Topography, Astronomy and Dynastic History in the Alignments of the Pyramid Fields of the Old Kingdom

MediterraneanArchaeologyandArchaeometry,Vol.10,No.2,pp.5974Copyright2010MAA

PrintedinGreece.Allrightsreserved.

TOPOGRAPHY,ASTRONOMYANDDYNASTICHISTORYINTHEALIGNMENTSOFTHEPYRAMIDFIELDS

OFTHEOLDKINGDOMMagli,G.

FacultyofCivilArchitecturePolitecnicodiMilano,PiazzaLeonardodaVinci32,20133Milan,Italy

Received:12/10/2009Accepted:16/01/2010

Correspondingauthor:[email protected]

ABSTRACTItisknownsincethe19centurythatinthelayoutofthepyramidfieldofthepharaohsofthe4th

EgyptiandynastyatGiza,amainaxisexists.Indeed,thesoutheastcornersofthesemonumentsaligntowardsthesiteofthetempleofHeliopolis,whichwasplainlyvisibleinancienttimes.ItwaslaterdiscoveredthatasimilarsituationoccursinthemainpyramidfieldofthesubsequentdynastyatAbuSir.Here,thenorthwestcornersofthreechronologicallysuccessivepyramidsagainvoluntarilyalign towardsHeliopolis.However, the templewas in thiscasenotvisible,due to therockoutcroptodayoccupiedbytheCairocitadelwhichblockstheview.Inthepresentpaper,ainterdisciplinaryapproachbasedonhistorical,topographicalandarchaeoastronomicalanalysisisdevelopedinanattemptatunderstandingsuchpeculiarfeatures,whichgovernedfromtheverybeginningtheplanningofthesewonderfulmonuments.Ageneralpatternactuallyarises,whichappearstohaveinspiredthechoiceofthesitesandthedispositiononthegroundofalmostallthefunerarycomplexesof thekingsduring theOldKingdom. Inparticular, thispatternhelps to explain thechoicesinthelocationofthefunerarycomplexesofNiuserreinAbusir,ofUnasinSaqqaraandofthekingsofthe6thdynastyatsouthSaqqara.

KEYWORDS:Gizapyramids,Abusirpyramids,Saqqarapyramids,SacredLandscape,AncientAstronomy

MAGLI,G.60

1.INTRODUCTIONAninterestingfeatureexistsinthelayoutsof

thepyramidsofGizaandAbuSir:thepresenceofamainaxisdirected to theareawhere theancient temple of the sun of Heliopolis oncestood,on theoppositebankof theNile.Theseaxes are connected with a process of solarisation of the pharaoh which probablystarted with Khufu, the builder of the GreatPyramid.AtGiza, theaxis (alreadydiscoveredin the 18th century) runs across the southeastcorners of themain pyramids (Goedicke 2001,Lehner 1985a,b, Magli 2009a,b). In Abu Sir, astraight lineconnects thenorthwestcornersofthepyramidsof three successivekings (Verner2002).However,theviewtoHeliopolisisblockedhere by the rock outcrop of the Cairo citadel(Jeffreys 1998). This alignment is, therefore,quitemysterious:itsintentionalityandrelationshipwiththecultofthesunarecertain,but itsaim seems to be failed.As a consequence, theverychoiceofthesiteofAbusirbythekingsofthe 5th dynasty remains unexplained (see discussionsinKrejci2001,Goedicke2001).

Theresearchpresentedhereoriginallystartedasanattemptatsolvingthisenigma;however,itturnedoutthatwhatwasnecessarywasacarefultopographical and archaeoastronomical analysisofthewholesetofmonumentsbuiltbetweenthe4thandthe6thdynasty,accuratelytakingintoaccountwhatwasknownhistoricallyandarchaeologically about them. In the course of such ananalysis, a general pattern was seen to arise,which inspired the location andplanningof almostallmonumentsand inparticular led to thechoice of the Abu Sir plateau. This pattern whichIwillcall,forareasonwhichwillbeclearsoon,symbolicinvisibilityisbasedonasetofradial lines(diagonals)whichconnect ina idealwayHeliopoliswith the sacred landscape builton theoppositebankof theNile. ItsoriginsappearstobeconnectedwiththeancientEgyptiansreligion, funerary cult and dynastic lineage, aswellaswith theirastronomicalknowledge.Further,asweshallsee,theideaofconnectinganewfunerary monument with the existing ones bymeans of topographical alignments remainedalivealsointhelatertimesoftheOldKingdom,duringtheendofthe5thdynastyandtherunof

the6th. It lost,however, thesolarcharacterofconnectiontowardsHeliopolis.

Fig.1MapofthemainGizapyramidswiththeGizaaxishighlighted(fromLehner1999)

Fig.2MapoftheAbuGorabAbuSirareawiththeAbuSirdiagonalshighlighted(numberingofthemonumentsinchronologicalorder).1UserkafSuntemple,2/3/4PyramidsofSahure,Neferirkare,Neferefre5unfinishedpyramidofShepseskare6a/6b/6cNiuserrePyramid,MastabaofPtahshepses,Niuserre

Suntemple.

2.THEGEOGRAPHYOFTHEMEMPHITEAREAANDTHEHORIZONFORMULA

Thegeographicalareaof interest in thepresentpaper canbevisualizedasa triangle (Fig.3). Thewestern leg of the triangle follows therockridgeofthedesertonthewestbankoftheNile from Abu Roash (A) to Abu Gorab (B)crossing Giza (G) and Zawiet el Arian (Z) ,whiletheothertwolegsaredefinedbytheposi

TOPOGRAPHY,ASTRONOMYANDDYNASTICHISTORYINTHEALIGNMENTSOFTHEPYRAMIDFIELDS 61

tion (H) of ancient Heliopolis, located on theeast bank of the Nile (to visualize this area Ihave chosen asA the center ofDjedefrePyramid,asBthecenteroftheUserkafSunTempleand asH thepositionof theMiddleKingdomobelisk in theHeliopolis area).The legHA isabout 25Kms long and crosses the flatland oftheNilevalley; the legHB is about 27.5Kmslong; it also crosses the Nile flatland but it istangent to the rock outcrop located at thenorthwestextremeoftheMoqattamformation.This outcrop is today occupied by the Cairocitadel (K). Important in what follows will bealsoAbu Sir (S)which is aplateau located ontheridgeofthedesertsomehundredsofmeterssouth of Abu Gorab, the pyramid field ofSaqqara (Q) further south, and the site of theOldKingdomcapitalMemphis(M).

