ARCHITECTURE AND MATHEMATICS IN ANCIENT EGYPT

ARCHITECTURE AND MATHEMATICS IN ANCIENT EGYPTARCHITECTURE AND MATHEMATICS IN ANCIENT EGYPT
In this fascinating new study, architect and Egyptologist Corinna Rossi analyses
the relationship between mathematics and architecture in ancient Egypt by explor-
ing the use of numbers and geometrical figures in ancient architectural projects and
buildings. While previous architectural studies have searched for abstract ‘universal
rules’ to explain the history of Egyptian architecture, Rossi attempts to reconcile
the different approaches of archaeologists, architects and historians of mathematics
into a single coherent picture. Using a study of a specific group of monuments, 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
and figures, this book is essential reading for all scholars of ancient Egypt and the
architecture of ancient cultures.
Dr Corinna Rossi is a Junior Research Fellow in Egyptology at Churchill College,
Cambridge.
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Reprinted with corrections 2006
Printed in the United Kingdom at the University Press, Cambridge
First published 2003
List of tables xiii
List of abbreviations xix
Part I Proportions in ancient Egyptian architecture Introduction to Part I: Harmony and proportions in architecture 2
1 In search of ‘the rule’ for ancient Egyptian architecture 7
Triangles and other figures 7
Three triangles for ancient Egypt 7 Viollet-le-Duc, Babin and the primeval pyramid 11 Choisy and the introduction of the Golden Section 16
The Golden Section 23
The origin and definitions of the Golden Section 23 The Golden Section and ancient Egyptian art and architecture 28 The theory of Alexander Badawy 32
2 Mathematics and architecture in ancient Egypt 57
Ancient Egyptian mathematics 57
The mathematical sources and their language 57 On , and other anachronisms 60
Intention, coincidence or tendency? 68
Triangles and architecture 68 Psychological experiments and involuntary trends 78 Cases from ancient Egypt 80
Conclusion to Part I: Ancient mathematics and practical
operations 87
vi Contents
Part II Ancient Egyptian sources: construction and representation of space Introduction to Part II: Tradition and variations in ancient
Egyptian art and architecture 92
3 Documents on the planning and building process 96
Architectural drawings 96
Representations of buildings and working drawings 96 Drawings with written dimensions: the problem of the scale 101 Full-size geometrical sketches of architectural details 113 The use of square grids and the idea of a module 122
Architectural models 128
Projects and works in the Nineteenth and Twentieth Dynasty
royal tombs 139
Documents on the works 139 Recording the progress: from the project to the survey 142
4 Foundation rituals 148
Building Texts 161
The dimensions of the primeval temples 161 The dimensions of the temples at Edfu and Dendera 166
Conclusion to Part II: From the plan to the building 174
Part III The geometry of pyramids Introduction to Part III: Combining the knowledge 178
5 Symbolic shape and constructional problems 180
The form 180
Pyramidal form and solar cult 180 Benben and benbenet 182 As high as possible 184
The technique 185
Seked, side-length, diagonals and corners 185 Methods for obtaining the slope 188 Dimensions and proportions 196
Contents vii
Analysing true pyramids 200
A list of true pyramids 204
Available data 204 Pyramidia as alternative sources 205
7 Pyramids and triangles 212
Geometrical models 212
Approximation and seked 212 Equilateral and b = h triangles 214 Seked 5 1
2 palms, generally called 14
11 triangle 215
Pythagorean triplets 216
The evolution of the form 221
Old Kingdom pyramids 221 Middle Kingdom pyramids 228 New Kingdom and Late Period pyramids 231
Conclusion to Part III: Interpreting the slope of pyramids 236
An overview 239
Appendix List of Old and Middle Kingdom true pyramids 242
Bibliography 255
Index 271
1 Early nineteenth-century reproductions of Egyptian monuments. page 8 2 Equilateral and ‘Egyptian’ triangles according to Viollet-le-Duc. 12 3 Construction of the vertical section of the pyramid of Khufu by means
of the 3-4-5 triangle according to Viollet-le-Duc. 13 4 Proportions obtained by means of equilateral and ‘Egyptian’ triangles in
four temples. 14 5 ‘Egyptian’ triangle in the design of the facade of the Parthenon according
to Viollet-le-Duc. 15 6 Equilateral and ‘Egyptian’ triangles in the design of the Basilica of
Constantine according to Viollet-le-Duc. 16 7 Proportions of the section of the Cathedral of Amiens according to
Viollet-le-Duc. 17 8 Dimensions of various elements of a pyramid. 18 9 Proportions of the section of the Great Temple of Paestum according
to Babin. 19 10 Relationship between the dimensions of some Greek temples and the
