thesouthside, thesideof lightand life,whileeverythingontheleftsidewasnorth,darknessanddeath;. . .between the two is themiddle toposof the throneorsanctuaryoftherulinggodwhichmustalwaysbeexactlyinthemiddle;thispowerinthecenter,Amun,isinvokedinmomentsofdistresswhenthemanorgodisindanger.23Itisthegodwhoissummonedtotherescueonthehypo-cephali(Fac.2,figs.811)whoisOsiris.Thesituationisthatof the lioncouch, theRe/Osiris scenario,whereRecomestorescueOsirisfromthedepths.Thebasicteachingofthealchemistsisthatindestructiblespiritfromaboveisunitedwith indestructiblematter (materia prima) frombelow intotheindestructiblebodyoftheresurrection.24

The vertical and horizontal divisions together makethefamiliarquadrateofthehypocephali.Thisiswell-nighinevitable:Sincethedawnoftimemenhavebeenwonttorepresentgraphicallytheearthinitsfourquartersandthecircleoftheheavensincombination(as)agraphicexpres-sionofunityandperfection;theyalsosymbolizethespiri-tual and temporal power.25 Sethe found in the heavenlychartsoftheEgyptianstypesofacommonbasiccomposi-tionthatportraytherisingofthesun.26Hecomparessome

22. See, for exampleBM8445, Brussels E 6319,Wien 253 a/2,Ash-molean1985-1095,andCairoSR10688,wherethefeathersarecutoffen-tirely. 23. Marquis de Rochemonteix, Le temple dApet o est engendrlOsiris de Thbes,RT 3 (1881): 79; seeHughNibley,An Approach to the Book of Mormon,3rded., CWHN 6(SaltLakeCity:DeseretBookandFARMS,1988),271,fig.34. 24. Jafaral-Sadiq,LettertoHisSon,inJuliusRuska,Arabische Al-chemisten(Heidelberg:Winter,1924),6162. 25. OlovR.T.Janse,QuelquesreflexionsproposdunBoldetypemgaren,trouvauVietNam,Artibus Asiae25/4(1962):281. 26. Sethe,Altgyptische Vorstellungen,11.


sixteendrawingswiththesunscourseintheheavensandtheunderworld.Henotes that theEgyptiansconceivedofthesunasadiskfloatingontheprimevalwatersofNunthatencircledtheearth,goingthroughtheskyaboveandalsoundertheearth.27Wethinkatonceoftherimofthehypo-cephalus(theearthsurroundedbyOceanusasdepictedonthe shield of Achilles). Sethe explains that the four-sidedopen area within the primeval waters of Nun in whichthesolarbarkisdepictedisanattemptatportrayingthreedimensionsonatwo-dimensionalplane,aflatprojectioninthemannerofourmodernmaps,28withtheupside-downarrangementoftheupperandlowersectorsshowingasitweretheantipodes[oppositethefeet]or,morecorrectly,theantikephalia[oppositethehead]ofourworld.29

The cosmicwheel, thequadrata, and theglobal shapeallindicateunmistakablythatthestructureisinmotion,aswecanseefromtheJanus-facedgodslookingbothaheadandbehind,fittedoutwithstridinglegsandwalkingstick,or seated in the various solar and lunar ships, cruisingthrough the skyon the hypocephali.Also in the JosephSmithhypocephalus,wearefacedwithaprocess,adevel-opment,oraroyalprogress.Fortheupperpartofthehypo-cephalus is readily recognized as celestial, and the lowerpart as chthonianof the earth, earthy.Theprocess indi-catedisthatofbringinglifeandlightfromtheupperworldtothelowerworldinordertoeffectresurrectionourbasicRe/Osirisscenario.

ThisvitalprocessisbroughttothemindoftheinitiateashebeginstorecognizebasicdesignshiddenspecificallyintheJosephSmithhypocephalus.Itishightimetoaddresstheindividualsectionsofthedisk,firstasgeometricalfig-uresonly.

27. Ibid.,3. 28. Ibid.,10. 29. Ibid.,7.

The PanelsEdith Varga has noted that there are eight standard

frames in thehypocephali (seep. **,fig.16). In the JosephSmith hypocephalus there are only seven, but the low-est section in the various hypocephali is filledwith any-thing the artist chooses toput there.Theyare apparentlynotprescribedandofficialpanels,butarerequiredbytheextremelyimportantdimensionstheygivetothecircle.Thesevenpanelsareconsistentwith thesevenheavensof theascension texts, the seven chambers of theHekaloth, etc.Suchpossibilities,thoughunproven,shouldnotbeignoredinanobjectlikethehypocephalus.

BycountingthesquaresonruledpapertoapproximatethevariousareaswithinFacsimile2,wecandeterminethattheareaofthelowestofthethreedivisionsisjust1/3ofthewholecircle,anditinturnisdividedintothreelevels,thetoponebeing1/4theareaof thewholecircle.Theupper-mostof the threemaindivisions is in turndividedverti-cally into three sections of equal area, each section being1/10of thewholecircle.Thewidthof thecentralpanel isapproximately2/3ofthepanelsoneithersideof it. Inthemiddlezone,wherethecentralfiguredominatesthescene,thethreepanelsareallequalinarea,andtheheightofthetwopanelscontainingtheJanusfigures,numbers1and2,isthesame.

Inallofthis,oneisfreetodetectresonancesoftheSefer Yetzirahwiththedominantruleofthree,aswellasthefavor-ite fraction of the Egyptians, 2/3, being the only fractiontheyusewhichdoesnothaveoneasanumerator.30Wealsorecall the3-to-2relationshipofmaleandfemale inEgypt,especiallyinthestoryofJoseph and AsenathandintheKab-bala.WolfhartWestendorfhasnotedthatthedivisionofthecosmosintomaleandfemaleistobetakenforgrantedifitis

30. Alan H. Gardiner, Egyptian Grammar: Being an Introduction to the Study of Hieroglyphs, 3rdrev.ed.(Oxford:OxfordUniversityPress,1966),197.


tobeperpetuatedinwhatJanAssmanncallsthemarriageofeternityandtime.31ThedoctrineisexpressedbyErikHornung:TheBeyondisalsoadistortedmirrorimageofthehereandnow.32SelimHassanmakesasimilarobserva-tion:Themostprominentbeliefwasthatwehaveacom-pleteuniverseof skyandearthwhichhas its counterpartreversedbelowit(seep.**,fig.42B).33

Insomehypocephali,thelowerchthonianorlunarsec-tionislargerthanthesolar.Otherhypocephaliaredividedequally;andstillothersarecarefullydivided,liketheFlor-encehypocephalus,intoequalthirds,withtheupperregiontakinguptwoofthethirds;however,theoppositearrange-ment also occurs. In a Louvre papyrus the proportion isfouruptothreedown,andintheBritishMuseumtherearethosewithKabbalisticproportionsofthreetosevenandseventothree,withthecowdominating.