Fig.3Thegeographicalareaofinterest.H=Heliopolis,K=CairoCitadel,A=AbuRoash,G=Giza,Z=ZawietelArian,B=AbuGorab,S=AbuSir,Q=Saqqaramain

field,M=Memphis

Weshallconcentratehereontheissueofmutual visibility/invisibility between Heliopolisand the abovementioned sites.Since it is easytofindnonscholarlypublicationswhereancientmonuments are allegedly supposed to be connectedbyinvisible lines, it isperhapsworthclarifying that thepresentworkhasnothing todowithmostsuchspeculations.Usuallyindeedthey have little sense either because the supposed lines aredrawnon aplane chart (whiletheearthhappenstoberound)orbecausetheyconnect siteswhichdonothave anyhistoricalconnectionswhatsoever(orbecauseofbothreasons together). In the present paper, monumentsbuilt ina relatively shortperiodof time

bydynasticallyrelatedrulersarediscussed,andthe specific issue of intervisibility is carefullyaddressed every time a interconnecting line isproposed. To this aim,we first notice that, asmentioned, theMoqattam formationblocks theview between H and any zone further souththanAbuGorab, thefirstof the invisiblezonesbeing theAbuSirarea.The legHB is thereforethe last possible intervisibility line fromH tothewestbank lookingsouth.ThetriangleHABincludesonlyflatlandandthereforethereexistsa theoretical intervisibility between all itspoints, in particular betweenH and places ontheABleg.Duetotherelevantdistanceswhichcome intoplayhowever,wehave to take intoaccount earths curvature. To this aim, webrieflyrecall thesocalledhorizon formula(actuallyastraightforwardconsequenceofPythagorastheorem,seeFig.4).

Fig.4Thehorizonformula

Given a point P located at vertical height hwith respect to theearth surfaceP, the farthermostpointWthatcanbeseenfromPisthepointwherepassesthetangenttotheearthcircumference from P. Clearly then the distance d=PWequals(2rh+h)whereristheearthradius.Iftheheightsarevery smallwith respect to rwe canputd~(2rh).Since2r~13000Km,wearriveatthesocalled horizon formula: the value of d inKmsapproximatelyequalsthesquarerootof13hif h is expressed in meters. It follows, for instance, that aperson 2meters tallhas a visiblehorizonslightlygreaterthan5Kms.However,ifthe object sighted has a non negligible height,then the twosumup.Asaconsequence,asunreflectingsignallocatedinHeliopolisatsay20meters above ground would have been easilyvisiblefromthewestridge,locatedatanaverage

MAGLI,G.62

ofsome20metersabovetheplain.Suchasignalmight have been located e.g. on the Petriesmound, a symbolic primeval mound whichwasprobablytheresincethefirstdynasties(foracomplete discussion on Heliopolis see Quirke2001andreferencestherein)or,moresimply,onanobservationpostlocatedatthetopofaprovisionalwoodenstructure.Oncethebuildingsitesof thepyramids reacheda sufficientheight, theuppercoursesbecameintervisiblewithHeliopolis. The fascinating view of the Giza pyramidsfromHeliopoliswasstillenjoyableat theendofthe19century,as confirmedby somepaintingsofthattimes;attemptsatadirectobservationaredoomed to failure today,due tomodernbuildingsandpollutionwhich strongly limitsvisibility.

3. GEOGRAPHICAL AND CHRONOLOGICALORDEROFTHEMONUMENTS

Inwhat followsweshallbe interested inalmost all the pyramidal complexes constructedduring theOldKingdom.Thegeographical location of these monuments and their chronologicalorderwillbothbeessential (Fig.5).Beforegoingfurtherhowever,itisworthrecallingthatthechronologyoftheOldKingdomissubject to debates about the length of each singlereignand the successionofa fewof thekings.Asa reliableworking framework I followherestrictly the accession dates as given by BainesandMalek(1984)andadopted,for instance,byLehner(1999).

Fig.5Thepyramids,theSunTemplesandtheHeliopolitandiagonalsdiscussedinthepaper:(1)AbuRoashdiagonal,(2)Gizadiagonal,(3)Abusirdiagonal.Thedottedstraightlinedenotesinvisibility.Thesteppedline

denotesthemaximalNileflood.

The Age of the Pyramids starts with the

Step Pyramid in Saqqara, built by the first 3thdynastypharaohDjoser (26302611).Afterhim,other pyramids will be built by kings of thesame dynasty: the first is Sekhemkhets inSaqqara.After,thesocalledLayerPyramid(Z1)wasbuilt atZawiet elArian and anothervery

far south, inMeidum.The attribution of thesetwoisuncertain;thefirstmighthavebeenbuiltbyakingcalledKhaba, thesecondbyHuniorbySnefruinthefirstyearsofhisreign.Alsouncertainistheattributionofanunfinished,partlymudbrick pyramid which is located in AbuRoash, the so called Lepsius 1 pyramid (A1);


anyhow thesubstructureexcavated in the rockpointstotheendofthe3th/beginningofthe4thdynasty (Swelim 1983).The 4thdynasty beginswithSnefru(25752551),thebuilderofthetwomagnificent pyramids located in Dashour(SouthSaqqara).WiththesonofSnefru,Khufu,weenterintotheperiodwhichisofdirectinterestforus.1) Khufu (2551 2528)builds theGreatPyra

mid(G1)inGiza.2) Djedefre (25282520) builds his pyramid

(A2)inAbuRoash.Djedefreaddsthesolarsuffixretohisname,formalizingaprocessofsolarisationofthekingwhichwascertainlyinitiatedbyhisfatherKhufu.

3) Khafre (2520 2494) builds or finishesbuildingthesecondpyramidofGiza(G2)1

4) Someone,perhapsanundocumentedshortreign king, starts the building of a hugeproject,comparabletothatofG1,inZawietelArian(Z1).

5) Menkaure ( 2490 2472) builds the thirdpyramidofGiza(G3).

6) Shepsekaf(24722467)appears tobreak thetradition.Indeedhedoesnotbuildapyramid,butagiantMastabaatSaqqaraSouth(called todayMastabaElFaraun); theexistence of a break in the funerary and religiouscustoms isreflectedalso inthepharaohstitle,whichdoesnotbringthesolarsuffix (for a discussion of different viewpointsseeVerner2002).

7) Userkaf (2465 2458) returns to the tradition.Hebuildsapyramid inSaqqara, significantlylocatedascloseaspossibletothewallof the firstpyramidever constructed,Djosers. Userkaf also builds the first suntempleinthezoneofAbuGorab(B1).Thistemplewascomposed, like thestandardpyramids twotemples / causeway complex, by a building set lowerdown and amonumentalaccessrampslopingupwards,givingaccesstoamonumentalarea.Here,ahugestonebasementwassustainingahigh(nonmonolithic)obelisk.Suchamonumentwascertainlydevoted to theSunGodanditsplanwasprobablyconceivedasacopyofthetempleinHeliopolis.

8) Sahure (24582446). With Sahure, whosenamemeansClosetoRe,wereturntoso

larisedkings.ThepyramidofSahure is locatedinAbuSir(S1).

9) Neferirkare (2446 2426) builds his pyramidinAbuSir(S2)

10) Neferefre builds his (unfinished) pyramidinAbuSir(S3)

11) Shepseskare probably builds his pyramidinAbu Sir (S4) aswell (the attribution ofthisunfinishedmonumentisnotcertain).Itis not completely clear if really this kingwas the successor of Neferefre or ratherviceversa,although,asweshallsee,theresultsof thepresentpaperstronglysuggestthe first option. In any case both reignswereveryshort,lastingasawhole10yearsorless(24262416).

