258 79 59MB
English Pages 362 [389] Year 1990
THE REASONING ARCHITECT MATHEMATICS
AND
SCIENCE
Garry Stevens
IN
DESIGN
The Course of Western Architecture
1400
1500
Christopher Wren
Leon Battista Alberti
——————
Philibert De LOrme
a
Filippo Brunelleschi
etn
Andrea Palladio
VL
Guarino Guarini
VERSITY/OF INIA ‘ESVILLE ES
ee)
Donato Bramdmnte
Francesco Borromini
Giorgio Vasari 1600
nation’s band represents the proportion of great architecture constructed in that nation at each point in time. It can be seen that Italy completely
1700
1800
1900
Etienne-Louis Boullée nee
Auguste
Choisy
William Morris
eee see John Ruskin
Batty Langley
UK
Augustus Pugin
GERMANY
a
Bernado Vittone
1700
1800
4
1900
dominates western architecture until the early seventeenth century. The United Kingdom dominates the nineteenth century, and the United States the twentieth. Superimposed on this chart are timelines for most of the architects and architectural thinkers mentioned in the text
THE REASONING ARCHITECT MATHEMATICS AND SCIENCE IN DESIGN
THE REASONING ARCHITECT Mathematics
and Science in Design Garry Stevens University of Sydney
McGRAW-HILL PUBLISHING COMPANY
New York St. Louis San Francisco Auckland Bogoté Caracas Hamburg Lisbon London Madrid Mexico Milan Montreal New Delhi Oklahoma City Paris San Juan Sao Paulo Singapore Sydney Tokyo Toronto
THE REASONING
ARCHITECT: Mathematics and Science in Design
Copyright © 1990 by McGraw-Hill, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 1234567890
ISBN ISBN
HAL
HAL
89432109
O-O7-0b1391-5 {hard cover} Q-07-0b13%b-b {soft cover}
This book was set in Serif by the College Composition Unit in cooperation with York Graphic Services, Inc. The editors were B. J. Clark and Jack Maisel; the designer was Amy E. Becker; the production supervisor was Denise L. Puryear. Arcata Graphics/Halliday was printer and binder. Library of Congress Cataloging-in-Publication Data Stevens, Garry. The reasoning architect: mathematics and science in design/Garry Stevens. Pp. oom. Includes index. ISBN 0-07-061391-5 1. Architectural design. NA 2750.87 1990
720'.1'S51—de20
2. Architecture—Mathematics.
3. Science—Methodology.
L Title. 89-12598,
About the Author arry Stevens is lecturer in architectural sci-
ence at the University of Sydney. His first degree
is in architecture,
and
he
holds
graduate qualifications in architectural computing
and the philosophy and sociology of science. He has taught computer-aided architectural design,
and science and technology studies for eight years. The Reasoning Architect is his second book to be published by McGraw-Hill.
To the thousand students I taught: Thanks, guys, I had a lotta fun.