proportions of some triangles according to Babin. 20 11 Design of the facade of the southern peripteral chapel at Elephantine
according to Choisy. 22 12 Triangles used by the Egyptians according to Choisy. 22 13 Constructions of equilateral and ‘Egyptian’ triangles according
to Choisy. 23 14 Subdivision of a segment according to the Golden Section and two
geometrical constructions of the same proportion. 24 15 Visualisation of the relationship among elements of a continuous proportion
and of the Golden Section. 25 16 Gnomonic growth visualised as a
√ 5 spiral (reprinted by permission
of the author). 27 17 Scene from the east wall of the chapel of the Ptolemaic tomb of Petosiris
(drawn after Lefebvre, Petosiris, pl. 32). 29 18 Two interpretations of the geometrical figures in a scene from the tomb of
Petosiris by Lawlor and Lamy (reprinted by permission of the author). 30
viii
List of illustrations ix
19 ‘Traces harmoniques egyptiens’ according to Ghyka (reprinted by permission of Editions Gallimard). 32
20 Sketch of the proportions of the pyramid of Khufu, according to Ghyka (reprinted by permission of Editions Gallimard). 33
21 Parallel between the proportions of the human body and of the temple of Luxor according to Schwaller de Lubicz (reprinted by permission of Editions Caracteres). 34
22 Gnomonic expansion of the temple of Luxor (reprinted by permission of the author). 35
23 An interpretation based on of the plan of the Osireion (Nineteenth Dynasty) by Lawlor and Lamy (reprinted by permission of the author). 36
24 Vertical section of the pyramid of Khufu showing the use of the eight ‘Ratios of Divine Harmony’ according to Fournier des Corats (reprinted by permission of Tredaniel Editeur). 37
25 Application of the eight Ratios of Divine Harmony to a New Kingdom brooch (above) and to the body of the goddess Nut from the Ptolemaic Dendera Zodiac (below) according to Fournier des Corats (reprinted by permission of Tredaniel Editeur). 38
26 Application of the eight Ratios of Divine Harmony to some columns of the temple of Amon at Karnak according to Fournier des Corats (reprinted by permission of Tredaniel Editeur). 40
27 Pillars 1:2, 1:4, 1:8 and prismatic pillar according to Badawy. 41 28 Method to design triangles by means of cords according to Badawy. 44 29 Analysis of the plan of the mortuary temple of Khafra (Fourth Dynasty),
according to Badawy. 45 30 Analysis of the reconstructed plan of the temple of Senusret I at Tod
(Twelfth Dynasty), according to Badawy. 46 31 Actual archaeological remains of the temple of Senusret I at Tod (Twelfth
Dynasty) and reconstruction of the original plan by Arnold (reprinted by permission of DAIK). 47
32 Analysis of the plan of the Sanctuary of the Great Aten Temple at Amarna (Eighteenth Dynasty), by means of a network of 8:5 triangles according to Badawy. 48
33 Plan of the actual archaeological remains of the Sanctuary of the Great Aten Temple at Amarna (Eighteenth Dynasty) according to the 1986 survey (courtesy of the Egypt Exploration Society). 49
34 Analysis of the plan of the temple of Luxor (Eighteenth Dynasty), according to Badawy. 50
35 Analysis of the plan of the temple of Karnak, New Kingdom, according to Badawy. 51
36 Analysis of the plan of the Ptolemaic temple at Dendera according to Badawy. 52
37 Analysis of the plan of the Ptolemaic temple at Kom Ombo according to Badawy. 53
38 Analysis of the plan and reconstruction of the western facade of the chapel of Hakoris at Karnak (Twenty-ninth Dynasty) based on a network of 8:5
x List of illustrations
triangles, according to Lauffray (copyright: ADPF-ERC – Ministere des Affaires etrangeres, Paris; reprinted with permission). 55
39 Examples of Dieter Arnold’s studies: the temple of Mentuhotep at Deir el-Bahari (reprinted by permission of DAIK) and the Middle Kingdom temple of Qasr el-Sagha (reprinted by permission of DAIK). 62
40 The calculation of the area of the circle, based on RMP problem 48, according to Robins and Shute, and Gillings (reprinted by permission of Massachusetts Institute of Technology Press). 65
41 Interpretation of some marks as traces of an equilateral triangle in the plan of the Roman temple at Kalabsha according to Siegler (reprinted by permission of Gebruder Mann Verlag). 72
42 Interpretation of the design of a facade and a section of the Roman temple at Kalabsha according to Siegler (reprinted by permission of Gebruder Mann Verlag). 74
43 Interpretation of the plan of the Small Aten Temple at Amarna (Eighteenth Dynasty), according to Badawy and Mallinson (reprinted by permission of CAJ). 76
44 The development of the square grid system according to Legon (reprinted by permission of the author). 82
45 Plan and section of the Amarna Royal Tomb (courtesy of the Egypt Exploration Society). 84