12) Niuserre (24162388)buildshispyramid inAbuSir (S5)andasun temple (B2) inAbuGorab.Niuserrewas the lastking tobuildhis funerary complex inAbu Sir. The lastkingsof the5thdynastywill indeed returnto Saqqara, as well as all the subsequentpharaohs(Section8).

4.VISIBILITY,INVISIBILITY,ANDSYMBOLICINVISIBILITY

Theabove listmay,ata firstglance, lead toconcludethateachkingdecidedinasomewhatrandomway toputhispyramid in one of theavailablesitesonthewestbankoftheNile,providedjustthatitwasnottoofarfromthecapitalMemphis.Actually,theplacementofpyramidalcomplexesdidhavetotakeintoaccountaseriesofpracticalfactors(Barta2005)suchaspresenceof nearby stone outcrops to be transformedinto quarries and accessibility of materials.Perhaps also thepresenceof the still active building site of the pyramid of the precedingpharaohmayhave influenced thechoice.However,there isnopossibledoubtonthefactthat,at least inmany cases, the pyramidswere notconstructed were reasonableness would havewanted. A clamorous example is that of theMenkaurepyramid(G3),whichhasbeenvoluntarilybuiltveryfarinthedesert;similarly(andindeedfortheverysamereason,asweshallsee)thisholdsfortheNeferefrepyramid(S3).Moregenerally, itmust be remembered thatwe arespeakinghereaboutnothing less than someof

MAGLI,G.64

the most magnificent, complicated and hugebuildings of thewhole human history.We doknowthatsuchmonumentsweredesignedwithmaniacalaccuracy(especiallyduringthe4thdynasty)and that theirconstructiondeserved theefforts of thousands of specialized, well feed,skilled workers (Lehner 1999, Hawass 2006).Thepyramidswerethesymbolofthepowerofthepharaohoverthedeathand,inreflection,ofthecontinuityof lifeof thewholecountry. It istherefore natural to think that their placementwascarefullychoseninordertosatisfyalsotoasetofreligious,symboliccriteria.

To investigate on such criteria our startingpointisthesocalledGizadiagonal(Fig.1).Itis an ideal line which connects the southeastcorners of the threeGiza pyramidswith goodaccuracy(alinebetweentheG1andG3cornersleaves the G2 corner about 20 meters to thenorth).Theexistenceofthislinetellsustwoimportant things.The firstone is just that thesiteof Giza was intervisible with Heliopolis andthatthisdoesnotoccurbychance:witnessistheplacement of the three monuments in accordance with such a Heliopolisdirected diagonal. The importance of this observation hasbeenenhancedbyacomprehensivestudyoftheintervisibility between Heliopolis and thepyramids (Jeffreys 1998). In this study it isshownthatthesitesofthe4thdynastypyramidswhoseowners first declared an affinitywiththe sun cult through their monuments and/ortheir nameswere chosen in such away to bevisiblefromHeliopolis.Inthesamestudyhowever itwas also noticed for the first time thatAbu Sir another burial place of solarisedkingsisnotvisiblefromHeliopolis:itactuallystands justa fewhundredsmeters to thesouthof the lastvisiblepoint. In spiteof this,aAbuSir diagonal very similar to that ofGiza doesexist.Itconnectsthenorthwestcornersofthepyramidsbuiltby Sahure,Neferirkare andNeferefreanditdoesindeedpointtoHeliopolis, inspiteof the fact that theplace is in itselfinvisible.The intentionalityof thisalignmentwhich was, therefore, obtained using signalslocatedhalfwayontheoutcrophasbeencontrolledbeyondanypossibledoubtusingtransitsurveymeasures(Verner2002).

The invisibilityof theAbu Sirdiagonal which governed from the very beginning thetopography of three pyramid building sitesthere is a firsthint that something of importance has yet to be understood in all this scenario.Anotherhintcomesfromasimpleobservation. In consequence of the alignment, lookingfromHeliopolis(orfromanyotherpointoftheaxis)theimagesoftheGizapyramidscreatea perspective effect: they contract on eachotherandmergeintothatoftheGreatPyramid.Therefore, it can be said that the G2 and G3pyramids were planned respecting two constraints(Magli2009):1) The chosen place is in plain view from

Heliopolis2) Inthisplace,thetwobuildingsG2andG3are

locatedintheuniqueavailablepositionswhichareactuallyinvisiblefromthetown,duetothepresenceofG1Whilepoint(1)canbeexplainedby thewill

of the ruler of writing in the stones his closeconnectionwiththeSunGod,point(2)seemstoawait for an explanation.As far asAbu Sir isconcerned,thesituationisevenmoreintriguing.Indeed,herewehave1) The chosen place is the first available goingsouthtobeinvisiblefromHeliopolis

2) In thisplace, the second and the third successivebuildingsaredisposedinsuchawaytobesymbolicallyinvisiblefromHeliopolis. Inotherwords, lookingalong theAbuSir diagonal they become almost invisible(in thesecondphaseofconstructionS2wasreplannedtobehigherthanS1).To explain this chain of occurrences, I pro

posethattheycorrespondtoawayofinterpreting andmodeling the sacred landscapewhichruledtheplacementsofthepyramidsofthesolarisedkingsup to theendof the5thdynasty.Thismodel,whichIproposetocallsymbolic invisibility,requiredthepyramidsofdynasticallyrelatedpharaohstobebuiltalongalineofsightconnectingakeyelementofthelayoutandthenideallyextendinguptothetempleofHeliopolis.In thisway, the direct lineage of the Pharaohfrom the Sun God was actualized on the


ground (see the final section for further insights).

To illustratehow themodelworks,wewillnow rerun the succession of the kings mentioned earlier, putting in evidence if and howthe locationof their respectivemonuments fitsintotheframework.1) KhufuselectsGizaasthebuildingplacefor

hispyramid.TheGizaplateau is certainlysuitable for the enterprise, but there arehintsthattheplacemighthavebeenchosenalsobecause itwassacred to thesunsincethepreviousdynasties.

2) Thefirsttoconformtothesymbolicinvisibility conceptwasDjedefre,Khufus son.Indeed, ifweplot the linewhich connectsHeliopolis with his pyramids southwestcorner,weseethatitcrossesnearthesouthwestcorneroftheAbuRoashpyramidA2,the so called Lepsius 1, which sits at theeasternmost end of the Abu Roash hills.Rather than forming an Akhet sign (twocoupledmountainswith the sun settingatthe center at the solstice) as has been recentlysuggested(Shaltoutetal.2007)thesetwopyramids thus forma AbuRoashdiagonalpointingtoHeliopolis.

3) After,Khafredecidedtoconformhimselftothe model as well, but returning to Giza;perhaps he appropriated and finished theconstructionof thesecondpyramid,but inanycasehewantedhistombtobeattachedmodestlytoapreexistingproject,thatofKhufu,respectingthesymbolicinvisibility.

4) The very same choice was made byMenkaure,whoarranged the layoutofhisfunerary complex in order to harmonizewith the preexisting one. The dynasticlineage of Khufu was thus actualized atGizabythesuccessionwhichactuallyproceeds into the desert of the southeast cornersofG1,G2andG3

5) TheunattributedpyramidatZawietelAriandoesnotfitthisscenario;someauthorsattributeittothe3thdynasty(anattributionwhich of coursewould solve the problemhere)butthesimilarityofitsprojectwith4thdynastypyramids is reported tobeapparent2(forfurtherdiscussiononthispyramidseesections7and8).

