Discussions in Egyptology 28,1994 ISSN 0268-3083
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Architecture and Mathematics in Ancient Egypt
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. -
Astronomy, Topography and Dynastic History in the Age of the Pyramids
Astronomy, topography and dynastic history in the Age of the Pyramids. Giulio Magli Faculty of Civil Architecture - Politecnico di Milano Piazza Leonardo da Vinci 32, 20133 Milan, Italy It is known since the 19 century that in the layout of the pyramid field of the pharaohs of the 4 th Egyptian dynasty at Giza, a “main axis” exists. Indeed, the south-east corners of these monuments align towards the site of the temple of Heliopolis, which was plainly visible in ancient times. It was later discovered that a similar situation occurs in the main pyramid field of the subsequent dynasty at Abu Sir. Here, the north-west corners of three chronologically successive pyramids again voluntarily align towards Heliopolis. However, the temple was in this case not visible, due to the rock outcrop- today occupied by the Cairo citadel - which blocks the view. In the present paper, a multi-disciplinary approach based on historical, topographical and archaeoastronomical analysis is developed in an attempt at understanding this peculiar feature, which governed from the very beginning the planning of such wonderful monuments. A general pattern actually arises, which appears to have inspired the choice of the sites and the disposition on the ground of almost all the funerary complexes of the kings during the Old Kingdom. 1. Introduction An interesting feature exists in the layouts of the pyramids of Giza and Abu Sir: the presence of a “main axis” directed to the area where the ancient temple of the sun of Heliopolis once stood, on the opposite bank of the Nile. These axes are connected with a process of “solarisation” of the pharaoh which probably started with Khufu, the builder of the Great Pyramid. -
24 HAWASS Page 379 Thursday, July 22, 2004 1:29 PM
001-a Contents vol. 1 Page i Thursday, July 22, 2004 1:55 PM Studies in Honor of William Kelly Simpson ¡%¢ …d¢ i 001-a Contents vol. 1 Page ii Thursday, July 22, 2004 1:55 PM William Kelly Simpson 001-a Contents vol. 1 Page iii Thursday, July 22, 2004 1:55 PM tudies in onor of illiam elly impson Volume 1 Peter Der Manuelian Editor Rita E. Freed Project Supervisor Department of Ancient Egyptian, Nubian, and Near Eastern Art Museum of Fine Arts, Boston 1996 001-a Contents vol. 1 Page iv Thursday, July 22, 2004 1:55 PM Front jacket illustration: The Ptolemaic Pylon at the Temple of Karnak, Thebes, looking north. Watercolor over graphite by Charles Gleyre (1806–1874). Lent by the Trustees of the Lowell Institute. MFA 161.49. Photograph courtesy Museum of Fine Arts, Boston Back jacket illustration: Palm trees at the Temple of Karnak, Thebes. Watercolor over graphite by Charles Gleyre. Lent by the Trustees of the Lowell Institute. MFA 157.49. Photograph courtesy Museum of Fine Arts, Boston Endpapers: View of the Giza Pyramids, looking west. Graphite drawing by Charles Gleyre. Lent by the Trustees of the Lowell Institute. MFA 79.49. Photograph courtesy Museum of Fine Arts, Boston Frontispiece: William Kelly Simpson at the Museum of Fine Arts, Boston, 1985 Title page illustration: A document presenter from the Old Kingdom Giza mastaba chapel of Merib (g 2100–1), north entrance thickness (Ägyptisches Museum Berlin, Inv. Nr. 1107); drawing by Peter Der Manuelian Typeset in Adobe Trump Mediaeval and Syntax. Title display type set in Centaur Egyptological diacritics designed by Nigel Strudwick Hieroglyphic fonts designed by Cleo Huggins with additional signs by Peter Der Manuelian Jacket design by Lauren Thomas and Peter Der Manuelian Edited, typeset, designed and produced by Peter Der Manuelian Copyright © Museum of Fine Arts, Boston, 1996 All rights reserved. -
Pyramids of the Giza Plateau Pyramid Complexes of Khufu, Khafre, and Menkaure