46 Geometrical method to double a 100-square unit area. 89 47 Eighteenth Dynasty (?) drawing of a portable shrine on papyrus (reprinted
by permission of Oxford University Press). 93 48 Proportions of Egyptian columns. 94 49 Representation of the royal granaries and storehouses of Amarna
(Eighteenth Dynasty) from the tomb of Meryra (courtesy of the Egypt Exploration Society). 97
50 Representation of an Amarna royal palace from the tomb of Meryra, Eighteenth Dynasty (courtesy of the Egypt Exploration Society). 98
51 Akhenaten rewarding Meryra from the Window of Appearance, Eighteenth Dynasty (courtesy of the Egypt Exploration Society). 100
52 Slate tablet from Heliopolis representing a temple and two reconstructions by Ricke (reprinted by permission of Akademie Verlag). 102
53 Ostracon BM 41228 (Eighteenth Dynasty), reconstruction of the plan by Glanville (courtesy of the Egypt Exploration Society) and by Van Siclen (reprinted by permission of GM). 106
54 Sketches of a subterranean tomb, from the tomb of Senenmut (Eighteenth Dynasty). 108
55 Plan on a wooden board (Eighteenth Dynasty), (courtesy of the Egypt Exploration Society and the Metropolitan Museum of Arts); reconstructions of the plan by Badawy and by Davies (courtesy of the Egypt Exploration Society and the Metropolitan Museum of Arts). 110
56 Plan of the peripteral temple of Tuthmosis III (Eighteenth Dynasty) facing the Sacred Lake in the temple of Karnak. 111
57 Sketch of a Twentieth Dynasty elliptical vault. 114
List of illustrations xi
58 Diagram of a curve (Third Dynasty), with hieroglyphic transcription. 116 59 Construction of an ellipse by means of a 3-4-5 triangle. 118 60 Sketch of a Ptolemaic column at Philae and of a Roman capital, and
reconstruction of their proportions according to Borchardt (reprinted by permission of Akademie Verlag). 119
61 Sketch of a capital at Gebel Abu Foda, Roman (copyright: Petrie Museum of Egyptian Archaeology, University College London; reprinted with permission). 123
62 Plan of the temple of Qasr el-Sagha with a superimposed 1-cubit grid (reprinted by permission of DAIK). 124
63 Plan of the Small Aten Temple at Amarna (Eighteenth Dynasty), with 20-cubit grid and with 18-cubit grid (reprinted by permission of CAJ). 125
64 Plan of the basement of the model of a temple of Seti I (Nineteenth Dynasty) and frontal view of Badawy’s reconstruction of the entire model. 130
65 Fragment of a model of the northern corner of the first hall of the Ptolemaic temple at Tod. 132
66 Plan and elevation of the model of a step pyramid (date unknown). 134 67 Model of the funerary apartment in a late Twelfth Dynasty pyramid
(reprinted by permission of DAIK). 136 68 Plan and elevation of the model of the pyramid of Amenemhat III at
Hawara (?) (Twelfth Dynasty). 137 69 Ostracon Cairo 25184 and plan of KV 6, the tomb of Ramses IX,
Twentieth Dynasty. 143 70 Ostracon Cairo 51936, Nineteenth Dynasty. 145 71 Papyrus Turin 1885 and plan of KV 2, the tomb of Ramses IV,
Twentieth Dynasty. 146 72 Scene from a foundation ceremony from the reign of Khasekhemwy,
Second Dynasty. 150 73 Fragmentary foundation scene from the valley temple of Snefru at Dahshur,
Fourth Dynasty. 151 74 Scene from a foundation ceremony from the sun temple of Neuserra, Fifth
Dynasty (reprinted by permission of Akademie Verlag). 152 75 Scene from the foundation ceremony of the Ptolemaic temple of Dendera. 152 76 Land-surveyors from the Eighteenth Dynasty tomb of Amenhotepsesi
(courtesy of the Egypt Exploration Society). 154 77 Figures based on the 3-4-5 triangle according to Lauer (reprinted by
permission of IFAO). 158 78 3-4-5 triangle in the plans of the valley temple of Snefru at Dashur and of
the funerary temple of Khufu (Fourth Dynasty), according to Lauer (reprinted by permission of IFAO). 160
79 3-4-5 triangle in the plan of the funerary temples of Teti, Pepi I and Pepi II (Sixth Dynasty), at Saqqara according to Lauer (reprinted by permission of IFAO). 161
80 Final stage of the Primeval Temple of the Falcon according to the Edfu texts. 163
81 Predynastic temple of Satet at Elephantine. 164
xii List of illustrations
82 Plan of the Predynastic ceremonial centre at Hierakonpolis, named HK29A (reprinted by permission of R. Friedman). 165
83 Plans of the Ptolemaic temples of Edfu and Dendera (reprinted by permission of IFAO). 168
84 RMP problem 57: the height of a pyramid is calculated from the base-length and the seked (slope) (reprinted by permission of MAA). 185
85 Seked of a sloping face according to the Rhind Mathematical Papyrus. 186 86 New Kingdom ostracon from Soleb representing two pyramids, plan of
pyramids 14 and 15 at Soleb and sketch of a pyramid on a Meroitic jar. 187 87 Petrie’s drawings of the diagrams at the four corners of Mastaba 17 (Third