6) MenkaurewassucceededbyhissonShepsekaf,who broke the solarised traditionbuildinghis tombasaMastabaatSaqqaraSouth, not far from Dashour and completely out of sight fromHeliopolis; actually, this huge monument appears to beconceivedasasortofgiantbaselineforthedoublemountains symbol of the Snefrupyramids, if seen from the ancient accesswaytotheStepPyramidplateau.

7) Afterhim,Userkafactualizedanambitiousbuilding program, which consisted in exhibitingareturnandacloselinktooldtraditionsofboththe3thandthe4thdynasties.Indeed, to show is closeness to the3thdynastykingshebuilthispyramidascloseaspossibletothatofDjoser.ToexhibitexplicitlyhisconnectionwithHeliopolisand thesolarised kings tradition of the 4th dynasty,Userkafbuilt thesun temple inAbuGorab. Interestingly enough, the templewas constructed very near the southernmostavailablepointofthewestbankoftheNile from which Heliopolis is still visible(Kaiser1956,Jeffreys1998).

8) WithSahurewedefinitively return to solarised kings.According to the proposalsof thepresentpaper,we can infer that thenaturalchoiceforSahurespyramidwouldhavebeenGiza,with theconstructionofafourthmonument aligned along theGizadiagonal. However, building a pyramid complexevenmorefarawayinthedesertwithrespect to Menkaures would have beennearly impossible. Thus, the architect hadtofindanewideatoallowhiskingtoconform to the symbolic invisibilitymodel.This ideawas toplace thepyramidofSahure in the first available location in thesouth fromwhichHeliopolis isnotvisible:AbuSir.

9) Immediatelyafter,Neferirkareinaugurateda new dynastic diagonal in this place,putting thenorthwest corner ofhispyramidon the ideal lineofsight toHeliopoliswhich crosses the corresponding cornerofthe Sahure pyramid (the apparently exaggeratedistancebetween themonuments isduetothenecessityofavoidingaslopelocated in between, see Krejci 2001). Al

MAGLI,G.66

thoughAbuSirisinvisiblefromHeliopolis,anypersonat the templewouldhavebeenaware,lookingatthewesternhorizon,thatthebrilliantobeliskoftheUserkafsuntemple indicated the beginning of the sacredarea,wherethekingsofthe5thdynastydecidedtobeburied,andthattheirpyramidswereactuallydisposedalongadiagonalpointingtoHeliopolisalthoughmodestlyinaninvisibleway.

10) ThefactthatalsotheunfinishedmonumentofNeferefre isalignedon the samediagonalisastronghinttotheideathatthiskingwas the first successor of Neferirkare(Verner 2001). After him, his successorShepseskare had to confront to the sameproblem which faced Sahure at Giza,namely, it was impossible to attach hispyramid to the preexisting 3pyramidsdiagonal without going very far in thedesert. Thus, the architect of the pharaohprobablyplannedthepyramidinthespaceleftbetweenUserkafs sun templeandSahurespyramid, remaining in thiswaynotjust invisible, but symbolically invisiblefromHeliopolis.

5.THENIUSERREPROJECTANDTHEMEANINGOFTHESUNTEMPLES

Theproblemoffindingasymbolicallyadequateplaceforthepyramidofthenextking,Niuserre,becamedramatic. Inordernot togo too faroffin the desert the planners of his monumentfoundnootherwaytoexhibitthelineageofthekingthanplacinghispyramidtotheeastsideofanexistingone,thatofhisfatherNeferirkare,aquiteunique exampleofintrusivedesign. Inthis way although it may seem incredible they also managed to inaugurate yet anotherdiagonal (Fig. 2). This secondary Abu Sirdiagonal is a line (first discovered by Lehner1985b)whichdynasticallyconnectsthesoutheastcornerofNiuserrespyramidwiththoseofthepyramidsofNeferefreandNeferirkareandtouches also the corner of the Mastaba ofPtahshepses, a very important personagewhobecameasoninlawofNiuserre.However, themain problem was still standing namely, thefactthattheconnectionofthekingwiththesun

god was not symbolically expressed by hisfunerary complex. We can therefore speculatethatperhapsexactly thiswas the reasonwhichinspired the construction by Niuserre of hisownSunTemple,locatednorthoftheUserkafsoneandthereforeinplainviewofHeliopolis.Inthisway (and although itmay really seem incredible)hisarchitectsalsomanagedtoinaugurateyetanotherdiagonal.Thisdiagonalisatleasttothebestoftheauthorsknowledgepresentedhere for the first time. It is a 7km lineoriented ~45 south of east which originatesfromthecenteroftheGreatPitinZawietelArian and intersects the southwest cornerof the3th dynasty pyramid in the same site. It thencrosses over Abu Gorab intersecting with impressiveprecisionthebasisoftheobeliskintheNiuserreSunTempleandthesouthwestcornerofUserkafstemple,andendsattheapexoftheNiuserrepyramid.

Fig.6TheZawietelAriandiagonal

The functionof thisdiagonal is to exhibit a

close linkbetween the funerarycomplexof theking and his Sun Temple. Actually, the truemeaning and origin of the SunTemples hasneverbeen explained satisfactorily.Firstof all,accordingtoexistingtextsoftheepoch,itseemsthat also Sahure, Neferefre, Neferirkare andMenkahour constructed one such monuments.However,thesetempleshaveneverbeen found.Asa consequence, some scholars have proposedthatthetextsactuallyrefertorenewalsmadebysuch pharaohs to Userkafs temple, and thusthat the unique other sun temple effectively


constructedafterUserkafswasthatofNiuserre(foradiscussionseeVerner2002,Nuzzolo2007andreferencestherein).Actually,at least inthepresent authors view, it is almost impossiblethat themissing templeswere completely dismantledastosafelyescapetothearchaeologicalinvestigationoftheveryrestrictedareainwhichtheyshouldhavebeen located,since it isobviousthattheyhadtobeonthewestbankandinview fromHeliopolis (the idea, sometimesputforward in thepast, that they couldhavebeenlocatedon theeastbank isat least in theauthors view definitively incompatible withtheir symbolic meaning of counterparts ofHeliopolis).