Pyramids of the Giza Plateau Pyramid Complexes of Khufu, Khafre, and Menkaure Charles Rigano AuthorHouse™ 1663 Liberty Drive Bloomington, IN 47403 www.authorhouse.com Phone: 1-800-839-8640 © 2014 Charles Rigano. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted by any means without the written permission of the author. Published by AuthorHouse 05/14/2015 ISBN: 978-1-4969-5249-3 (sc) ISBN: 978-1-4969-5248-6 (e) Any people depicted in stock imagery provided by Thinkstock are models, and such images are being used for illustrative purposes only. Certain stock imagery © Thinkstock. Because of the dynamic nature of the Internet, any web addresses or links contained in this book may have changed since publication and may no longer be valid. The views expressed in this work are solely those of the author and do not necessarily reflect the views of the publisher, and the publisher hereby disclaims any responsibility for them. Front Cover: - Khufu’s Great Pyramid at dusk as seen from near the Sphinx. Photo by the author. Back Cover: Top - The three Giza Pyramids from close to far: Menkaure and subsidiary pyramids, Khafre, and Khufu. Photo by author. Bottom - Author Charles Rigano with Menkaure’s Pyramid in the background. Author’s photo. Contents Preface The Gizeh Plateau Area Surrounding the Giza Plateau Giza Plateau Pyramid Complexes Chapter 1: The Giza Plateau Chapter 2: Khufu’s Pyramid Exterior Chapter 3: Khufu’s Pyramid Interior Chapter 4: Khufu’s Mortuary Complex Chapter 5: Khufu’s Pyramid – Designing, Closing, and Opening Chapter 6: Khafre’s Pyramid Site Chapter 7: Khafre’s Pyramid Interior Chapter 8: Khafre’s Mortuary Complex Chapter 9: The Sphinx Complex Chapter 10: Menkaure’s Pyramid Exterior Chapter 11: Menkaure’s Pyramid Interior Chapter 12: Menkaure’s Mortuary Complex Appendix: The Tuthmosis IV Stele Between the Sphinx Paws Photo Credits Suggested Reading and References Preface I remember the moment in 1970 that my fascination for Ancient Egypt started. -
Hawass, Zahi. “The Discovery of the Pyramidion of the Satellite Pyramid
IUBILATE CONLEGAE STUDIES IN MEMORY OF ABDEL AZIZ SADEK Part I In; ( \ . -c-!''3. _ ,;HjA~I~~~ Varia Aegyptiaca Vol. 10, Nos. 2-3 August-December 1995 IUBILATE CONLEGAE STUDIES IN MEMORY OF ABDEL AZIZ SADEK Part I edited by Charles C. Van Sic len III San Antonio, Texas 1995 [1997] Varia Aegyptiaca Charles C. Van Siclen III, Editor Vol. 10 * Nos. 2-3 * August-Decmber 1995 Editorial front cover Varia Aegypriaca is interested in articles pertaining to all facels of ancient Egypt, and especially articles dealing with objects, lexicography, epigraphy, and the results of field Editorial work. This issue of Varia Aegypriaca is the Varia Aegypriaca is published three IiIlles a year first of several devoted to the memory of in April, August and December by Van Siden Books, I I I Winnetka Road, San Antonio, Texas my late friend AzlOUZ (Abdel Aziz 78229-3613, USA. Sadek). It was originally conceived as a volume in his honor which he might live Domestic subscription rates (vol. II, 1996): to see, but such was not the case. The $35.00 (individuals, direct), $45.00 (libraries, dealers, etc.). Other countries: add $5.00 to article included herein were written for cover additional postage via surface mail. Air the most part in 1995. The delay in mail service is not available. Subscriptions may publication is due to problems of the he paid in advance or at time of publication. Please make all remittances payable to Van editor. My sincerest apologies are Siden Books, in United States currency only, by offered to the authors who have waited postal or international money order or bank so long to see their works in print. -
Architecture, Astronomy and Sacred Landscape in Ancient Egypt Giulio Magli Frontmatter More Information
