architecture and mathematics in ancient egypt

300 Pages·2010·4.71 MB·English

ARCHITECTURE AND MATHEMATICS IN
ANCIENT EGYPT
In this fascinating new study, architect and Egyptologist Corinna Rossi analyses
therelationshipbetweenmathematicsandarchitectureinancientEgyptbyexplor-
ingtheuseofnumbersandgeometricalfiguresinancientarchitecturalprojectsand
buildings.Whilepreviousarchitecturalstudieshavesearchedforabstract‘universal
rules’ to explain the history of Egyptian architecture, Rossi attempts to reconcile
thedifferentapproachesofarchaeologists,architectsandhistoriansofmathematics
intoasinglecoherentpicture.Usingastudyofaspecificgroupofmonuments,the
pyramids, and placing them in the context of their cultural and historical back-
ground, Rossi argues that theory and practice of construction must be considered
as a continuum, not as two separated fields, in order to allow the original plan-
ning process of a building to re-emerge. Highly illustrated with plans, diagrams
andfigures,thisbookisessentialreadingforallscholarsofancientEgyptandthe
architectureofancientcultures.
DrCorinnaRossiisaJuniorResearchFellowinEgyptologyatChurchillCollege,
Cambridge.

ARCHITECTURE AND
MATHEMATICS IN ANCIENT EGYPT
CORINNA ROSSI
cambridgeuniversitypress
Cambridge,New York, Melbourne, Madrid, Cape Town, Singapore, Sa~o Paulo
Cambridge University Press
TheEdinburghBuilding,Cambridge,CB28RU,UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521690539
(cid:2)C CorinnaRossi2003
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2003
Reprinted with corrections 2006
First paperback edition 2007
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
ISBN-13 978-0-521-82954-0hardback
ISBN-13 978-0-521-69053-9 paperback
Cambridge University Press has no responsibility for the persistence or accuracy
of URLs for external or third-party internet websites resferred to in this publication,
and does not guarantee that content on such websites is, or will remain,
accurate or appropriate.
Contents
Listofillustrations page viii
Listoftables xiii
Preface xiv
Acknowledgments xvii
Listofabbreviations xix
PartI ProportionsinancientEgyptianarchitecture
IntroductiontoPartI:Harmonyandproportionsinarchitecture 2
1 Insearchof‘therule’forancientEgyptian
architecture 7
Trianglesandotherfigures 7
ThreetrianglesforancientEgypt 7
Viollet-le-Duc,Babinandtheprimevalpyramid 11
ChoisyandtheintroductionoftheGoldenSection 16
TheGoldenSection 23
TheoriginanddefinitionsoftheGoldenSection 23
TheGoldenSectionandancientEgyptianartandarchitecture 28
ThetheoryofAlexanderBadawy 32
2 MathematicsandarchitectureinancientEgypt 57
AncientEgyptianmathematics 57
Themathematicalsourcesandtheirlanguage 57
On(cid:2),(cid:3)andotheranachronisms 60
Intention,coincidenceortendency? 68
Trianglesandarchitecture 68
Psychologicalexperimentsandinvoluntarytrends 78
CasesfromancientEgypt 80
ConclusiontoPartI:Ancientmathematicsandpractical
operations 87
vi Contents
PartII AncientEgyptiansources:constructionand
representationofspace
IntroductiontoPartII:Traditionandvariationsinancient
Egyptianartandarchitecture 92
3 Documentsontheplanningandbuildingprocess 96
Architecturaldrawings 96
Representationsofbuildingsandworkingdrawings 96
Drawingswithwrittendimensions:theproblemofthescale 101
Full-sizegeometricalsketchesofarchitecturaldetails 113
Theuseofsquaregridsandtheideaofamodule 122
Architecturalmodels 128
Votiveobjects 128
Workingmodels 135
ProjectsandworksintheNineteenthandTwentiethDynasty
royaltombs 139
Documentsontheworks 139
Recordingtheprogress:fromtheprojecttothesurvey 142
4 Foundationrituals 148
Foundationceremonies 148
Theritualsequence 148
Cordsandgeometry 154
BuildingTexts 161
Thedimensionsoftheprimevaltemples 161
ThedimensionsofthetemplesatEdfuandDendera 166
ConclusiontoPartII:Fromtheplantothebuilding 174
PartIII Thegeometryofpyramids
IntroductiontoPartIII:Combiningtheknowledge 178
5 Symbolicshapeandconstructionalproblems 180
Theform 180
Pyramidalformandsolarcult 180
Benbenandbenbenet 182
Ashighaspossible 184
Thetechnique 185
Seked,side-length,diagonalsandcorners 185
Methodsforobtainingtheslope 188
Dimensionsandproportions 196
Contents vii
6 Theproportionsofpyramids 200
Analysingtruepyramids 200
Numerologicaltheories 200
Lauer’ssimpleratios 202
Alistoftruepyramids 204
Availabledata 204
Pyramidiaasalternativesources 205