A Well-Known SurpriseOnemayfreelyreadonesowninterpretationintoallof

this,butthereisoneanomalywecannotoverlook.TryasonewilltodividetheJosephSmithPapyrusintoanyoftheaboveverticalproportions,theresultisfrustration.Themainhori-zontaldividersdisplaynovisiblesymmetryorcommensu-ratedivision.Oneisdisturbedtofindnotevenanapproxima-tiontotheotherhypocephalithedisproportionispositivelyglaring.Couldithavebeenintentional?Thefirstimpulseforamathematicianwould be to test it by the golden sectionorphi (, )proportion,which is theFibonacci series, and

31. WolfhartWestendorf, RaumundZeit als Entsprechungen derbeidenEwigkeiten,inFontes atque Pontes:Eine Festgabe fr Hellmut Brun-ner, ed.ManfredGrg(Wiesbaden:Harrassowitz,1983),430. 32. ErikHornung,The Valley of the Kings: Horizon of Eternity, trans.DavidWarburton(NewYork:Timken,1990),74. 33. SelimHassan,Excavations at Gza(Cairo:GovernmentPress,1946),6.1:319.

this operation gives instant satisfaction; Facsimile 2 sticksthroughouttothesacredgoldensection!

Compass and SquareTheproblemofdimensionsandanglesisraisedinthe

dawnof timeat the temple.The terriblequestion,Is thisallthereis?receivesafavorableansweronlywhenoneisguaranteedasafehomeontheotherside,unitedwitheter-nity,aunionwhichcantakeplaceonlythroughthetemple,ascalemodeloftheuniverse,forthetemplewasamodelof theworld writ small.34Theconcept is shared intheGreek principle, that the trio Universe-Temple-Man is anunimpeachablewhole, all following the sameproportionsoranalogia indescendingscale.35Hence frommegalithictimesthetemple,thefirstbuildingonearth,hadtobebuiltandrebuiltwithaccuracyandprecision.

Toconstructaperfectrightanglewasthefirstrequire-mentforthebuilderoftheEgyptiantemple,adelightfullysimpleexercise.Hewouldsimply tieknots inastringatintervalsof 3, 4, and5unitsany sizeunitwilldo.Theknots in the loop of string,when stretched tight,wouldmarktheanglesofaperfectrighttriangle,theEgyptiantriangle, which not only gives us our square but dem-onstrates theawesomePythagoreantheoremat thesametime:32+42=52,or9+16=25(fig.70).This3-4-5trianglewas considered sacred in antiquity; it was particularlyvaluable for establishing architectural plans and struc-turessinceitpermittedeasycontroloftherightangle;...theuseofthistriangleappearsattheverydawnofthehis-toryofEgyptintheplanofthegreattomb...ofNegadah,

34. William J.Murnane,United with Eternity: A Concise Guide to the Monuments of Medinet Habu (Chicago:Oriental Institute,University ofChicago,1980),6. 35. MatilaGhyka,The Geometry of Art and Life(NewYork:SheedandWard,1946),112.


whichwasperhapsthatofNarmer-Meneshimself.36ThemainEgyptianconcernwithsacredwholenumbersinrit-ualisseenintheembalmingoftheApis-bull,inwhichthewashingvesselshadarimdiameterof5palms,abaseof3palms,andaheightof4palms.37

Butthegroundplanoftheancienttemplewasnotthesquarebutthehalfsquare,the2x1rectangle.Thepropor-tionsofthefoundationofthetemplerequire,asVitruviusstates, that thewidthbehalf the length.38The rule alsoappliestotheholyplaceofMosestabernacle(Exodus26:18,20,2223)aswellasSolomonstemple(1Kings6:2)(fig.71).

After the perfect cooperation of the three sides of theEgyptiantriangle,32+42=52,oneturnsinanticipationtothe

36. Jean-Philippe Lauer, Le triangle sacr dans les monuments delAncienEmpire,inActs. First International Congress of Egyptology, Cairo, Oc-tober 210, 1976, ed.WalterF.Reineke(Berlin:AkademieVerlag,1979),72. 37. R. L. Vos,The Apis Embalming Ritual, P. Vindob. 3873 (Louvain:Peeters,1993),5455. 38. Vitruvius,On Architecture3.4.3,4.4.12.

Figure70.TheancientEgyptianscouldestablisharightanglebyform-ingarighttrianglewithunitsof3,4,and5.ThefamousproofofthePythagoreantheoremshowsthesquaresofthethreesidesinwhich9+16=25.TheFreemasonscontinuetousethisgraphicdemonstration.

Holy of Holies

Holy Place

20 cubits

30 cubits

40 cubits

20 cubits

20 cubits

Holy of Holies

Holy Place

10 cubits

10 cubits

10 cubits 20 cubits

TohonortheLordshouse,Solomondoubledthelinearmeasuresofthetabernaclebutkeptthesamefloorplan.ThustheholyplacealsoagreedwiththeruleofVitruvius.

Holy of Holies

Holy Place

20 cubits

30 cubits

40 cubits

20 cubits

20 cubits

Holy of Holies

Holy Place

10 cubits

10 cubits

10 cubits 20 cubits

Figure71.TheLordinstructedMosestomakethetabernacle,aportabletenttemplewithwoodenwallscoveredwithgold.Theholyofholieswasaperfectcubeoftencubitswhiletwosimilarcubesformedtheholyplace.


diagonalofthesacred2x1rectanglewithhighexpectations.According to Giorgio de Santillana the resulting propor-tionscameasastunningsurpriseandagrievousblowtotheancients.39Forthediagonalofthe2x1rectanglewasthehypotenuseofaproperrighttriangle,butitssquarerootwasanirrationalnumber, 5,12+22=5.Thiswasreallyablessing,forthoughtheso-calledsacredtrianglewitharatioofsidesof3-4-5wasusedintemples,tombs,andpyramidsasearlyastheOldKingdom,itisessentiallystatic.40ItisthedynamicgoldensectionthatbreaksawayfromtheconfinesoftheCurieprinciplethatis,thatwhenthingsareinperfectequilibriumnothinghappens.