Clearly,theapproachpresentedinthispapersupportsthethesisthatthemissingtemples(oratleastthoseofSahure,NeferefreandNeferirkare,whosepyramidsareknown)neverexisted.In fact, I propose that Niuserre constructed anew temple near Userkafs one because hispyramidwasnot symbolically linked toHeliopolis,beingoutof theAbuSirdiagonal. If thistheory is correct, then at leastoneof the symbolicmeaningsof these temples canbe furtherspecified as follows: they were witness of theroyal linkwiththeSunGodwheneverthis linkwasnot explicitly stated by the location of the funerarycomplex of the king. The theory proposed thuspointstoanextremelycloseconnectionbetweentheconstructionofsun templesand thatof thefunerary complexes.Actually, the architecturalcomplexcomposedby thestandardelementsValley Temple/Causeway/Funerary Templefinds its first complete expression at Giza (inpart it is already present at Meidum andDashour). It starts therefore together with thesolarised kings funerary complexes. If, asmentioned(Quirke2001)theSunTempleswerea sort of replica of theHeliopolitan one, thenalsothepyramidstemplescomplexwiththeirsimilar two buildings + causeway scheme mighthavebeenconceivedonthesamebasis.Inthisconnection itmaybenoted thatgeologicalhintsata slight (a fewcenturies)predatingofpartsoftheSphinxcomplexandKhafremortuary temple to earlier dynasties exist (Reader2001). If this currentlymuchdebated,seee.g.Vandecruys (2006) hypothesis will be confirmedby future research then thesestructures

might refer to a earlier sun temple visuallyconnectedwithHeliopolis,wherethesuncultisknowntobeactiveatleastfromthe2thdynasty.

6.THESOLARISEDKINGSANDTHEPYRAMIDSGEOMETRY

Asalreadymentioned,Niuserrewasthelastking tobuildhis funerary complex inAbuSir.Thesuccessivekingsofthe5thdynasty,Menkahour,DjedkareandUnas,movedtoSaqqara3aswell as the very last kings who constructedpyramids in theOldKingdom, thoseof the6thdynasty(Section8).

Table1.Theslopesofthepyramidsofthe4th,5thand6thdynastiesstartingfromKhufusgreatPyramid.4

KHUFU 14/11DJEDEFRE 14/11KHAFRE 4/3MENKAURE 5/4USERKAF 4/3SAHURE 6/5NEFERIRKARE 4/3NEFERENRE ?NIUSERRE 14/11MENKAHOUR ?DJEDKARE 14/11UNAS 3/2TETI 4/3PEPII 4/3MERENRE 4/3PEPIII 4/3

Theendof the timeof thesolarkings isper

hapsreflectednotonlyinthedefinitiveabandonofthesolarsuffixandofanytopographicalconnectionwithHeliopolis,butalsoinastandardization in the geometry of thepyramids themselves.Toexplorethisissue,werefertoTable1,inwhichtheslopesofthepyramidsfromKhufuonward are reported in terms of the likely rationaltangentwhichwasusedtoconstructthem.WiththeexceptionoftheverysteepslopeofthepyramidofUnas(alikelyreasonforsuchaslopewill bediscussed below), these values vary between5012(Sahure)and5310(Khafre).Ithasbeen recently noted that these slopes roughlycorrespondtotheheightofthesunwhenpassingthe first vertical at the summer solstice duringtheOldKingdom.Thismeansthataspectacularilluminationeffect tookplace fora fewdays in

MAGLI,G.68

cludingthesummersolstice:thewestandsouthfacewerealmostsuddenlyandfullyilluminatedby the ascending sun at themoment it crossedthefirstvertical(BelmonteandZedda2007).Thisobservationfurnishesanadditionallinkbetweenthe pyramids and the solar cult and leads tospeculate that different slopes were experimentedinthecontextofthissunandshadoweffect. Among these slopes, that of the secondGizapyramid is theuniquewhich correspondstoaPythagoreantriangle(345,allintegerlegs).Whether this trianglewasconsideredsacred aspassedonbymuchlater,classicalwritersornot,thereisnodoubtthatPythagoreantriangleswerethebestsuitedforconstruction,becausethebuilderscouldimmediatelycheckthecorrectnessoftheslopeofthecasingblocks.Fromthetableitisapparentthat,withtheendofthetimeofthesolarkingstheslopeofthepyramidsbecomesafixedstandard,actuallythemostsuitableamongthosepreviouslyexperimented,i.e.the4/3slope.

7.POSSIBLEASTRONOMICALREFERENCESOFTHEHELIOPOLITANDIAGONALS

The symbolic invisibility framework provides a feasible explanation for the diagonaldispositionof thesuccessiveplansof themonuments of the solarised kings. However, nodoubt the funerary cult comprised, besides thesolar, also a stellar component. This isshown, for instance,by theaccuracyoforientationtotruenorthofthepyramidsbasesandbytheorientationof the shafts in theKhufupyramid,aswellasbeingconfirmedslightlylaterbythe Pyramid Texts (see Magli 2009, Magli andBelmonte 2009 for complete overviews on thissubject). It is, therefore, worth analyzing if thechoiceof thesitesmayhavehadalsoanocturnalconnectionwithHeliopolis,where,asiswellknown,astronomywaspracticed (clearly, if existing, such a connectionmighthave been conceived and used also as an help in tracing thediagonals).Toinvestigatethisissuewecheckifthe azimuths of the diagonals correspond tothesettingofbrightstarsasviewedfromHeliopolis(Fig.5),andtheresultsareasfollows:1) The Abu Roash diagonal (~28 south of

west) isalreadyknown tobeanastronomi

callysignificantsight line (seediscussion inShaltoutetal.2007).Itpointsquitepreciselytothewintersolsticesunset;italsopointstothe setting of Sirius (within one degree ifDjedefreaccessedin2528).Perhapsthiswasconnected to the name of the pyramid,whichwasDjedefreisastarSehedu(wedonotknowthemeaningofthetermSehedu).

2) The Giza diagonal (~45 south of west)points to the settingof thebrightestpartoftheMilkyWay.At those times anobserverlooking from Heliopolis would have seenthestarsoftheSouthernCross,followedbytheverybrightstarsofCentaurus,flowtogetherwith thegreatcelestialriveranddisappear from view behind the apex of theGreat Pyramid. In particular if Khufu isdated 2551 then the bright starRigil sat inoptimalalignmentwiththediagonal.

3) TheAbuSirdiagonal(~71southofwest)corresponds to the setting of the starCanopus.Asalreadymentioned,thesitewasjust invisible fromHeliopolis, and thereforethestarwasviewedtosetovertheoutcropof theCairocitadel. Inparticular ifSahure isdated 2458 the agreement iswithinonedegree.Of course, such occurrences might be acci

dental. In support of their being perhaps notaccidentalthefollowingargumentcanbecited.All the above mentioned stars were Decans(Belmonte 2001; about Canopus see howeveralsoBelmonteandLull2006).TheDecanswere36 celestial objects whose heliacal rising happened in successive weeks of 10 days. The(ritual)useof theDecans tocount thehoursofthenight isdocumented inEgypt from the 9thDynasty (2154 BC) but probably dates before.We thushave threedecanal starsorgroupsofstars Sirius, CruxCentaurus, and Canopus which respectively sat in (approximate) alignmentwith the three diagonals AbuRoash,Giza, Abusir when viewed from Heliopolis.Further,thesestarshadheliacalsettings(thelastday inwhichthestar isbarelyvisible, immediately after sunset, and then sets) in successiveperiodsgoingfromnorthtosouth(Table2).