Cambridge University Press 978-1-107-03208-8 - Architecture, Astronomy and Sacred Landscape in Ancient Egypt Giulio Magli Frontmatter More information ARCHITECTURE, ASTRONOMY AND SACRED LANDSCAPE IN ANCIENT EGYPT This book examines the interplay between astronomy and dynastic power in the course of ancient Egyptian history, focusing on the fundamental role of astronomy in the creation of the pyramids and the monumental temple and burial complexes. Bringing to bear the analytical tools of archaeoas- tronomy, a set of techniques and methods that enable modern scholars to better understand the thought, religion and science of early civilisations, Giulio Magli provides in-depth analyses of the pyramid complexes at Giza, Abusir, Saqqara and Dahshur, as well as of the Early Dynastic necropolis at Abydos and the magnifi cent New Kingdom Theban temples. Using a vari- ety of data retrieved from studies of the sky and measurements of the build- ings, he reconstructs the visual, symbolic and spiritual world of the ancient Egyptians and thereby establishes an intimate relationship among celestial cycles, topography and architecture. He also shows how they were deployed in the ideology of the Pharaoh’s power in the course of Egyptian history. Giulio Magli is Full Professor at the Faculty of Civil Architecture of the Politecnico di Milano, where he teaches the fi rst archaeoastronomy course established in an Italian university. He holds a PhD in astrophysics, and his research today focuses on archaeoastronomy and the relationship between architecture, landscape and astronomy in ancient Egyptian, Incan and Bronze Age Mediterranean cultures. In addition to his many papers in this fi eld, he is the author of Mysteries and Discoveries of Archaeoastronomy (2009) and is one of the co-authors of the recent UNESCO-IAU thematic study on astronomy and cultural heritage. -
Diagrams in Ancient Egyptian Geometry Survey and Assessment
Available online at www.sciencedirect.com Historia Mathematica 36 (2009) 321–373 www.elsevier.com/locate/yhmat Diagrams in ancient Egyptian geometry Survey and assessment Gregg De Young Department of Mathematics and Actuarial Science, The American University in Cairo, P.O. Box 74, New Cairo, Egypt Available online 20 October 2009 Abstract This article surveys and catalogs the geometric diagrams that survive from ancient Egypt. These diagrams are often overspecified and some contain inaccuracies in their construction. The diagrams accompany algorith- mic texts and support the mathematical programme of their authors. The study concludes with a brief compar- ison with the diagram traditions of ancient Babylon, early India, and Greece. Ó 2009 Elsevier Inc. All rights reserved. Résumé Cet article étudie les diagrammes géométriques qui survivent dans l’Egypte antique. Ces diagrammes sont plus spécifiques qu’il n’est nécessaire pour le problème et certains contiennent des inexactitudes dans leur con- struction. Les diagrammes accompagnent des textes algorithmiques et soutiennent le programme mathématique de leurs auteurs. L’étude se termine par une comparaison brève aux traditions diagrammatiques à Babylone et en Grèce. Ó 2009 Elsevier Inc. All rights reserved. MSC: 01A16 Keywords: Diagrams; Egyptian geometry 1. Preliminary disclaimer I am not a specialist in the study of ancient Egypt. I do not write about ancient Egyptian diagrams and mathematical texts based on my linguistic or mathematical background. I have chosen to focus on ancient Egypt because there has come down to us a clearly defined and limited body of diagrams for which good-quality photographs are readily E-mail address: [email protected] 0315-0860/$ - see front matter Ó 2009 Elsevier Inc. -
The Layer Monuments