The top of the vertical divider, the line between thefeathersonfigure2,marks thehighpointof thedayandthe year, representing what, according to Santillana, theancients called the vernal equinox: the Sun-carrier, andthemainpillarofthesky...determiningthefirstdegreeofthesunsyearlycircle,andthefirstdayoftheyearthatis,theclassicJanusmotif.41Thisagreeswellwithwhatwehaveobservedsofar.Usingthefoundationplanofthe2x1rectangleasabuildingblock,theverticallinedividedinthephiproportionlaysthefoundationofourcosmicchartwithagoldenrectangle(fig.72).Theawkward12+22=( 5)2righttriangle establishes the golden section as the appropriateinterfaceofupperandlowerworlds,thesolarandthechtho-nian,themaleandthefemale;itistheever-generativephiproportion,divineproportion,goldenmean,goldensection,golden ratio, etc. JohannesKepler called the Pythagorean

39. GiorgiodeSantillana,The Origins of Scientific Thought: From Anaxi-mander to Proclus, 600 B.C. to 300 A.D (Chicago:UniversityofChicagoPress,1961),6970. 40. AlexanderBadawy,TheHarmonicDesignoftheUpperTempleatTetiPyramid,BiOr33(1976):3. 41. GiorgiodeSantillanaandHerthavonDechend,Hamlets Mill: An Essay on Myth and the Frame of Time(Boston:Gambit,1969),59.

theoremandthephiproportionthetwogreattreasuresofgeometry,42andherewehavethemboth.

Thegoldenratioexpressestherelationshipthatthesumoftwoquantitiesistothelargerquantityasthelargeristothesmaller.Thegoldenratioisthefollowingalgebraicirra-tionalnumberwithitsnumericalapproximation:

The figure of a golden section illustrates the defininggeometricrelationship.Expressedalgebraically:

Thegoldensectionisalinesegmentsectionedintotwoaccordingtothegoldenratio(fig.73).Thetotallengtha + bistothelongersegmentaasa istotheshortersegmentb.Other names frequentlyused for or closely related to thegolden ratio are golden section (Latin sectio aurea), goldenmean,goldennumber,andtheGreekletterphi().

ButwastheEgyptiandraftsmanawareofthephipropor-tion?Thedividingupoftheworldcouldhavebeenahappyaccident, forpolls takenworldwideovermanyyearshaveshownthatthehumaneyejustnaturallyprefersthegoldensection to any other proportion in a rectangular panelarchitectureandart.GustavFechner,givingthousandsofpeopletenpropergeometricallyproportionedrectanglestochoosefrom,foundthat75.6percentchosethegoldensec-tionbeforeallothersasthemostpleasingexperience.43Didtheartistsimplydrawthelineswheretheylookedgoodtohim?

Theobjectinthehypocephaluswastodrawthenextmostimportanthorizontalinterface,thelinebetweentheHyperionrealmsonhighandthecenterofeverythingwheretheGreatRulersitsfacinginalldirectionsandmaintainingabalance

42. QuotedinH.E.Huntley,The Divine Proportion: A Study in Mathe-matical Beauty(NewYork:Dover,1970),6265. 43. Ibid.

1+5 1.6180339892

=

a+ba

==ab


between everything above and below, thus providing, likethe temple, theall-important linkbetween the twoworlds,givingmanaplaceinboth,andsoassuringhimimmortal-ity.AtthepointwherethetwosidesoftherighttriangleABIstandintherelationshipof2to3,wemarktheupperdivisionI(seep.**,fig.72).Thismethoddependsonthe2to3ratio.Wehaveseenthatthefraction2/3wasagreatfavoritewiththeEgyptiansandalsonoted,inthecaseofAsenathandothers,that2/3istheproportionwhichmaintainstheunityofthesexesitisthedynamicandcreativefraction.

Wenowhavethemaincosmichorizontalsectionsofourhypocephalusinplace.EgyptologistshavedeniedthattheEgyptianswereawareofthephiratiosinceitisnotmen-tionedintheliterature.AndyetElseKielland,whonotesthattheyhavenot lookedveryhard,has amplydemonstrated

A

C B

AD

C B

E

F

H

G

F

I

Thephicurve(ubiquitousinthenaturalworld)appearsherewithinthegoldenrectangleABGHasitissuperimposedonFac.2,ChurchHistoryLibrary;itevokesthespiralramshorns(seep.262,fig.28),thesupremesignofAmun,whoappearsinfigure1andwhosenamepredominatesinthehypocephalitextsingeneral.PointFgivesusthebaselinefordividingtheupperandlowerportionsofthehypocephalus.PointImarkstheupperdivision.

howtheEgyptiansmadeconstantuseofit,beginningwiththePaletteofNarmeritself,whichshowsabsolutedepen-denceontheconstruction.44

PrinceHardedeftellsthestoryofhowhisfatherKhufusought to obtain the number of the secret chambers ofthesanctuaryofThothso thathemight include themin

44. ElseC.Kielland,Geometry in Egyptian Art(London:Tiranti,1955),99.

A

C B

AD

C B

E

F

H

G

F

I

Figure72.The1x2rectangleABCD,greatlyfavoredinEgyp-tiantemplearchitecture,cangiveusthegoldensectioninanelegantgeometricdance.WithcompasspointatC,drawanarcfromBtothediagonalatE;thenmovethecompasstoAanddrawanarcfromEtoF,givingusthephiproportion:AB:AF=AF:BF.


the construction of his tomb in the Great Pyramid.45 Nostructure has been more thoroughly studied or revealedmoreimpressivephi-proportionsthanthetombchamberofKhufuintheGreatPyramid(seep.**,fig.4). Itsfloorandceilingaredoublesquares2x1rectangles;itsheightis1/2ofthediagonalofthoserectangles;thelengthofthecham-ber is4unitsandthediagonalof theendwallsexactly3,whilethegreatdiagonal(betweenthemostdistantcornersofthechamber)is5,thusgivingustheEgyptiantriangle,3by4by5(fig.74).46AcommonformofEgyptiantombisthegoldensolidinwhichthegreatdiagonal(betweenthemostdistantcorners)istwiceitswidth,whichinturnisrelatedtoitsheightinthephiproportion.47

Thephiratiogeneratesthephispiral(logarithmicspiral,equiangular spiral, whirling squares, spira mirabilis, etc.),andthephipentagramandFibonacciseriesareallintimatelyrelated,ofcourse.48Whythenwasthediscoveryofthesecon-nectionsmadecenturiesapart?Whymustthefindingsbesounexpected? No mathematician created these numbers,writesH.E.Huntley,whocravestoknowhowitallhap-pened.49Or,asMorrisKlineputs it,Wemustfacethefactthatthereisnouniversallyacceptedcorrespondencebetweenmathematicsandphysicalreality.50

How conscious were the Egyptians of these dimen-sions?DidtheysimplyreactlikethepublicinFechnersdaytowhattheyfoundpleasingtotheeye,orweretheycaughtupinthecelestialandimponderablequalitiesofthesemea-surements? PascalsTriangle isperhaps themost famous

45. Westcar Papyrus = P. Berlin 3033 7/58, English translation inMiriamLichtheim,Ancient Egyptian Literature: A Book of Readings,3vols.(Berkeley:UniversityofCaliforniaPress,197380),1:219. 46. Ghyka,Geometry of Art and Life,62. 47. Ibid.,60. 48. Huntley,Divine Proportion,100. 49. Ibid.,154,36. 50. MorrisKline,Mathematics and the Search for Knowledge(NewYork:OxfordUniversityPress,1985),210.