Contents FOREWORD BY MARIO SALVADORI FOREWORD BY HENRY J. COWAN ACKNOWLEDGMENT 1 THE REASONING 1-1
Ideas
ARCHITECT:
Mathematics,
Science, and Art in Architecture
3
1-2 Science and Art 5 1-2-1 Achievement in Sciences and Arts
1-2-2 The Content of Our Culture 9 1-2-3 Science and Art in Society 11 1-2-4 Science and Art in Architecture
1-3 The Life of Mathematics 1-3-1
Mathematics as Art
18
1-3-3 Mathematical Issues
22
5 15
18
1-3-2 The Discipline of Mathematics
19
2 PRIMITIVE NOTIONS: Mathematics in Preclassical Civilization
2-1
Before History
27
2-1-1 The Prehistory of Culture
27
2-1-2 Neolithic Architecture and Mathematics
2-1-3 Counting
27
34
2-2 Preclassical Mathematics
30
38
2-2-1 Egyptian Fractions 38 2-2-2 Babylonian Rationals 39
2-2-3 Greek Irrationals
2-2-4 How Many Reals? 3 THE WHOLE
44
45
HEAVEN A NUMBER:
3-1 Greek Mathematics
49
3-1-1 The Impact of the Greeks 49 3-1-2 Pythagorean Numbers 50
3-1-3 Rational and Irrational 3-2 The Critical Tradition
55
57
Greek Philosophy and Mathematics
49
xX
Contents
3-2-1 3-2-2 3-2-3 3-2-4
Beginnings 57 The Theory of Forms 58 The Arch of Knowledge 61 The Activity of Philosophising
62
4 THEORY INTO PRACTICE: Greek Geometry and Classical Architecture
4-1 Euclid’s Geometry
69
4-1-1 The Elements of Euclid 4-2 Polyhedra
69
69
74
4-2-1 The Platonic Solids 74 4-2-2 Other Families of Polyhedra 76 4-2-3 Relationships in Polyhedra 77 4-3 The Proportions of Classical Architecture
81
5 BY FORCE OF REASON: Logic and Deduction
5-1 Understanding the World
93
5-1-1 Analogy, Induction, and Deduction 5-1-2 The Grand Inquiry 95 5-1-3 The Place of Reason 97 5-2 Rational Argumentation 5-2-1 Deduction
5-2-2 5-2-3 5-2-4 5-2-5
The The The The
6 ORDER
93
93
98
99
Laws of Thought 101 Deductive Argument 104 Case of Euler’s Formula 119 Nature of Reason 122 FROM
CHAOS:
Mathematics
6-1 The Medieval Worldview
127
6-1-1 The Medieval Aesthetic
129
6-2 The Mason’s Geometry
131
6-3 Order in Pattern
137
6-3-1 Transformations
137
6-3-2 Translation
141
6-3-3 Rotation and Reflection
142
and Architecture in the Middle Ages
127
Contents
6-3-4 Tesselations
144
6-4 Order in Chaos: A Medieval City 7 THE SENSE 7-1
OF PROPORTION:
Renaissance in Florence
153
Renaissance Mathematics
and Architecture
161
161
7-1-1 The Rediscovery of Antiquity
161
7-1-2 The Triumph of the Architect 7-2 Proportion
Xi
162
165
7-2-1 Renaissance Proportion 7-2-2 Symmetry 170
165
7-2-3 Modern Proportioning Systems 7-3 Dimensional Coordination
171
180
8 A NEW PERSPECTIVE: Art and Science in the Renaissance 8-1 Linear Perspective
185
8-1-1 Alberti’s Method 187 8-1-2 Properties of Perspective 8-2 A New Way of Seeing 8-2-1 Science and Art Diverge 8-3 Wider Geometries
189 192 195
198
8-3-1 The Isometries 198 8-3-2 The Similarities 199 8-3-3 Perspectivity and Projectivity
200
8-3-4 Maps and Drawings as Projections
8-4 Graph Theory
201
205
8-4-1 Representing Plans by Graphs 8-5 A Hierarchy of Geometries 9 THE ENTERPRISE
185
205 211
OF SCIENCE: The Scientific Revolution and the Baroque
9-1 The Scientific Revolution
219
9-1-1 The Beginnings of Modern Science 9-1-2 Science as Revolution 224
220
219
xii
Contents
9-1-3 Ancients and Moderns
9-2 Analytic Geometry
227
229
9-2-1 Polar Coordinates 233 9-2-2 SemiLog Coordinates 233 9-2-3 Logarithmic Coordinate Systems
237
9-3 Transformations in Cartesian Terms
239
10 AN ORDERED UNIVERSE: Enlightenment
10-1 The World as Machine
247
10-1-1 First Reactions of Architecture to Science 10-2 Groups of Symmetries 10-2-1
Groups
253
10-3 Matrices
262
247
250
253
10-2-2 Point Symmetry Groups 255 10-2-3 Line Symmetry Groups 258 10-2-4 Other Groups 260 10-3-1
Transformations as Matrices
10-3-2 Figures as Matrices 11
THE WORLD
264
266
IN A BOX: Mathematical
Models and Nineteenth-Century Science
11-1 Science and Architecture in the Nineteenth Century 11-1-1
Science Emulated
11-1-2 Science Rejected
271
275
11-2 How Does Science Know?