THE LAYER MONUMENTS Six layer monuments namely Hebenu, Sinki, Nubt, el Kula, el Ghenimiya and Elephantine (see entries) were built by a king or kings unknown to us and serve a purpose which is hitherto unclear. The investigations concerning them are incomplete, but some light however has been cast: they are almost of the same construction, dimensions, and seem to date to the 3rd dynasty; they have no substructures nor temples and are not within a cemetery of that period. To the 6 layer monuments some monuments could be added or related: Two pairs of tumuli at Naqada, investigated by Petrie. A square structure of rough masonry truly oriented to the cardinal points, of a side measurement of 11 m, suggest that it was a core of an unfinished or destroyed layer monument. It was excavated under the Middle Kingdom chapel of Ay at Abydos by C.T. Currelly. A benben named Seketra, and perhaps a pyramid named Nefer, recorded in ink on stone vessels discovered under the step pyramid at Saqqara. The pyramid of Snofru at Seila (see entry). Two step pyramids of queens of Menkaura at Giza (# G3b and G3c), differ in structure and function. Two benbens dating to the 5th dynasty in the sun temples of Userkaf at Abu Sir and Neuserra at Abu Ghurab. Four 5th dynasty benbens of other kings known from texts. With the exception of Ghenimiya which has never been excavated the other 5 layer monuments and Seila have only been partly investigated. The nuclei of Hebenu, Sinki and Nubt were built of quarried limestone and selected concretions from the slopes of the nearby plateau, Seila and el Kula were built of quarried limestone, el Ghenimiya was built of quarried sandstone and Elephantine was built of quarried granite and selected bolder. -
Mystery of the Great Pyramid Missing Capstone Share When You Look up at the Great Pyramid, It’S Apex Is Missing
http://blog.world-mysteries.com March 3, 2011 Mystery of the Great Pyramid missing capstone Share When you look up at the Great Pyramid, it’s apex is missing. It is flap topped and not pointed like a pyramid should be. Usually, when a pyramid was constructed, the top part, or capstone (also called top-stone), was the last thing to be placed on it. It was considered the most important part of the pyramid and was made of special stone or even gold. The capstone was usually highly decorated. Was the great pyramid always without a capstone or was it stolen, destroyed, etc? No one knows but the accounts of visitors to the pyramid from the ancient past (as far back as the time of Christ) always reported that the pyramid lacked a capstone. It is possible that it was never finished. Another possibility is that capstones were sometimes made of gold and maybe the first thing looted. The only problem is that this would be a very large capstone. If you climbed to the top, you could walk around very freely on the pyramid as many have done. It is about 30 feet in each direction. Thus, this capstone would have been huge and weighed a tremendous amount. Also on the summit you would see something that looks like a mast or flagpole. Actually it was placed there by two astronomers in 1874 to show where the Pyramid’s actual apex would have been if finished. On the back of a dollar bill, you can see a pyramid with a flat top. -
The Layer Monuments
THE LAYER MONUMENTS Six layer monuments namely Hebenu, Sinki, Nubt, el Kula, el Ghenimiya and Elephantine (see entries) were built by a king or kings unknown to us and serve a purpose which is hitherto unclear. The investigations concerning them are incomplete, but some light however has been cast: they are almost of the same construction, dimensions, and seem to date to the 3rd dynasty; they have no substructures nor temples and are not within a cemetery of that period. To the 6 layer monuments some monuments could be added or related: Two pairs of tumuli at Naqada, investigated by Petrie. A square structure of rough masonry truly oriented to the cardinal points, of a side measurement of 11 m, suggest that it was a core of an unfinished or destroyed layer monument. It was excavated under the Middle Kingdom chapel of Ay at Abydos by C.T. Currelly. A benben named Seketra, and perhaps a pyramid named Nefer, recorded in ink on stone vessels discovered under the step pyramid at Saqqara. The pyramid of Snofru at Seila (see entry). Two step pyramids of queens of Menkaura at Giza (# G3b and G3c), differ in structure