B

CD

A

EF

G

a + b is to a as a is to b

a + b

a b

Figure73.Whenacirclecircumscribesourfamiliarsacredtwosquares,itestablishesthegoldensection.Thismethodistheclassicwayofconstructingthegoldenrectangleusingacompassandstraightedge:(1)ConstructaunitsquareABCD.(2)DrawalinefromthemidpointEofsideCDtoanoppositecornerB.(3)UsethatlineastheradiustodrawanarcthatdefinesthelongdimensionDFoftherectangleAGFD.


ofallnumericalpatterns,writesHuntley,yetPascalhim-selfwasapparentlynotawarethathistrianglecontainstheFibonacciseries,whileFibonacci,equallynaive,mayhavestumbledontheseries . . . throughanexaminationoftheChineseTriangle (fig.75).51Howcould theyhavemissedthesevitalconnections?Wehaveonlytoberemindedthatthedevelopmentofthecapacityfor logical thinkingis innoway connectedwith [highermathematics].52 The dis-coveryof these remarkable relationships ismarkedby an

51. Huntley,Divine Proportion,131,134. 52. Ibid.,4.

5 units

4 units

3 un

its

a bc

d

e

f

to AlphaDraconis

to Orion

10 royal cubits20 royal cubits

Startingwithdoubledsquares,thepriestsseemedtohavebasedthedesignonthe3-4-5righttriangleAEF.

unexpectednesswhichmaystimulatesurprise,andevenasenseofwonder.53

Spira MirabilisProgressionoftheFibonacciseriescanbeillustratedby

joining together squares having sides related in the Fibo-nacciprogressioninwhich,torepeat,eachnewtermisthesumofthetwoprecedingterms(fig.76).Soa1-unitsquareisjoinedtoa2-unitsquarewhichgivesa3-unitsquare;thisaddedtoits2-unitpredecessorgivesusa5-unitsquare,and

53. Ibid.,149.

5 units

4 units

3 un

its

a bc

d

e

f

to AlphaDraconis

to Orion10 royal cubits20 royal cubits

Figure74.ThehugepolishedgraniteblocksofKhufusburialchamberwerecarefullyplacedinaccordancewithaplandevisedlongbefore.


thenan8-unit,13-unit,21-unit,etc.Eachnewsquareinturnisboostedbythesizeofthesquarethatwentbeforeit.Tak-ingtheinmostcornerofeachsquareasthecenterofacircle,and the sideof that squareas its radius,one subscribesaquartercirclewhichmeetswiththesamearconthepreced-ingsquare,andsoon.Theseconnectedarcshaveallexactlythe sameshape; theygiveus themagnificent spiral. ThegoldensectionandpentagramofPythagorasandtheFibo-nacciseriesofLeonardoofPisaareallassociatedwiththisremarkablecurve,writesHuntley.54

Thusthereisadirectrelationshipbetweenthephipro-portionandthefamousFibonacciseries,withthetwodraw-ing closer together as the numbers get larger and larger,thoughtheseriesnevergetsallthewaytophi.Buttheplaceofphi inourhypocephalus is themost important step inintroducingtheworldoflivingthingsintothecosmos.

Weshallseethatfigures7,5,and6inthehypocephalusdramatize,inthatorder,theplanting,birth,andperpetua-tionoflifeupontheearthfromabove.ThatthisthemewascarriedoutdeliberatelyintheJosephSmithhypocephalusis

54. Ibid.,100.

55

34

21

13

8

5

Figure76.Thephispiral,orwhirlingsquares,wasderivedintheeighteenthcenturyfromthestudyoftheFibonacciseries.Thesidesofeachsquarestandinthephiratiototheadjacentsquares.

madeevidentbyastunningsymbol,thephiproportionofPythagorasinallitssplendor.

Like the curve, theFibonaccinumbers are full of sur-prises; their emergence in a beehive or rabbitwarren, inasunflowerorasea-shell,iscertainlyunexpected.55ItisPhiappearingoutof theblue.56Yet itwasonlywhen thedigitalvalueofphiwasconsideredthatsomeofitsstrikingparadoxesappeared.Tosquareit,allyouhavetodoisadd1toit;togetitsreciprocal,youjustsubtract1fromit;everytimeyouraise itspower itgetsnearer torationalitythatis,asnumbersriseintheFibonacciseries,theratiobetweensuccessive numberswhich bracket the phi number in theseries gets closer to phibut never arrives.57 No wonderfrom ancient times [it] has been enveloped in a halo ofmysticism.58

55. Ibid.,142. 56. Ibid.,99. 57. Ibid.,26,4650. 58. Kielland,Geometry in Egyptian Art,11.

111

1211331

1464115101051

1615201561172135352171

18285670562881Figure75.NamedafterBlaisePascal(162362),thegreatFrenchmathematiciananddevoutCatholic,thistrianglehadbeendescribedcenturiesearlierinChinaandPersia.EachtermofthePascaltriangleisthesumofthetwonumbersaboveit.


