Table Of ContentMathematical
Excursions
to the World’s
Great Buildings
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Mathematical
Excursions
to the World’s
Great Buildings
A l e x A N D e R J. H A H N
PRINCeTON UNIVeRSITY PReSS
PRINCeTON AND OxFORD
Copyright © 2012 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 6 Oxford Street,
Woodstock, Oxfordshire OX20 1TW
press.princeton.edu
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Hahn, Alexander
Mathematical excursions to the world’s great buildings / Alexander J. Hahn.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-691-14520-4 (hardcover : alk. paper) 1. Architecture—Mathematics.
2. Architectural design. 3. Architecture—Data processing. I. Title.
NA2750.H325 2012
720.1’51—dc23
2011028356
British Library Cataloging-in-Publication Data is available
This book has been composed in ITC New Baskerville
Printed on acid-free paper. ∞
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Contents
Preface vii
1
Humanity Awakening: Sensing Form
and Creating Structures 1
2
Greek Geometry and Roman Engineering 12
3
Architecture Inspired by Faith 53
4
Transmission of Mathematics and
Transition in Architecture 97
5
The Renaissance: Architecture and
the Human Spirit 138
6
A New Architecture: Materials,
Structural Analysis, Computers, and Design 205
7
Basic Calculus and Its Application to
the Analysis of Structures 265
Glossary of Architectural Terms 301
References 307
Index 311
Photo Sources 318
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Preface
The Mathematical Excursions into the world of architecture that this book un-
dertakes are organized around two historical narratives. The primary nar-
rative has a focus on aspects of architectural form (the role of geometry,
symmetry, and proportion) and structure (matters of thrusts, loads, tensions,
compressions) of some of the great buildings of western architecture from
the pyramids of Egypt to iconic structures of the twentieth century. Some of
the high points of this narrative are the Parthenon in Athens, the Colosseum
and Pantheon in Rome, the Hagia Sophia, historic mosques, great Roman-
esque, Gothic, and Renaissance cathedrals, Palladio’s villas, the U.S. Capitol,
the Sydney Opera House, and the Guggenheim Museum in Bilbao. (This
narrative is not comprehensive, but instead seeks to illustrate important ar-
chitectural features with compelling examples.)
A second narrative develops current elementary mathematics from a his-
torical perspective. This includes selected aspects of Euclidean geometry,
trigonometry, the properties of vectors, coordinate geometry in two and
three dimensions, and (at the very end) basic calculus. It is the raison d’être
of this book to intertwine these two stories and to demonstrate how they
inform each other. The mathematics provides clarifying insights into the
architecture, and, in turn, the architecture is a stage that gives visibility to
applications of abstract mathematics. To be clear, the two narratives meet
around topical issues and not when they happen to be aligned chronologi-
cally. In fact, the chronological alignment between mutually informing ar-
chitecture and mathematics is rare. (Greek geometry and architecture is an
exception.) The reality is that the elementary mathematics that might have
clarified the understanding of a complex structure was almost always beyond
the reach of the builders of the time.
The collage of historic buildings (all at the same scale) of Plate 1—refer to
the section of colorplates after chapter 4—provides a snapshot of this book.
It studies many of these buildings and focuses its mathematical analysis on
their domes, arches, columns, and beams.
The prerequisites that you will need to bring to this book are a working
knowledge of some basic high school math (such as elementary algebra and
a little geometry) as well as an interest in learning about architecture and
its vocabulary (as the Glossary presents it). You will be able to use the book
in different ways. To a large extent, your choices will be influenced by your
viii Preface
background in mathematics. If you are not on particularly friendly terms
with this discipline, I encourage you to be patient and persistent (and work
through the details) when you take up the Euclidean geometry, the few ele-
ments of trigonometry, and the basic study of vectors of Chapter 2. You will
then have the architectural narratives of Chapters 1, 2, 3, and 5 open to you.
A little more patience and persistence will allow you to absorb the basic his-
torical mathematics and the elementary introductions to coordinate systems
in two and three dimensions that Chapter 4 presents. My hope and expec-
tation is that your reward will be an engaging journey through the first six
chapters of the book. The two sections on perspective that conclude Chapter
5 and the two historical sections on structural engineering in Chapter 6 are
a bit technical, but they may be skipped as they do not directly impact the
rest. If you wish a mathematically more challenging expedition, take these
four sections on and also have a thorough go at Chapter 7. This chapter re-
views the basics of calculus and applies its methods to the analysis of domes
and arches. Each of the seven chapters concludes with a Problems and Dis-
cussion section. Many of the 200-plus problems and 18 discussions focus on
particulars of the topics presented; others go off on tangents that expand
their scope. For a lighter and quicker journey through the book they can all
be skipped. Most of the problems are designed to facilitate the understand-
ing of the mathematics. Some of them are challenging. An instructor of a
course using this text should assign them with care.
Acknowledgments
This book has drawn much benefit from a number of my friends and col-
leagues at the University of Notre Dame. Those of the Department of Math-
ematics deserve particular mention for the supportive environment that they
have provided over the years. A special word of thanks to Timothy O’Meara,
mathematician and former provost, for reading much of an earlier version of
the book and for providing important suggestions that improved it. Thanks
go to Neil Delaney of the Department of Philosophy, for many enriching
conversations about the creative structures of Frank Gehry and Santiago Ca-
latrava, and many other topics in which we share an interest. Thanks also
to David Kirkner of the Department Civil Engineering and Geological Sci-
ences, who always found the time to answer the questions about structural
engineering that I brought to his attention. A word of thanks to Michael
Lykoudis, Dean of the School of Architecture, for stimulating my interest in
architecture, and to his colleagues Richard Bullene, Norman Crowe, Dennis
Doordan, Richard Economakis, David Mayernik, John Stamper, and Carroll
Westfall for supplying information about various matters that arose in the
preparation of this text. My gratitude also to my friends and colleagues at
Preface ix
Notre Dame’s Kaneb Center for Teaching and Learning for always remind-
ing me that teaching must be measured not by the quality of the exposition
but by the learning experience of the students.
I am most grateful to Marc Frantz, Department of Mathematics at Indiana
University, Bloomington, for his positive response to my text. The critical,
but always well reasoned, insightful, and constructive remarks of his review
enhanced several discussions of the text. Thanks also to Robert Osserman,
director of special projects for the Mathematical Research Institute at Berke-
ley, for clarifying a point about the Gateway Arch in St. Louis.
A warm word of gratitude to Princeton University Press, especially Vickie
Kearn, my editor, and her assistants Stefani Wexler and Quinn Fusting; and
to Sara Lerner in production. They were always quick to respond, very help-
ful, encouraging, and consummately professional.
To my wife Marianne, thank you so much for your love and support, and
for not minding too much that my mind always seemed to be on something
else. Finally, to my father, Dr. George Hahn, thank you for teaching me the
values of commitment and discipline with the example of your life.
Alexander Hahn
Department of Mathematics
University of Notre Dame
September 2011