Itmay benoticed that theunfinishedpyramid at Zawiet el Arian fits fairly well in the


above described astronomical picture. Indeed,the sight linebetweenZ2andHeliopolisbearsan azimuth ~56 south of west. This azimuth,besides identifying the risingof thebright starArcturusoverHeliopolis (Shaltout et al. 2007),points to thesettingof thedecanalstarFomal

haut as seen from the sacred city (within onedegreeata fiducialdateof,say,2500).Further,theHeliacal setting of Fomalhaut in earlyNovember fits in between the dates for CruxCentaurus(Giza)andCanopus(AbuSir).

Table2.Thecorrespondencebetweenthealignmentsofpyramidsofthe4thand5thdynastiesandthedecreasingsettingazimuthsofdecanalstarsasviewedfromHeliopolisatthe(likely)timeofconstructionofthemonuments

LOCATION RULER DATE AZIMUTH STAR HEL. SETTING (JUL.) MAP

ABU ROASH DJEDEFRE 2528 BC 28 S. OF WEST SIRIUS EARLY MAY A2GIZA KHUFU 2551 BC 45 S. OF WEST RIGIL END JULY G1ZAWIET EL ARIAN ? 2500 BC 54 S. OF WEST FOMALHAUT EARLY NOV. Z2ABU SIR SAHURE 2458 BC 71 S. OF WEST CANOPUS EARLY APRIL S1

8. THE UNAS PROJECT AND THESAQQARADIAGONAL

Thesymbolic invisibility thus led,at leastaccording to the thesismaintainedhere, to thebirth of three ideal lines which connectedHeliopolis with the pyramid fields of AbuRoash, Giza and Abu Sir respectively. WithNiuserre, two further ideal lines were conceived,which however lost the Heliopolitancharacter.Interestinglyenough,theideaofdynastic alignments (representing lineage, orcloseness of religious ideas, or more simplyhintstopasttraditions)remaineduptotheendofthePyramidsage.Itis,indeed,clearlyvisiblealsoatSaqqara.

The existence of a Saqqara diagonal hasbeen discovered by M. Lehner (1985b), whohoweverdidnotattempttodiscussitsmeaning.It isa lineoriented roughlySWNE (it isdifficult to ascertain its azimuthprecisely, becausethecornersofthepyramidsarenotcleared,butitcanbeestimatedasbeing~48northofeast).ItmighthavebeeninauguratedbyUserkaf.HeplacedhispyramidnearthenortheastcornerofDjosers precinct. In this way he aligned thesoutheastcornerofhispyramidwiththesoutheast cornerofDjoserspyramidand thenorthwest corner of Sekhemkhets unfinished pyramid. It should benotedhowever thatperhapsthisinhomogeneousalignmentwasnotintentional:theSekhemkhetspyramidoriginallyhada high enclosure wall, and the pyramid itselfmight have been already buried atUnas time

(Fakhry 1974).Userkafmighthave just chosenthebestpositioni.e.theonenearesttoedgeofthe plateau among all the possible locationsnearDjosersprecinct.Who,however, certainlydidnotchooseagoodpositioniskingUnas,thelastkingofthe5thdynasty.Heobligedhisarchitects toarrange thediagonalofhispyramidonthe line connecting the southeast corners ofDjosers andUserkafspyramids.There can benodoubtonthefactthatthisalignmentwasdesigned for symbolical, and not practical, reasons.Toachieve thegoal thearchitectswere infact obliged to build the pyramid near thesouthwestcorneroftheprecinctofDjoser,thusveryfarinthedesert.Consequently,theyhadtoconstruct also avery long (more than 700meters)causewayand,evenworse,hadtoclearthezonenearDjoserssouthwallwhichwasoccupied by many preexisting mastabas. Some ofsuch tombs were thus interred, others evendismantled.YetanotherproblemwastheheightofthewalloftheSteppyramidcomplex,whichwould have obstructed the view of the Unaspyramid toapersonapproaching thediagonalalignment; it is for this reason (at least in theauthors view) that the pyramid was plannedwith a very steep slope, actually the steepestslopeofall theOldKingdompyramids.To increase visibility the line was further tracedalongthediagonalofthepyramid,notalongthesoutheast corner, again a unique case amongthevarious diagonalswhichusually connectthesamestructuralelements.

MAGLI,G.70

Fig.7TheSaqqaradiagonal(numberingofthemonumentsinchronologicalorder).1DjoserStepPyramid,2UnfinishedpyramidofSekhemkhet,3PyramidofUserkaf,4PyramidofUnas5PyramidofTeti

TheplacementofUnaspyramid closely resemblesthatofMenkaurespyramidinGiza,tothe point that the resemblance can hardly beconsideredcasual.Asimilaritywasalreadynoticed someyears agobyGoedicke (2001),whoobservedthatanunobstructedlineofsightconnects theUserkafpyramidwithKhufus.Actually, if similar connecting lines are traced between thesummitsofDjosersandUnaspyramids with the apexes of Khafres andMenkaures monuments respectively, it becomesclearthattheplacementofUnaspyramidwasconceived torealizeasortof (rough)copyof the arrangement of theGiza pyramid field.Theselinesareinfactabout14.5Kmslong(andtherefore allowed a direct intervisibility) and,althoughtheyarenotparallel,theirrelativedeviation stays within 2. Some years later, theSaqqara diagonal was further continued byUnas successor Teti. An oldfashioned symbolic invisibilitywould have required Teti tobuildhispyramid inapositionvery far in thedesert,along thediagonal to the southwestoftheUnasone.Thiszonewaspartlyoccupiedbythe Sekhemkhets unfinished complex, and/orthe original symbolic meaning of the diago

nalswas lost; inanycaseTetiavoidedthis inconvenientpositionandactuallychoosetobuilthis complex in the most favorable positionalong theSaqqaradiagonal,aligning thenorthwestcornerofhispyramidandplacingittothenortheast of the Userkafs one, very near theridgeoftheSaqqaraplateau.

Prolongation of the line further northeastshows that itpassesveryclose to the southernlimitof theareaof theearlydynasticmastabas(moreprecisely,the linepassessome70meterssouth of the southernmost Mastaba 3507) andthen crosses over the (unexcavated) zone ofarchaic Memphis. People ascending on theSaqqaraplateaufromthisdirectionthusexperiencedavisualsuperpositioneffectresemblingthatvisibleapproachingGizafromHeliopolis.

ThepyramidbuildingsiteofTetis immediate successor, Userkare, has never been individuated (seenextsection).AfterUserkare, theking Pepi I built his pyramid in a position insouth Saqqara, very near the ridge of the plateau, while his successor Merenre moved furthersouthwest(Fig.8).Inthisway,healignedthediagonalofhismonumenttothatofPepiI.Unfortunately,however,weshallneverknowif


PepiII,successorofMerenre,wouldhavelikedtoaddhismonument tothisnewdiagonal,because the correspondingposition to the southwestwouldhavebeenlocatedinaWadi(driedriver), absolutely not suitable for building apyramid.Perhaps as a consequenceof this,hechoose a position immediately south of thesameWadi,nearShepsekhafsmonument.