279
11-2-1 The Problem of Consensus
279
11-3 Making Models
281
11-3-1 Physical and Symbolic Models 11-3-2 Models in Architecture
284
282
11-3-3 The Virtues of Theories and Models 11-4 Algebra
287
11-4-1 Vectors 288 11-4-2 Imaginary Numbers
291
285
271
271
Contents
12
FINDING THE BEST: Our Century and Mathematics
12-1
The Modern
298
12-2 Walter Gropius’ Housing Problem 12-2-1 Defining the Model 303 12-2-2 Analyzing the Model 304 12-2-3 The Case of Tower Apartments 12-3 Finding the Best
301
306
308
12-3-1 Generate and Test Procedures
12-3-2 Improvement Procedures 311 12-3-3 Linear Programming 312 12-3-4 Other Optimization Techniques 13 THE POSTMODERN
311
CONDITION:
13-1 Language in the World
317
13-1-1 The Analytic Tradition
318
13-2 The World in Language
314 Reason and Architecture after Modernism
321
13-2-2 Structuralism 322 13-2-3 The Poststructural 325 13-2-4 The Postmodern and the Poststructural 13-3 The Question of Science
Science and Society
327
328
331
13-4 Science and Art in Architecture
332
Scholarship in Science and Architecture
13-4-2 The Critical Turn
317
320
13-2-1 Phenomenology in Architecture
13-4-1
295
295
12-1-1 Modernist Architecture
13-3-1
xiii
334
BIBLIOGRAPHY AND REFERENCES INDEX
332 343 354
Foreword Te
essential purpose of this foreword is to forewarn readers that they do not have in their hands a book on architecture, or a book
on science, or on mathematics, or on philosophy, but what I can only describe as an encyclopedia for the modern cultured person. I feel that such a warning is required by the nature of the book and is, I hope, useful. By misunderstanding architectural tradition the layperson, as well as the starting architectural student, often believes that architecture is a discipline
to be exclusively listed among the arts. By misun-
derstanding the essence of science, the same two
categories of people are often taught that while they are capable of understanding art, they are physiologically unable to penetrate the mysteries of science unless specifically trained in it. As a consequence of these misunderstandings they pass wrong judgments about architecture, unaware that this most demanding of contemporary disciplines has both an (obvious) artistic component but also a less obvious, and not less essential,
scientific component. The masters of architecture, those of the past as well as those of the present, have been deeply aware of both components but have seldom raised their voices to emphasize the “duality” of architecture. Garry Stevens has gone to the printing presses as possibly the first architect daring to explain the two faces of architecture. To achieve his complex goal he presents a complete picture of the development of science and of the innumerable facets of mathematics, as well as an explanation of the interaction between
science and mathematics
in
the growth of architecture. Many are the authors who have tried to explain the need for an understanding of basic science in
the practice of architecture, but Professor Stevens
is unique in the width and depth of his presentation. While most popularisers of science try their hardest
to make
it sound
simple
and
intuitive,
twisting through easy metaphors most of its real complexities, Stevens has grabbed the bull by the horns and gone widely into all phases of mathematics, from number theory to algebra, from the
calculus to probability theory, and from topology to logic. He has simply been honest and honestly been simple, presenting each of his subjects through clear examples of their practical importance as well as through their historical and logical origins. Such an approach makes his book unique and in concert with the requirements of our society: at long last the reader is allowed to learn a total approach to the scientific aspects of architecture and to an understanding of how science, the
basis of modern knowledge, is the natural issue of our entire cultural heritage. Because of the book’s honesty and thoroughness it is impossible for a foreword to give even a vague idea of the contents. By the time they will have finished Stevens's opus those readers willing to learn from it will possess a new understanding not only of architecture but of all the disciplines that challenge them to become twenty-firstcentury persons. If at first some of the varied topics may appear complex, let the reader be told that the satisfaction to be obtained from this book will more than repay the time and concentration it requires.