LeonardoofPisa,thesonofBonacci(henceFibonacci),attheageof27broughttheseriesfromNorthAfricaalongwiththerevelationofArabicnumeralsandtheirsuperioritytotheoldRomansystem,successfullypromotedbyabookLiber Abaci(Book of Calculation),agreatadvanceinthetech-nologyofcomputing,antiquatingeventhehighlyefficientabacus.Theserieshelearned,somesayinEgypt,astherab-bitproblem,theprogressiveseriesbywhichrabbitsrepro-duce.59Alongwiththerabbit

N1 - pt-sign of heaven

V17 - sa-knot of protection

D30 - Nhb-k3w

djed-pillar glyph

knot glyph

rabbit

bee

Miscellaneous Glyphs for One Eternal Round

,thebasicEgyptianhiero-glyphic symbol for something that exists, goes the bee

N1 - pt-sign of heaven

V17 - sa-knot of protection

D30 - Nhb-k3w

djed-pillar glyph

knot glyph

rabbit

bee

Miscellaneous Glyphs for One Eternal Round

,theoriginalinhabitantofEgypt.InthegenealogyofthedronebeewefindtheFibonacciseriesflawlesslyrepeatedinthemaleandfemalelinesandinbothcombined.60

Phideterminesthelawofbiologicalgrowth,whetherofaplantorananimal,oranypartofthem,calledtheexpo-nential law,ofwhat the famousbiologist,DArcyThomp-son,calledgnomicgrowth.61Wecanconceivenosimplerlawthanthis,hewrites,andthissimplestoflawsisthatwhichNaturetendstofollow,62fromthemicroscopicsea-shellforaminiferatothenoblestfossilofthemall,thelordlyAmmonite,sonamedbecauseitsmajesticcurlmatchesthatof the Ram-headedAmunwhich fittingly takes us backtoourhypocephalus.Fromtherewetakeoffanddiscoverthe familiar spiral laid out on a scale of light yearsthatis, in thebest-knownandfirstdiscoveredSpiralNebulas,Messier 51. The curve and the series are best known innaturefromstudiesofthesunflowerflorets,withtwosetsofmiraclespiralsmovinginoppositedirections,whileeveryseedbelongstobothsystems(fig.77).63Pineapplesandpineconesdisplaythesamearrangementintheirbosses.64

59. Huntley,Divine Proportion,15759. 60. Ibid.,15961. 61. Ghyka,Geometry of Art and Life,14. 62. Huntley,Divine Proportion,169. 63. Ibid.,164. 64. Ibid.,165.

Aproperreasonforintroducingthegoldensectionintoour hypocephalus is its connectionwith themusic of thespheres(seep.**,fig.50);itisthegoldenratiothatintroducesthehumaneartothatmusic.Bythesamesortofpolltak-ingasthatfoundinthepublicspontaneouslychoosingthegoldensectionbyeye,so, judgingbyear,humanityvastlyprefers thosemusical intervals,discoveredbyPythagoras,whichareinexactphiproportiontoeachother.Thepollsshow the two most pleasing musical combinations werethe2to1thatis,theoctave,thebasicfoundationofevery-thing,and,mostpleasingofall,themajorsixthor512vibra-tionscombinedwith320,anexactphiproportionof8to5.65

Two StarsThemostimpressivedemonstrationofthepowerofthe

goldensectiontogenerateitselfisseenintheconstructionof the star pentagon or pentagram.66 So, asMatilaGhykanotes,weshallnotbesurprisedtoseethepreponderanceofpentagonalsymmetryinlivingorganisms,especiallyinbotanyandamongstmarineanimals(starfishes,jellyfishes,sea-urchins).67 Along with the phi curve and Fibonacciseries, the pentagon turns up everywhere in natureinplants,allblossoms,mollusks,seashells,starfishes,jellyfish,water lilies,allroses,honeysuckles,carnations,etc.GhykacreditsthePythagoreanswithextraordinaryintuitioninchoosingthepentagramorpentalphaastheawesomesym-boloftheirsociety.68Itwastheprofoundsecretbywhichthemembersrecognizedeachotherbythefive-pointedtokeninthehand.Membersofthebrotherhoodsworetosufferdeathratherthantorevealit.Theprecautionwaswelladvised,foronceitgotout,crackpotsandoccultistsclaimedthepenta-gramastheirown,evenintheexerciseofblackmagic.

65. Huntley,Divine Proportion, 5455. 66. Ghyka,Geometry of Art and Life,17. 67. Ibid.,18. 68. Ibid.,114.


Thephicurvewasnotdiscovereduntiltheeighteenthcentury, yet it is the star pentagram which from ancienttimesrepresentedthelifeforcesoftheuniverse,asGoethesFaustremindsusinthefirstact:IchshauindiesenreinenZgen, Die wirkende Natur vor meiner Seele liegen(Ibeholdinthesepurelinestheworkingsofnaturerevealeduntomymind) (fig. 78).69 If thephi curvegoeson forever

69. JohannWolfgangvonGoethe,Faust: Der Tragdie erster Teil,lines44041,asopposedtothemorefamousviewofFaustinHughNibley,OneEternalRound,inTemple and Cosmos, CWHN 12:398,fig.52.

forming56spirals,thecurvesareactuallymadeof196straight-sidedplaquesofblackandwhitemarble.28indiameter.Tholos,sanctuaryofAsclepiosatEpidauros,ca360b.c.PanagitsKavvadias,Fouilles dEpidaure (Athens,1891),1:plateIV.5.

expandingorcontracting,soevenmorespectacularlydoesthepentagram(fig.79).Byextendingthesidesofapentagon,wemakeastarpentagram;eachpointofthestaristheso-calledsublimetriangle,an isosceles trianglewithanapexof36degrees.Byconnectingthepointsofthestar,wemakealargerpentagon,byextendingthesidesofwhichwegetalargerstarpentagram,andsoforth.Itgoesonforeverinbothdirections,while the intersecting linesalwaysdivideeachotherintophiratios.Nootherpolygoncanbeprojectedinsuchamanner.

Figure77.Thestrikingdesignofthefloretsintheheadofeverysun-flowershowsthebeautifulefficiencyoftheplacementoftheseedsinopposingspirals.PhotobyArtPollard.Amongtheearliestknownexamplesofthespiramirabilis,thistemplefloorshowsgreatsophis-tication.Basedontheheptagon,adifficultpolygontoconstruct,and


Figure78.In1651,Rembrandtengravedthissceneinwhichthealche-mist,perhapsFausthimself,seestheshiningrevelationofhisquest:acirculardiagramofmysticwords.ThecenterdisplaystheGreekletterforChristoswiththeLatinlettersINRI,theabbreviationforJesusofNazareth,KingoftheJews.Victoria&AlbertMuseum,London/ArtResource,NY.

Whilethephicurvelongawaitedtherecognitionofthelearned,thestarpentagramisfoundonancientmonumentsandpapyritogetherwiththenamesofGod(fig.80).70Thevictory of Antiochus Soter in Palestine over his enemieswas attributed to a figure of the pentagram, which wasalsofoundononeofhiscoins.71TherivalwhomAntiochus

70. MaxGrunwald,ShieldofDavid,inThe Universal Jewish Encyclo-pedia: An Authoritative and Popular Presentation of Jews and Judaism since the Earliest Times,ed.IsaacLandman,10vols.(NewYork:UniversalJewishEncyclopedia,193943),9:507;seeHughNibley,TheBookofAbrahamandtheBookoftheDead,inAbraham in Egypt,2nded.,CWHN 14(SaltLakeCity:DeseretBookandFARMS,2000),53,fig.9. 71. F.Secret,Pentagramme,PentalphaetPentaclelaRenaissance,Revue de lhistoire des religions180(1971):11617.