9. MERIDIAN ALIGNMENTS ATSAQQARASOUTHANDDASHOUR

TheFrenchEgyptologistGeorgeGoyondiscovered several meridian (i.e. northsouth)alignments between pyramids and previouslyconstructedmonumentsbelongingtoanancestor or sacred sites (Goyon 1977); themost famousofGoyonsalignmentisthatofGizawiththesacredsiteofKhem (Letopolis) locatedduenorth.A search formeridianalignments in thepyramidfieldseffectivelygivesquitesurprisingresults, which can hardly be attributed to achance.Further,theideaofmeridianalignmentsappears beforeGiza, because it is alreadypresentatDashour.Indeedwehave(inchronologicalorder):1) The Red and the Bent pyramid of Snefru

are constructed in such a way that theirwestsidesalignwiththetwosouthcornersof the socalledGisr elMudir, a huge rectangularenclosurelocatedsouthwestoftheStepPyramidandprobablyconstructedbya king of the first two dynasties, perhapsKhasekhemwy (Goyon already noticed atleastoneof suchalignments,whichhe referstotheDeMorganthediscovererof

the Gisr elMudir precinct). This alignmentactuallyhelpstoexplainwhytheRedPyramidwas constructed so far in thedesert,whiletherelativedistancebetweenthetworemainssomethingofamystery.

2) Thecenterof thevalley templeof theBentpyramid aligns with the apex of the StepPyramid.

3) AmeridianalignmentconnectstheapexofUnas with the west side of ShepsekhafsMastaba.OnthislinealsotheapexofMerenrespyramidwaslaterconstructed.

4) AmeridianalignmentconnectstheapexofPepiIwiththatofUserkaf.

Itappears,therefore,apossibleintentionalityinthesuccessiveplanningofthePepiIMerenrepyramids,whichideallyconnectsthemwiththeUnasproject(Magli2010).Atapurelyspeculative level, one might think that it wasUserkare to initiate thisnewpyramid field. Inthiscase,hemighthavelocatedhispyramidinmeridian alignment with that of Djoser, andthereforethismonumentmightstillbeburiedintheareabetweenPepiIandMerenrepyramids.Interestingly, a necropolis of dignitaries of the6thdynastyhasbeenrecentlydiscoveredinthesocalled Tabbet alGuesh, namely the zonewhichseparatestheSaqqaracentralgroupfromthe Pepi I complex, giving many hints at thepossibilitythatapyramidhasstilltobediscoveredatSaqqarasouth (Dobrev2008).Thisareaisactuallycrossedbyyetanotherpossiblynoncasual connecting line, since the northwestsoutheastdiagonalofPepiIpointstothesoutheastcorneroftheGisrelMudir.

Fig.8MeridianalignmentsbetweenDashour,SaqqarasouthandSaqqara(numberingofthemonumentsinchronologicalorder).1GisrelMudir2DjoserStepPyramid,3a/3b/3cRedPyramid,BentPyramid,Valley

TempleoftheBentPyramid4MastabaElFaraun5PyramidofUserkaf,6PyramidofUnas7PyramidofPepiI8PyramidofMerenre

MAGLI,G.72

10.DISCUSSIONMirceaEliadeoncesaid(1978)thatthesym

bolismcontained in thesacredspace issooldandsofamiliarthatitmaybecomeimpossibletorecognize it.Thepresentworkrepresentsanattemptatindividuatingtheexistenceofageneral pattern which inspired the humanmadelandscapeduringoneofthegreatestseasonsofhumancivilization.Wecancall thisperiod thetimeof the solarkings, thosegreat rulerswhowereclosetotheSunGodasthenameofSahurehasit.

Thisperiodwas inauguratedbyKhufu.According to authoritative scholars indeed (Stadelmann 1991, Hawass 1993) Khufu depictedhimselfastheincarnationofRa.Inthisconnection Hawass notices that no relief is knownshowingKhufumakingofferings toGods,andperhapsthisisnotduetothelackofdocumentationbutrather is inaccordancewith thenewcultwhichwouldhaveequatedthekingwiththegodhimself. If this is true, then thesolarised kings would have had the necessity ofshowing in explicit, monumental terms theirdirectlineagefromtheSunGodinthefunerarycomplexes.Inthepresentpaper,ithasbeenproposedthatthewayinwhichthiswillwasactualized was the intentional alignment of structuralfeaturesofsubsequentpyramids.Thishadthe consequenceof creatingvisual,perspectiveeffects, inwhich themonumentsprogressivelymergeeachother.

Ofcourse,wecannotattempthereatdevelopingacompleteanalysisofthepossibleconsequences in termsofdynastic (orsimplyinspiration) relationshipsbetween successivekingswho decided to connect visually their monumentstothoseofpreviouspharaohs.However,at least a few observations in support of thenonrandomness of the topographical orderwearediscussingherecanbemade.

First of all, the ubiquitous (Giza, Zawiet elArianAbu Sir, Saqqara) existence of surveylinesorientedquartercardinallyinthepyramidfields isreflected ina factwhichemergedveryclearly from the recently completed survey oftheorientationsofvirtuallyalltheancientEgyptiantemples(Belmonteetal.2009).Theanalysisof thedata indeed led theauthors to identifyaseriesoffamiliesoforientation;amongothers(likee.g.,ofcourse,asolstitial family)alsoaquartercardinal family of orientations isclearly discernible, and comprises temples ofanyepoch.

As iswell known,no contemporary textualevidence is available regarding the pyramidsprojects and design. However, an echo of aHeliopolitan symbolism of progressive ascendancecanperhapsbeseen inapassageof thepyramidtexts,whereitissaid(PT307)MyfatherisanOnite,andImyselfamanOnite,borninOn[=Heliopolis]whenRawasruler.Perhapsanother echo appears in the traditional talespassed on in the Westcar papyrus (Lichtheim1973). In this papyrus (dating to the MiddleKingdom) five storieswhichare told toKhufubyhissonsarereported.Inonesuchstories,thebeginningofthe5thdynastyisdirectlylinkedtothesuninaquiteintriguingway.Indeed,amagician called Dedi prophesies to Khufu thatthreesonsofthewifeofthechiefpriestofRainHeliopolis(andofRahimself,atleastaccordingtosometranslations)willreign,thefirstofthembeing Userkaf. Khufu gets upset of this, butDedi reassures the king that this will happenonly after thatKhufus son and grandsonwillreign.Ofcourse,Pap.Westcarhasnottobeconsideredasanhistoricaldocument;however,thetale clearly alludes to a discontinuity betweenthe4thand the5thdynastiesbyrescaling thelineage of the kings directly toHeliopolis, exactlyonewouldbetemptedtosayasithappensontheground.

ACKNOWLEDGMENTS

TheauthorgratefullyacknowledgesJuanBelmonteforacarefulreadingofthefirstdraftofthepresentpaperandmanyconstructivecomments.

REFERENCESBarta,M.(2005)LocationofOldKingdompyramidsinEgypt,CambridgeArch.Journal15177191.