7 Pyramidsandtriangles 212
Geometricalmodels 212
Approximationandseked 212
Equilateralandb=htriangles 214
Seked51 palms,generallycalled 14 triangle 215
2 11
Pythagoreantriplets 216
Theevolutionoftheform 221
OldKingdompyramids 221
MiddleKingdompyramids 228
NewKingdomandLatePeriodpyramids 231
ConclusiontoPartIII:Interpretingtheslopeofpyramids 236
Anoverview 239
AppendixListofOldandMiddleKingdomtruepyramids 242
Bibliography 255
Index 271
Illustrations
1 Earlynineteenth-centuryreproductionsofEgyptianmonuments. page8
2 Equilateraland‘Egyptian’trianglesaccordingtoViollet-le-Duc. 12
3 ConstructionoftheverticalsectionofthepyramidofKhufubymeans
ofthe3-4-5triangleaccordingtoViollet-le-Duc. 13
4 Proportionsobtainedbymeansofequilateraland‘Egyptian’trianglesin
fourtemples. 14
5 ‘Egyptian’triangleinthedesignofthefac¸adeoftheParthenonaccording
toViollet-le-Duc. 15
6 Equilateraland‘Egyptian’trianglesinthedesignoftheBasilicaof
ConstantineaccordingtoViollet-le-Duc. 16
7 ProportionsofthesectionoftheCathedralofAmiensaccordingto
Viollet-le-Duc. 17
8 Dimensionsofvariouselementsofapyramid. 18
9 ProportionsofthesectionoftheGreatTempleofPaestumaccording
toBabin. 19
10 RelationshipbetweenthedimensionsofsomeGreektemplesandthe
proportionsofsometrianglesaccordingtoBabin. 20
11 Designofthefac¸adeofthesouthernperipteralchapelatElephantine
accordingtoChoisy. 22
12 TrianglesusedbytheEgyptiansaccordingtoChoisy. 22
13 Constructionsofequilateraland‘Egyptian’trianglesaccording
toChoisy. 23
14 SubdivisionofasegmentaccordingtotheGoldenSectionandtwo
geometricalconstructionsofthesameproportion. 24
15 Visualisationoftherelationshipamongelementsofacontinuousproportion
andoftheGoldenSection. √ 25
16 Gnomonicgrowthvisualisedasa 5spiral(reprintedbypermission
oftheauthor). 27
17 ScenefromtheeastwallofthechapelofthePtolemaictombofPetosiris
(drawnafterLefebvre,Petosiris,pl.32). 29
18 Twointerpretationsofthegeometricalfiguresinascenefromthetombof
PetosirisbyLawlorandLamy(reprintedbypermissionoftheauthor). 30
viii
Listofillustrations ix
19 ‘Trace´sharmoniquese´gyptiens’accordingtoGhyka(reprintedby
permissionofEditionsGallimard). 32
20 SketchoftheproportionsofthepyramidofKhufu,accordingtoGhyka
(reprintedbypermissionofEditionsGallimard). 33
21 Parallelbetweentheproportionsofthehumanbodyandofthetempleof
LuxoraccordingtoSchwallerdeLubicz(reprintedbypermissionof
EditionsCaracte`res). 34
22 GnomonicexpansionofthetempleofLuxor(reprintedbypermissionof
theauthor). 35
23 Aninterpretationbasedon(cid:2)oftheplanoftheOsireion(Nineteenth
Dynasty)byLawlorandLamy(reprintedbypermissionoftheauthor). 36
24 VerticalsectionofthepyramidofKhufushowingtheuseoftheeight
‘RatiosofDivineHarmony’accordingtoFournierdesCorats(reprintedby
permissionofTre´danielEditeur). 37
25 ApplicationoftheeightRatiosofDivineHarmonytoaNewKingdom
brooch(above)andtothebodyofthegoddessNutfromthePtolemaic
DenderaZodiac(below)accordingtoFournierdesCorats(reprintedby
permissionofTre´danielEditeur). 38
26 ApplicationoftheeightRatiosofDivineHarmonytosomecolumnsofthe
templeofAmonatKarnakaccordingtoFournierdesCorats(reprintedby
permissionofTre´danielEditeur). 40
27 Pillars1:2,1:4,1:8andprismaticpillaraccordingtoBadawy. 41
28 MethodtodesigntrianglesbymeansofcordsaccordingtoBadawy. 44
29 AnalysisoftheplanofthemortuarytempleofKhafra(FourthDynasty),
accordingtoBadawy. 45
30 AnalysisofthereconstructedplanofthetempleofSenusretIatToˆd
(TwelfthDynasty),accordingtoBadawy. 46
31 ActualarchaeologicalremainsofthetempleofSenusretIatToˆd(Twelfth
Dynasty)andreconstructionoftheoriginalplanbyArnold(reprintedby
permissionofDAIK). 47
32 AnalysisoftheplanoftheSanctuaryoftheGreatAtenTempleatAmarna
(EighteenthDynasty),bymeansofanetworkof8:5trianglesaccordingto
Badawy. 48
33 PlanoftheactualarchaeologicalremainsoftheSanctuaryoftheGreat
AtenTempleatAmarna(EighteenthDynasty)accordingtothe1986survey
(courtesyoftheEgyptExplorationSociety). 49
34 AnalysisoftheplanofthetempleofLuxor(EighteenthDynasty),
accordingtoBadawy. 50
35 AnalysisoftheplanofthetempleofKarnak,NewKingdom,according
toBadawy. 51
36 AnalysisoftheplanofthePtolemaictempleatDenderaaccording
toBadawy. 52
37 AnalysisoftheplanofthePtolemaictempleatKomOmboaccording
toBadawy. 53
38 Analysisoftheplanandreconstructionofthewesternfac¸adeofthechapel
ofHakorisatKarnak(Twenty-ninthDynasty)basedonanetworkof8:5