and function. Two benbens dating to the 5th dynasty in the sun temples of Userkaf at Abu Sir and Neuserra at Abu Ghurab. Four 5th dynasty benbens of other kings known from texts. With the exception of Ghenimiya which has never been excavated the other 5 layer monuments and Seila have only been partly investigated. The nuclei of Hebenu, Sinki and Nubt were built of quarried limestone and selected concretions from the slopes of the nearby plateau, Seila and el Kula were built of quarried limestone, el Ghenimiya was built of quarried sandstone and Elephantine was built of quarried granite and selected bolder. -
A Spurred Spiral Ramp for the Great Pyramid of Giza
Brichieri-Colombi, Spurred Spiral Ramp PJAEE, 17(3) (2020) A SPURRED SPIRAL RAMP FOR THE GREAT PYRAMID OF GIZA Stephen Brichieri-Colombi# # E-Mail: [email protected] Brichieri-Colombi, Stephen. A Spurred Spiral Ramp for the Great Pyramid of Giza. – Palarch’s Journal of Archaeology of Egypt/Egyptology 17(3) (2020), 1-20. ISSN 1567- 214X. 20 pages + 13 figures, 3 tables + 4 frames. Keywords: Engineering, Ergonomics, Giza, Great Pyramid, Khufu, Optimization, Ramp configuration ABSTRACT An easier and equally feasible configuration of spiral ramps for the construction of the Great Pyramid of Giza (Brichieri-Colombi, 2015), would be for a spiral ramp extended as a spur tangential to the pyramid rather than orthogonal to it. The general arrangement, which could have been used for many other large pyramids as well, is similar to that proposed by Lehner (1985: 129-132), but without the mass of temporary works that Lehner envisaged. It avoids the need to create a trench over the body of the pyramid during construc- tion, as proposed by Arnold (1991: 98), while respecting the constraints imposed by the available tools, workforce capabilities and design features of the pyramid. Finding the ideal configuration would not have been easy for the ancient builders, but this paper demonstrates how they could have done so with models. It also addresses the key construction issues associated with spiral ramps. An analysis of the construction effort required demonstrates that a ramp slope of 1:6 (9.5⁰) would have minimised the work involved. This finding suggests that pyramid construction hypotheses should be evaluated in terms of both feasibility and optimality to assess which are the most likely to have been adopted by ancient Egyptians. -
Biography of a Great Pyramid Casing Stone
The Journal of Ancient Egyptian Architecture vol. 1, 2016 Biography of a Great Pyramid Casing Stone David Ian Lightbody Cite this article: D. I. Lightbody, ‘Biography of a Great Pyramid Casing Stone’, JAEA 1, 2016, pp. 39-56. JAEA www.egyptian-architecture.com ISSN 2472-999X Published under Creative Commons CC-BY-NC 2.0 JAEA 1, 2016, pp. 39-56. www.egyptian-architecture.com Biography of a Great Pyramid Casing Stone D. I. Lightbody1 In the collection of the National Museum of Scotland in Edinburgh is an unusual artefact; a block of limestone, roughly hewn and damaged in places, and weighing around half a metric ton. The stone is of remarkable historic and scientific significance. Around 2,600 B.C.2 it was cut from the bedrock, shaped and sized in the RA-Aw quarries, now known as the Tura quarries (they are still accessible in the hills overlooking the east side of the Nile valley just south of modern day Cairo). The stone was then transported 15 km across the river valley to the largest pyramid ever built in Egypt, the Great Pyramid of pharaoh Khufu at Giza, where it was raised and placed on its outer face. Incredibly, a written record of the journey these stones took to Giza has survived.3 In 2013 frag- ments of a 4th Dynasty papyrus from the 26th or 27th year of the reign of pharaoh Khufu were recovered from the Wadi al-Jarf on the west coast of the Red Sea, where an Old Kingdom port is being excavated. The papyrus fragments that the joint Franco-Egyptian team4 uncovered turned out to be the oldest ever found.