ProressoR Mario SALVADORI Co umsia Untversity
New York
Foreword eometry has been part of architectural design from the very beginning of building in masonry. The angle of inclination of the Great Pyramid was determined 4600 years ago by a geometric construction that involved the then fashionable problem of ‘‘squaring the circle.” The popularity of geometry was at least in part due to the limitations of measuring rods and chains, and of surveying instruments for measuring angles. It is relatively easy to lay out a straight line by stretching a rope and to draw a circle by scribing with one end of a rope around a fixed point. The straight line and the circle can then be used to multiply unit lengths from the cubit (the length of the forearm) or the foot. They can also be used,
as Garry
Stevens
demonstrates,
to subdi-
vide a length into nonstandard divisions and to set out angles. Thus geometry was the basis of both architectural design and construction for a large part of its history. Geometric constructions, based on direct sight-
lines and reflections, were the basis of Greek and Roman acoustic design, the only “architectural sci-
ence” that existed before the seventeenth century. Structural design was entirely empirical until that time,
and
the rules
used
were
all expressed
in
terms of geometry. Most important of all, the aesthetic rules of the Greek, the Roman,
and the Re-
naissance architects were based on various geo-
metric ratios. Thus the scientia of the Renaissance architects was the knowledge of those aesthetic rules, which could be applied only by those who had studied geometry. The ars (translated appropriately as the “useful arts”) was concerned with the craft rules of building construction, and they could be mastered without a mathematical education. Until the seventeenth century architectural mathematics was mainly geometry and some of it was, even by modern standards, very complex. Few people today. know enough of it to follow all
the
constructions
given
by
Guarino
Guarini,
a
seventeenth-century Baroque architect, in his Architettura Civile. Architecture was at that time one of the most mathematical branches of knowledge, lagging not far behind astronomy, then the queen of the sciences. The transformation of Christopher Wren from an astronomer into an architect was in the seventeenth century not as remarkable as it seems today. By the late nineteenth century when buildings
became
bigger,
more
numerous,
and
more
com-
plex, there was a science of structural design, using ever more mathematics. During the twentieth century the science of structural design was supplemented by mathematically based methods for the design of the thermal, luminous, and acoustic
environment. The architect more and more passed the responsibility for these mathematically based branches of the design of buildings to specialized consultants. Mathematics became the most unpopular course in architecture schools; in some it was simply abolished. Why are so many architects, particularly those with “artistic” inclinations, today positively hos-
tile to mathematics, while until the seventeenth century many were accomplished mathematicians (including some whom we admire for the artistry of their buildings)? There are a number of reasons for this change in attitude. Mathematics has changed from a science dominated by geometry, which can be demonstrated on a drawing board, to a combination of the more
abstract algebra and
of arithmetic,
de-
spised as mere “number-crunching.” Science has changed from a new and rebellious branch of knowledge, much admired by young
artists in the seventeenth century, to one allied to
the “military-industrial complex.” The artistically inclined architect, particularly while young, is now instinctively antiscientific.
xvili
Foreword
Then there is the question, why study mathematics when most of the technical problems requiring its use can be passed over to specialist consultants? The answer is, of course, that architects
must have at least a general knowledge of the various specialties involved in the building process if they are to retain control of it, and that they must understand the costing of a building, a highly complex process involving for the larger buildings a lot of number-crunching, until one reaches a cost of many millions of dollars. Most important of all, architectural design consists of an ounce of artistic inspiration and many tons of logical argument.
The historical approach should make the subject more palatable to architecture students who are inclined to be hostile, since it shows the con-
stant interaction between two long-established branches of human knowledge. I commend this attempt to reestablish the close link that once ex-
isted between architecture and mathematics.
Henry J. Cowan Proressor Emeritus OF ARCHITECTURAL SCIENCE University oF SypNEy, AUSTRALIA
Acknowledgments Ts
first person I must thank is Sue Stewart,
who
read the entire manuscript after each
revision and made tions, especially on the profession and recent structuralism, preventing
Cornell, after Bruce Allsop.