A B C D

36

Figure79.Thepentagonistheonlyregularpolygonthatcanre-createitself.NotethephiratioofAD:AC=AC:AB=AB:BC.


defeatedwas JudasMaccabaeus,whoused the same signas his notarikon.72 Johann Reuchlins De Arte CabalisticaclaimedtorestorethesymbolicphiofPythagorasinwhichare combined theLabarum ofConstantine, the pentagramofAntiochusSoterandthenotarikonofJudasMaccabee.73ItwassaidtobeengravedonthecornerstoneofSolomons

72. Ibid. 73. Ibid.

Figure80.(A)Earlyexamplesoffive-pointedstarsinEgyptoccuronFirstDynastycylinderseals,ca.2800b.c.(B)StarscoveredtheceilingoftheburialchamberofUnasandotherpharaohs,ca.2340b.c.(C)BythetimeoftheNewKingdom,thefivepointsofthestarwerethoughttorepresenttheextendedfivefingersofthehandsofstardeities.(D)Thishieroglyphrepresentednotonlyastarbutalsoworship.TheSumeriancuneiformsignUBusedasinglelinetodrawthestar,30002800b.c.(E)TheencircledstarsurroundedwithenigmaticlettersappearedonJewishjarsealings,4th5thcenturiesb.c.(F)ThiswoodcutdepictedmanasthemicrocosmwithintheZodiac,AgrippavonNettesheim,De occulta Philosophia (1531).

AB C

E

D

F

temple, and Pythagoras, Kabbala, Rosicrucians, and Free-masons all have laid claim to the pentagram.74 For othersectariesthestarrepresentedthefivewoundsofChrist,thehighestpointofthepentagrambeingtheheart.75Somehavewondered why the pentagrams on the Salt Lake Templepointdownward.TheexplanationcanbefoundintheNau-vooTemple,wherecorrespondingstarshaveonelongpointextendeddownward,indicatingtheprojectionortransmis-sionof light to theearth fromonhigh.76This idea is alsoexpressedintheEgyptiandesignationofthegoddessSes-hatinwhichastarisshowndescendingfromtheplaceofsevenbooksbringingknowledgetoearth(fig.81).

Since1570whenHuil,aprinterofCologne,adopteditashistrademark,thepentagramhasfiguredasthespecialmarkofthemedicalmysteryandassuchisafavoritedeviceofthealchemists.TheimmenseprestigewhichitachievesisexpressedinthefulltitleofaLatinworkbyoneValerianus,publishedin1639inLyons:ThePentagon,aStimulanttothePhilosopherortheNewArtofRecollectionwiththeInsti-tutions ofNaturalPhilosophy, and themore sublime andsecretartofmedicine,boththeoreticalandpractical,aswellasthelong-soughtKeyofallarcaneandnaturaltraditionsorwritingsofMacrocosmandMicrocosm,fromtheancientwise Philosophers, Physicians, Mathematicians, Hebrews,Chaldeans,Greeks,Latins,Arabs,Cabalists,Hermetics,Pla-tonicists,Peripatetics,andSelectedModernWorks,inanewworkforthispresentday.77

TheFrenchscholarsthroughoutthenineteenthcenturydebatedwhether thepentacleshouldbecalledapantacle,shiftingtheGreek(pente)=5,to(panta)=allembracing, encompassing the universe, synthesis of the

74. Ibid.,12526. 75. Ibid.,11920,124. 76. SeeHughNibley,TheMeaningoftheTemple,inTemple and Cos-mos, CWHN 12:17,fig.2. 77. Secret,Pentagramme,PentalphaetPentacle,120.


macrocosm.78ItwasalsoknownastheFlamingStar,sup-posedlyhandeddownfromHermesTrismegistoshimself.79The Masons adopted the Flaming Star in the eighteenth

78. Ibid.,114. 79. Ibid.,128.

Figure81.QueenHatshepsut,dressedasPharaoh,hammersinoneofthestakesthatstretchedthecordsusedtolayoutthetemplegroundplan.SheisassistedbythegoddessSeshatwearingtheseven-starredcrown.Ca.1460b.c.Granitechapel,Karnak.

century.Butnowheredoesitemergemoreboldlyorunex-pectedly than in the Joseph Smith hypocephalus, whereJosephcanhardlybeaccusedofstealingitfromtheMasons.

The Other StarEvenmore public,more controversial, but hardly less

sacredisthatotherstarthehexagon.Ingeometryandinnature,itisinseparablywedtothepentagram.ItdeservesattentiontooasarivalofthepentagonforthetitleofSealofSolomon.80EvenWebstersaysthetwoareinterchange-able.Eachconsistsofastarofwhicheverypointtouchesacircle.Itisnotonlyoneofthegreatmysticsigns,itisalsoasdominantinnatureasthepentagonitself.Asageneralrule,livingthingsrarelytakeahexagonalform,whereasminer-alsorcrystalsnormallytakethehexagonalratherthanthepentagonal form. The inorganic substances do not growfromwithinasorganismsdo,butbyagglutination, thatisbytakingonmaterialfromtheoutside,likeasnowflakeforming around a speck ofdust, alwayswith six sides orpoints.Ifthepentagramcontrolslivingandgrowingorgan-ismsof thebiosphere, thehexagon is thesubstanceof thelithosphere,thestructureofallcrystals,metals,ormineralswhichprovidetheonlysubstanceinwhichlifecantakeroot.Yetthelivingandthedead,biologicalandmineral,theoddandevenfiveandsix,areperfectlyreconciledandflourishtogether.Thusfiveandsixcometogetherintheclockandcalendar,thetwelvehoursaremarkedoffintofive-minutedivisions, and no confusion results. Five and six are alsoflawlesslyjoinedinPlatosmodeloftheuniverse:thedodeca-hedronwithitstwelvepentagons,six-times-twofive-sidedfiguresinacosmicentity(fig.82).81Vitruviustellsustherewere two schools of thought among those who acceptedPlatos10astheperfectnumber;oneschoolfavoredsixand

80. Grunwald,ShieldofDavid,507. 81. Plato,Timaeus55d56c.


calledthefraction5/6the (pentamoiros).82It isinterestingthat5/6ofpiequalsthesquareofphi!Thisequa-tion iswell known tomathematicians as one of themostextraordinarycoincidencesinmathematics:

AstoPlatos10,itisthenumberofthesacred(tetraktys),asummationbeginningatthetopwiththemonadfollowedbythedyad,triad,andtetrad,givingusthefourelements,formingthefigureknownasthesacreduncreatedtetraktys(fig.83).

