ARCHITECTURE 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. ARCHITECTURE AND MATHEMATICS IN ANCIENT EGYPT CORINNA ROSSI cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sa~o Paulo Cambridge University Press The Edinburgh Building, Cambridge, CB2 8RU, 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 C Corinna Rossi 2003 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2003 Reprinted with corrections 2006 First paperback edition 2007 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN -13 978-0 -521 -82954 -0 hardback 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 List of illustrations page viii List of tables xiii Preface xiv Acknowledgments xvii 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 Votive objects 128 Working models 135 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 Foundation ceremonies 148 The ritual sequence 148 Cords and geometry 154 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 6 The proportions of pyramids 200 Analysing true pyramids 200 Numerological theories 200 Lauer’s simple ratios 202 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 1 14 Seked 5 2 palms, generally called 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 Illustrations 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 fa¸cadeof 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 fa¸cadeof 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 ‘Trac´esharmoniquesegyptiens’ ´ 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 Caract`eres). 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 Tr´edanielEditeur). 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 Tr´edanielEditeur). 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 Tr´edanielEditeur). 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 Tˆod (Twelfth Dynasty), according to Badawy. 46 31 Actual archaeological remains of the temple of Senusret I at Tˆod(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 fa¸cadeof 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 – Minist`eredes Affaires ´etrangeres,Paris; reprinted with permission).