Figure 1-10, A. Hess
and the Society of Architectural Historians. Figure
many important suggesstate of the architectural developments in postme from lapsing into er-
1-12, Pion Ltd and B. Hillier. Figures 2-2, 2-3, and 2-7, Pion Ltd and C. Chippindale. Figures 2-4, 2-5, and 2-6, J. Barnatt and Turnstone Books. Figures 3-2 and 3-3, T. Burnes and Rhodos. Figures 3-7,
fessors Salvadori and Cowan for their forewords. I also wish to thank John Gero for his continuing support and guidance over the years, and Peter Smith for putting up with me. Jay Kappraff and Vedder Wright deserve my thanks for their interest and encouragement in the text. Many thanks also go to the following reviewers of this manuscript: Mark Gelernter, University of Colorado;
McGraw-Hill, from W. G. Lesnikowski, Rationalism and Romanticism in Architecture, (1982). Figures
ror on numerous occasions. I must next thank Pro-
William Glennie, Rensselaer Polytechnic Institute; Robert Heller, Virginia Polytechnic and State Uni-
versity; Ray Levitt, Stanford University; Michael Mahoney,
Princeton
University;
Jens
G.
Pohl,
California Polytechnic and State University; Jerald Rounds,
Arizona
State
University;
James
C.
Snyder, University of Michigan; Anne Griswold Tyng, University of Pennsylvania; and Gordon Varey, University of Washington. Finally, my thanks
to B. J. Clark, Jack Maisel,
Amy
Becker,
Marci Nugent, and all the other guys at McGrawHill for their efforts in the production of this book. Every effort has been made to trace the copyright owners of illustrations and quotations used in this book. However, in some cases the publisher or author seems to have disappeared off the face of the earth. If any copyright owner has been unacknowledged,
the
author
would
appreciate
the owner contacting him. I wish to gratefully acknowledge the following for allowing me to reprint illustrations: Chapter openers for Chapters 1, 2, 3, 4, 7, 9, 11, 12, and 13 and Figures 6-7 and 6-12 were kindly supplied by Mr. Tone Wheeler from his immense
slide collection. Chapter openers for Chapters 5, 6, 8, and 10 and Figure 6-1 were kindly supplied by the
inestimable
Ian
Fraser.
Figure
1-5,
David
9-6, 10-1, and 10-5, reprinted by permission of 4-4 and 4-8, reprinted by permission of Cambridge University
Press
from
J.
A.
Baglivo
and
J.
E.
Graver, Incidence and Symmetry in Design and Architecture (1983). Figure 4-9, G. E. Martin and
Springer-Verlag Inc. Figure 4-15, based on a figure in Coulton (1977). Figures 5-1 and 11-7, reprinted
by permission of Cambridge University Press from P. Steadman, The Evolution of Designs (1979). Fig-
ures 6-2, 6-3, 6-4, and 6-5, based on figures in Ackerman (1949). Figures 6-8, 6-9, 6-10, and 6-11,
based on figures in Shelby (1977). Figures 6-13 and 7-11, The British Architectural Library, Royal Institute
of British
Architects,
London.
courtesy of R. Oxman and A. D. 6-28, reproduced by courtesy of the British Museum. Figures 6-37, 6-41, and 6-42, reprinted with the
Figure
6-14,
Radford. Figure the Trustees of 6-38, 6-39, 6-40, kind permission
of Pion Ltd., F. E. Brown, and J. Johnson. Figure
7-1, reprinted by permission of Cambridge University Press from P. Scholfield, The Theory of Proportion in Architecture (1958). Figures 7-14 and 7-24, from R. Nagarajan (1976). Figures 7-22 and 7-23,
reprinted with the consent of The Architects’ Journal. Figure 7-25, based on a figure in Nagarajan (1976).
Figures
8-1,
8-4,
8-5,
8-17, reprinted by permission
8-7,
8-9,
8-10,
of Routledge
and
and
Kegan Paul from L. Wright, Perspective on Perspective (1983). Figure 8-29, reprinted by permission of Macmillan Publishers Ltd. from J. Mainwaring, An Introduction to the Study of Map Projection (1942).