82. Vitruvius,On Architecture3.1.58.

Figure82.Thedodecahedronisasolidwithtwelvepentagonalsides.Platosawitastheimageofthedivinecosmos.AlaterRomansilver-coatedleaddodecahedronisoneofthemorethanseventyfoundthroughoutwesternEurope.ThisobjecthasthenamesoftheZodiacengravedonitstwelvesides;itsfunctionremainsamystery.15/8indiameter,4thcenturya.d.,foundinGeneva,1982.MuseedArtetdHistoire,Geneva.

1+5 1.6182

=

3.142

5/6=25/62.618

22.618

Therefore,

Ithas the formofa treeandfiguresas the treeof lifein theSefer Yetzirah, in Solomons seal, or inDavids star.The two triangles, one upright and one inverted,wouldbe the symbol of evolution and involution, the inner linkofthevisibleandinvisibleworld,therepresentationoftheTenSefirothasthemathematicalfigureoftheTreeofLife.83TheRomanastronomerManilius, indescribingthediffer-entways inwhich theZodiacmaybedivided, says,twohexagonsinscribedinthecircleconnectthealternatesigns

83. Grunwald,ShieldofDavid,507.

Figure83.EquilateraltrianglesareveryrareinEgyptianart(A)TheearliestexampleswerefoundpaintedonthewallsofKhufusboatpit,ca.2530b.c.Theyarethoughttobeinstructionstothecraftsmen.(B)ThousandsofyearslatertheLadyTa-sherit-MinworshipsoneintheuniqueJSPVIIIvignette,ChurchHistoryLibrary.(C)ThePythagoreantetraktys(fourness)wasanarrangementoftendotsinanequilateraltriangle.Acompasscanbeusedtocreatethehexagonthatspacesthemequallyapart.


asfollows.1.AriesGeminiLeoLibraSagittariusAquarius.2.TaurusCancerVirgoScorpiusCapricornPisces(fig.84).84AlfredE.Housmanexplains:Thusonehexagonconsistsofthemasculineandtheotherofthefemininesigns.85Likethepentagram,theshieldofSolomondesignatestheemer-genceofthemicrocosm(man)fromthemacrocosm(theuni-verse), of theZeirAnpin (immediate aspectofGod) fromtheAbaVeumma(hiddenaspectofGod).86AbaVeummaistheHebrew ( abb w-imma)orfatherandmother,maleandfemale.

84. MarcusManilius,Astronomicon35884, inM. Manilii Astronomi-con, ed.AlfredE.Housman,5vols.(Cambridge:AcademicPress,1937),2:xiii,figureon2:xiv. 85. Ibid.,2:xiii. 86. Grunwald,ShieldofDavid,507.

Wearestillremindedofthehypocephalusinthesignifi-canceofthetermshield,whichMaxGrunwaldsayspointstoanastrologicalandmythicalconnectionandisreminis-centoftheshieldofMelkart(Hercules),withwhichthelat-terslewthegiantAntaeus.JustasthiswaspreservedinthetempleatGadeira,sotheshieldofDavid,accordingtoDavidReubeni,waspreservedinthesynagogueatBologna.87LetusrecallthatitwasAntaeussbrother,thepharaohBusiris,whotriedtosacrificestrangersonhisaltarfromwhichHer-culeswasdelivered,88 justasAbrahamwasfromthealtarofanotherpharaoh. Incidentally, thebest-known traditionconcerningAntaeusisthathehadtostayindirectcontactwiththeearthtoretainhisstrength.Isthisprincipleondis-playinthehypocephaliwherethecentralfigurestrangelynever sits on a throne but always squats directly on theground?ThestrangeshapeofthebodyofthecentralfigureofFacsimile2arisesfromplacingtwo seatedfiguresbacktoback,i.e.

Glyphs for OER

Glyphs for captions:

Figure 10. pt, heaven (also 7.33) N1

Figure 13. ankh S34

Figure 13. sa symbol of protection V17

8.15 D30

Chapter 6 Q7 N27

Chapter 7 U143 word for hour

Chapter 8 O10 T18 N1 lotus blossoum lion ram

V28-N5-V28 T25 O28

Chapter 9 G32

Chapter 13

his name (the unmistakeable six stalks of papyrus **glyph** that leap out at the student)

M8

Chapter 15

The strange shape of the body of the central gure of Facsimile 2 arises from placing two seated guresback to back, i.e. **glyph**

C4-C4.ThedoubleheadinFacsimile2isinfacttheheadoftheupperstandingfigure,sincethedrawingintheChurcharchivesshowsthatthatpartofthehypocephaluswasmissing.Inalllikelihoodthefigureoriginallyhadfourheads,asdomosthypocephali,andthisistheartistswayof trying toportray fourbodies, seated facingoutward inthefourcardinaldirections,inatwo-dimensionalmedium.

Macrocosm, MicrocosmThearchitectVitruvius says that symmetry,which is

harmonyandthebasisofarchitecture, isbestrepresentedin the measurement and balance of the human body.89The word cosmos, attributed to Pythagoras, meant origi-

87. Ibid. 88. SeeHughNibley,SettingtheStagetheWorldofAbraham,inAbraham in Egypt,CWHN 14:185,fig.31;andHughNibley,An Approach to the Book of Abraham, CWHN 18(SaltLakeCity:DeseretBookandFARMS,2009),294,fig.35. 89. Vitruvius,On Architecture1.2.34.

3. Gemini5. Leo

2. Taurus6. Virgo

1. Aries7. Libra

4. Cancer

10. Capricorn

9. Sagittarius 11. Aquarius

8. Scorpio 12. Pisces

1

2

Figure84.Thepopularfirst-centuryastronomerManiliushelpfullyincludesonewayinwhichtheZodiachousesarerelatedtoeachother.


nally Order,writesGhyka, and this order isperceivedas harmony, as consonance between ourselves and theUniverse.This ideawasdevelopedas the correspondencebetween theMacrocosmos (theWorld) and theMicrocos-mos,orMan,withsometimestheTempleaslink,aspropor-tionalmeanbetweenthetwo.90Jafar,thefatherofalchemy,followedthisprincipleexactlyinunitingtheextremesintoeternalessence,indestructiblemateria primawiththespiri-tualelixir91againthebasicquestofreligion, toestablishalinkbetweenmortalmanandtheimperishableworlds.