BainesJ.andMalek,J.(1984)TheCulturalAtlasoftheWorld:AncientEgypt,Oxford.Belmonte,J.A.(2001)Thedecansandtheancientegyptianskylore:anastronomersapproach,InsapIIIMeeting,

Palermo.Belmonte,J.A.,andLull,J(2006)AFirmamentAboveThebes:UncoveringtheConstellationsofAncient

Egyptians,JHA37,373392Belmonte,J.A.,Shaltout,M.,andFekri,M. (2009)On theOrientationofAncientEgyptianTemples:(4)

EpilogueinSerabitelKhadimandOverviewJ.H.A.39Belmonte,J.A.&Zedda,M.(2007)LightsandshadowsonthepyramidsInLightsandshadowsincultural

astronomyed.M.ZeddaandJ.Belmonte,Ass.ArcheofilaSarda,Isili,p.181211Dobrev,V.(2008)OldKingdomTombsatTabbetalGuesh(SouthSaqqara),inAbusirandSaqqaraintheYear

2005M.Barta,F.CoppensandJ.Krejcieds.AVCR,PragueEliade,M.(1978)Occultism,Witchcraft,andCulturalFashions:EssaysinComparativeReligionUniversityof

ChicagoPressFaulkner,R.(1998)TheancientEgyptianpyramidtexts,OxfordUniversitypress,OxfordFakhry,A.(1974)ThePyramidsUnivofChicagoPressGoedickeH.(2001)AbusirSaqqaraGizainAbusirandSaqqaraintheYear2000ed.M.BartaandJ.Krejci,

AVCR,Prague.Goyon,J.L.(1977)Thesecretsofthepyramidsbuilders,Pygmalion,Paris,1977.Hawass,Z.TheGreatSphinxatGiza:DateandFunctionInSestoCongressoInternazionalediEgittologia.Ed.

G.M.ZacconeandT.RicardidiNetro,pp.177195.Turin.Hawass,Z.(2006)MountainsofthePharaohs:TheUntoldStoryofthePyramidBuildersDoubleday,LondonJeffreys,D.(1998)ThetopographyofHeliopolisandMemphis:somecognitiveaspects,BeitragezurKulturge

schichtegyptens,RainerStadelmanngewidmet(Mainz)6371.Kaiser,W.(1956)ZudenSonnenheiligtmernder5.Dynastie,MDAIK14,104116.KrejciJ.(2001)TheoriginsandthedevelopmentoftheroyalnecropolisatAbusirduringtheOldKing

dom.InAbusirandSaqqaraintheYear2000ed.M.BartaandJ.Krejci,AVCR,Prague.Lehner,M.(1985a)ThedevelopmentoftheGizaNecropolis:TheKhufuproject,MitteilungendesDeutschen

ArchaologischenInstitutsAbteilungKairo,41Lehner,M.,(1985b)AcontextualapproachtotheGizapyramids,ArchivfurOrientforsch.31,136158Lehner,M.(1999)Thecompletepyramids,ThamesandHudson,London.Lichtheim,M.(1973)AncientEgyptianLiterature,UniversityofCaliforniapress,BerkeleyMagli,G.(2009)AkhetKhufu:ArchaeoastronomicalHintsataCommonProjectoftheTwoMainPyra

midsofGiza,Egypt.NNJArchitectureandMathematics11,3550.Magli,G.(2009)MysteriesandDiscoveriesofArchaeoastronomy,SpringerVerlag,NY(originallypublished

asMisterieScopertedellArcheoastronomiaNewton&Comptoneds.,Roma2005).Magli,G.(2010)Archaeoastronomyandarchaeotopographyastoolsinthesearchforamissingegyptianpyramid

Preprint,Dept.ofMathematics,PolitecnicodiMilano.Magli,G.,andBelmonte,J.A.(2009)Thestarsandthepyramids:facts,conjectures,andstarrytales.InIn

SearchOfCosmicOrder selectedEssaysonEgyptianArchaeoastronomyEditedby J.A.BelmonteandM.Shaltout,SCAeditions,Cairo,(inpress).

NuzzoloM.,(2007)TheSunTemplesoftheVthDynasty:AReassessment,SAK36,217247.Quirke,S.(2001)TheCultOfRaThamesandHudson,London.Reader,C.(2001)AgeomorphicalstudyoftheGizanecropoliswithimplicationsforthedevelopmentof

thesite,Archeometry43,p.149165Rossi,C.(2003)ArchitectureandMathematicsinAncientEgypt.London:CambridgeUniversityPress.Stadelmann,R.(1991)DieagyptischenPyramiden:VomZiegelbauzumWeltwunder:P.vonZabern,MeinzSwelim,N.(1983)SomeProblemsontheHistoryoftheThirdDynasty.Archaeol.Soc.OfAlexandriapress.Vandecruys,G. (2006)TheSphinx:dramatisingdataanddating.PalArchs JournalofArchaeologyof

Egypt1,113Verner,M.(2001)WhowasShepseskara,andwhendidhereign?inAbusirandSaqqaraintheYear2000ed.M.

BartaandJ.Krejci,AVCR,Prague.Verner,M.(2002)ThePyramids:TheMystery,Culture,andScienceofEgyptsGreatMonumentsGrovePress

MAGLI,G.74

NOTES1.RecenthintspointtothepossibilitythattheoriginalKhufuprojectenclosedalsothispyramidand

Khafrefinishedit(Shaltoutetal.2007,Magli2003,2009),butthisissueisnotofspecialrelevancehere.2.Themonumentliesinsideamilitarybaseandistheuniquesitementionedherethattheauthorhas

notpersonallysurveyed.3.TheMenkhaourpyramidisidentified,withoutcertainityhowever,withthepyramidLepsius29ly

ingeastoftheTeticomplex;thepyramidofDjedkarewasbuiltinaprominentpositionontheridgeinSouthSaqqara;theUnaspyramidliesinCentralSaqqaraandwillbediscussedatlengthinsection8.

4.Therecanbelittledoubtaboutthefactthattheslopesofthepyramidswerechosenintheplanningphaseandcontrolledduring theconstructionbymeansof rational tangents, i.e.choosing two integernumbers.Wethusreportherethesimplestrationalfractions(lowestpossiblenumbers)givingareasonablerationalapproximation(within1%)ofthemeasuredslopes.Itshouldbenoted,however,thatinmuch later sources (suchas theRhindPapyrus)aquiteawkwardwayofmeasuring the tangentofanangle,calledseked,appears.Theseked ishowmuchonehas toslide thebase inorder toachieveanheightofonecubit.Thismeasurehastheobviousdrawbackofbeingdependentontheunitofmeasure,while angles are of course nondimensional and therefore easytohandle physical quantities.AlthoughnoproofoftheuseofthesekedintheOldKingdomexists,itiscustomaryintheliteraturetotrytoexpresstheslopesofthepyramids inthisway(seee.g.Rossi2003).Sincethecubitwasdividedinto7palmsand thepalmwasdivided into4 fingers,even someeagerly simple slopes (suchas3/2)becomeutterly complicatedwhen expressed into the seked system.There isno reasonwhatsoever tobelievethatthebuildersofthepyramidsusedthiswayofworking,ifonlyitisobservedthatworkmeninchargeofcuttingcasingblockcouldusearbitraryunits tocheck theirwork if theslopewas insteadgivenasatwointegersnumbersinformation.

Documents

Topography, Astronomy and Dynastic History in the Alignments of the Pyramid Fields of the Old Kingdom