to Fourth Dynasty) at Meidum (copyright: Petrie Museum of Egyptian Archaeology, University College London; reprinted with permission). 189
88 Diagram of pyramid Beg. 8 at Meroe, as drawn on the wall of its chapel and its reconstruction (reprinted by permission of Akademie Verlag). 190
89 Geometrical relationships in a pyramid one cubit high. 193 90 Method for obtaining the slope in a pyramid according to Lehner
(reprinted by permission of the author). 194 91 Method for obtaining the slope in a Meroitic pyramid according to Hinkel
(reprinted by permission of Akademie Verlag). 195 92 Plan and section of the Bent Pyramid, Fourth Dynasty. 198 93 Relationship between face and vertical section in a pyramid. 208 94 West face of the pyramidion of Khendjer, Thirteenth Dynasty. 209 95 Equilateral triangle as vertical section and face of a pyramid. 210 96 Sekeds corresponding to an equilateral triangle, and to a triangle in which
base = height. 214 97 Seked of 5 1
2 palms, also called 14
11 triangle. 217
98 Pythagorean triplets possibly employed as models in the construction of pyramids (to different scales). 220
99 Diagram of the evolution of the slope of Old and Middle Kingdom true pyramids, with the addition of four New Kingdom pyramids. 222
100 Pyramids of Snefru (Fourth Dynasty). 224 101 Pyramids of Khufu, Djedkara, Khafra and Menkaura (Fourth Dynasty). 227 102 Pyramids of Pepi I, Pepi II, their satellites and their queens
(Sixth Dynasty). 237
Tables
1 Schematic chronology of ancient Egypt page xxi 2 List of ancient Egyptian units of measurements 61 3 Architectural sketches and drawings 104 4 Full-size geometrical sketches 113 5 Architectural models of funerary and religious monuments 128 6 Documents on the architectural work in the Nineteenth and Twentieth
Dynasty tombs 144 7 The dimensions of some chambers of the Ptolemaic temples of Edfu and
Dendera according to the Building Texts 170 8 Proportions of some pyramids according to Jean-Philippe Lauer 203 9 List of surviving pyramidia of pyramids 206
xiii
Preface
Mathematics has always played an important role in architecture, in the past just as
in the present. Despite this continuity, however, reconstructing exactly how the re-
lationship between architecture and mathematics worked in an ancient culture may
prove rather complicated. An investigation into the way architecture and mathe-
matics interacted in the past, in ancient Egypt as well as in other cultures, may
be misled by three main sets of tangled problems. The first is generated by our
expectations of the results of such a research; the second depends on the reliability
of the drawings used to test or ‘discover’ a theory; and the third stems from the way
mathematics is employed during the research.
Regarding the first point, it is evident that in ancient monuments people have
found all they wanted to find in terms of mathematical concepts and geometrical
figures. A small-scale plan, a ruler, a compass and a bit of imagination are enough
to ‘discover’ several mathematical relationships in the design of any building. This
does not imply, however, that the ancient architects based their reasoning on the
same points, nor that they were aware of all of the possible interpretations of their
plans.
A second point concerns the drawings employed in this type of study. The habit of
using mainly plans to analyse the proportions of buildings may produce a dangerous
distance between the actual monument and its schematic representation. A plan is a
useful and simple way to represent a building, but it includes just a few clues about
the elevation, and even less about the masses and materials involved. Defining a
building just by means of its plan may be reductive, and discussing its proportions
on this basis may be misleading. Another problem related to the use of drawings
is their reliability. Precision in architectural surveys has not always been a priority,
and graphic reconstructions sometimes have been based on the imagination more
than is appropriate.
A third, important point is the way scholars use mathematics. In their search
for a ‘rule’ that would explain the proportions of ancient Egyptian architecture,
xiv
modern scholars have generally ignored ancient Egyptian mathematics and have
based their theories on our modern mathematical system. In some extreme cases,
this line of research has led to complicated interpretations based on symbolic and
esoteric concepts. These theories do not necessarily provide any useful information
about the ancient culture to which they are supposed to refer, but on the other hand
they may play an important role in a study of the culture and the historical period
that produced them – that is, Europe in the last two centuries. The modern diffusion
of a scientific and logical way of thinking seems to have corresponded…