For making measurements, the skeleton is by far themostaccuratestructure(fig.85).Wemustnowregardthisframework of bone as the chief source of the most vitalprinciplesofdesign.92Thegoldensectioniscalledphi,infact,asatributetoPhidias(),thegreatestofsculp-torswhoembodiedthephiproportioninallhishumanfig-ures.Inparticularthearchitectureoftempleswasworked

90. Ghyka,Geometry of Art and Life,112. 91. Ruska,Arabische Alchemisten,6162. 92. Jay Hambidge, The Elements of Dynamic Symmetry (New York:Brentanos,1926),xix.

out,accordingtoVitruvius,afterthefashionoftheexactproportionsofthehumanbody.93

ArecentstudyofafamousGreektempleatDidyma(fig.87;seemaponp.**,fig.66)notesthatthemasterbuilderresponsible for thedesign[of the temple] . . .wasguided,butnotboundby,thestrictobligationsimposedonhimbygeometricdesign....Evidentlyarchitecturewaspermeateddowntothesmallestdetailbyastruggletoachievebeautyinthiswaytoo.94The120columnsof24fluteseachintheDidymatemplesuggesttheusualcosmicrelationships.Theancientsbelievedthatgeometrywasderivedfromthegodsandthatworkingwithgeometrywastheprivilegeofthesonsofthegods,i.e.thepriesthood.95InallthistheGreeksemployedtechniquesthatwerenotsignificantlydifferentfromthoseemployedbytheearlierEgyptianarchitects.96Thusthetopradiusofeverycolumnisrelatedtothebottomradiusintheexactproportionof5to6,thesacredPythago-reannumbersof10and12,etc.

Inthetemple,saysVitruvius,weseehowalldisciplinesareinterconnectedamongthemselves,bothwithrespecttosubstanceandrelationship,...composingtogetherthesev-eral numbers into a single unified corpus.97 This holismis carried on secretly but recognizably down to the pres-ent, as theheritageofPythagoraswas transmittedby theTherapeutae of Alexandria. Perhaps the oddest thingabout Modern Science, wrote Bertrand Russell, is thereturn toPythagoreanism.98 Thebasic senseof thewordharmony(,harmonia)isnotamusicalbutacarpenters

93. Vitruvius,On Architecture3.1.39. 94. Lothar Haselberger, The Construction Plans of the Temple ofApolloatDidyma,Scientific American,December1985,128B. 95. TonsBruns,The Secrets of Ancient Geometryand Its Use,2vols. (Copenhagen:Rhodos,1967),2:109. 96. Haselberger,TempleofApolloatDidyma,132. 97. Vitruvius,On Architecture1.1.12. 98. QuotedinGhyka,Geometry of Art and Life,168.

Figure85.ThephilosopherProtagorassaid,Manisthemeasureofallthings.Theidealproportionsofamanwerecarvedtofitexactlywithinthiselegantbas-reliefgable,nowcalledthemetrologicalrelief.Marble,ca.450b.c.AshmoleanMuseum,Oxford.


the aesthetics of scientific theories, . . . communicating asenseofinterrelationshipsamongstacomplexGestalt.101Ofthesefigures,wearetold,mostunderstandinghasvan-ished,thoughafewlinesorsimpleformsimplyamuchgreateramountofcommunicationthancouldotherwisebemade.102

Thephi ratios (see p. **, fig. 72) and star pentagram inthe Joseph Smith hypocephalus not only locate our pen-dantworld in thecosmosand inhabit itwith living thingsbutput the stampof eternityon thewholeplanaccordingtotheteachingsofHuntley:Naturesbeautydies.Thedaydawnswhen thenautilus isnomore.The rainbowpasses,

101. Derek J. de Solla Price, TheHexagram, Pentagram, andOcto-gram, andOtherGeometrical andScientificTalismans andSymbols,inChanging Perspectives in the History of Science: Essays in Honour of Joseph Needham, ed. Mikul Teich and Robert Young (London: HeinemannEducational,1973),250. 102. Ibid.,25052.

term,ameansofjoining,fastening;joint;framework,99thetriumphoftheworkbeingthetemple.ItissignificantthatoneofthefiguresintheJosephSmithhypocephalusistobehadintheHolyTempleofGod(Fac.2,fig.8,explana-tion).TonsBruns,whohashisownspecialinterpretationofdivineproportions,claimsthatancientgeometry...(was)wrestedfromthehandsofreligiousordersbyguildsofmedievalstonemasons,whichinessencewereapoorimita-tionoftheoncepowerfultemplebrethren.100

Thestrangeattractionofnumberstosomeismatchedbytheappealofvisualdesigntoothers:Thereexistsatypeofhumanmindtowhichthethreesymbols[hexagram,penta-gram,andoctogram]...speakwithouttheinterventionofwords....Suchnon-representationaliconography...formsalongandfiguratetradition....Itisavitalcomponentof

99. Henry Liddell and Robert Scott,Greek-English Lexicon (Oxford:Clarendon,1996),224. 100. Bruns,Secrets of Ancient Geometry,9,57.

adytum

naiskos

fluting

chord

contour arc

axis of symmetry

top radius

bottom radiuscolumn diameter

base

adytum

naiskos

fluting

chord

contour arc

axis of symmetry

top radius

bottom radiuscolumn diameter

base

Figure86.ThehugetempleofApollowasbuiltaroundthesmallancientnaiskostohonorthegodandtoprovidealureforpilgrims.Thearchitects(334b.c.)preservedtheirmasterplansbyengravingthemontheinte-riorwallsoftheadytumsincetheyknewtheywouldnotlivetoseethecompletionofthegreatwork.Twenty-fivecolumnswerestillnoterected

whenthetemplewasabandonedsixhundredyearslater.Toensurethatallthecolumnswereconsistent,exactdiagramswerelaidout,sometimesat1/16scale.Thecontourarcshowedthesubtlecurveofthe65-foottallcolumns,whichcreatedtheopticalillusionthatthecolumnswerestraight.Didyma,Turkey.AfterScientific American,Dec.1985.

632 ONEETERNALROUND

theflowerfades,themountaincrumbles,thestargrowscold.But thebeauty inmathematicsthedivineproportion, thegolden rectangle, spira mirabilisendures for evermore.103Thusbyintroducingtheseideasintothehypocephalus,theartistmightconsciouslyorunconsciouslyhaveputthestampofeternityonit.Itisadeeplyreligiousratherthanmagicalobject.This,aswehaveseen,isamplyconfirmedforJosephSmith by closer consideration of the figures in the JosephSmithhypocephalus.

103. Huntley,Divine Proportion,176.

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