Figure 8-34, A. Blunt and A. Zwemmer Ltd. Figure 8-35, from F. D. K. Ching (1980). Figure 8-38,
reprinted by permission of the McGraw-Hill Pubxix
XX
Acknowledgments
lishing Company
from
R. A. Class and
R. E.
Koehler, Current Techniques in Architectural Practice
(1976). Figures 9-4 and 9-5, reprinted by permission of Dover Publications, Inc. from H. L. Resnikoff and R. O. Wells, Mathematics in Civilization (1984). Figure 9-9, S. Edgerton and the Society of Architectural Historians. Figure 9-12, courtesy
of BIS-Shrapnel. Figure 9-13, based on a figure in Bon (1973). Figure 10-4, courtesy of Harold Dorn
and Robert tesy of the based on a 12, courtesy
Mark. Figure 11-3, reproduced courTate Gallery, London. Figure 11-6, figure in Echenique (1972). Figure 12of Building Services Ltd and J. Clarke.
Figure 13-1, reproduced with the consent of Louis
Hellman and The Architects’ Journal. Figure 13-2, based on a figure in Sterman (1985). Figure 13-5,
based on a figure in Barnes (1985). Figure 13-6, courtesy of Architecture Australia.
I wish to gratefully acknowledge the following for allowing me to reprint quotations: Extract by A. Placzek, reprinted with the permission of The Free Press, a division of Macmillan
Publishing Company, from the Macmillan Encyclopedia
of Architects,
Adolph
K.
Placzek,
editor.
Copyright © 1982 by The Free Press, a division of Macmillan Publishing Company; Addison-Wesley Publishing Company Inc for an extract from J. Potage, Geometrical Investigations; A. M. Heath and
Co Ltd for extracts from Carl Sagan, Murmurs of Earth (1978); MIT Press for extracts from Alberto
Piez-Gomez, Architecture and the Crisis of Modern
Science (1983); Curtis Brown Ltd for extracts from J. Brownoski, Science and Human Values (1962); Pion Ltd for extracts from P. Steadman, Architectural Morphology (1983); Dover Publications for extracts
from Henri Poincaré, The Value of Science (1958),
from Vitruvius’ The Ten Books on Architecture, translated by M. H. Morgan (1960) and from L.
Sullivan, Autobiography of an Idea; Cambridge University Press for extracts from G. Hardy, A Mathematician’s Apology (1969), from Roger Ascham’s
English Works, edited by W. A. Wright (1904), from Philosophy in France Today, edited by A. Montifiore (1983), from I. Lakatos, Proofs and Refutations (1979), and from The Works of Archimedes, edited by
T. L. Heath (1897); Oxford University Press for ex-
tracts reprinted
from
The Oxford
Aristotle, edited by W.
Translation
D. Ross (1941), M.
of
Kline,
Mathematics: The Loss of Certainty (1982), and from
T. L. Heath, A History of Greek Mathematics (1921);
T. Bruns and Rhodos for extracts from The Secrets of Ancient Geometry and Its Use (1967); David Fulton
Publishers for extracts from N. Prak, Architects: The
Noted and the Ignored, originally published by John Wiley and Sons (1984); Arthur Probsthain Publishers
for extracts
from
Mo-Tzu’s
Against
Fatalism,
translated by Y. P. Mei (1929); State University of New
York Press and H. Gans
for extracts from
Professionals and Urban Form, edited by J. R. Blau,
M. La Gory, and J. S. Pipkin (1983); Basil Blackwell
Ltd for the extract by R. Hare from Ratio (1960),
and from Ancilla to the Pre-Socratic Philosophers, ed-
ited by K. Freeman (1971); Boydell and Brewer Ltd
for extracts from John Aubrey’s Brief Lives, edited by R. Barber (1975); Penguin Books Ltd, Harmondsworth, for extracts from Plato: The Re-
public, translated by H. D. P. Lee (1977), copyright H. D. P. Lee, 1955, from Plato: Timaeus and Critias,
translated by H. D. P. Lee (1977), copyright H. D. P.
Lee,
1971,
from
N.
Pevsner,
An
Outline
of
European Architecture (1968), copyright N. Pevsner,
1943, from Sophocles: The Theban Plays, translated by E. F. Watling (1947), copyright E. F. Watling, 1974, from Plutarch: Makers of Rome, translated by I. Scott-Kilvert (1965), copyright I. Scott-Kilvert, 1965, and for the extract by Isocrates from
Greek
Political Oratory, translated by A. N. W. Saunders (1970), copyright A. N. W. Saunders, 1970; The
Johns from
Hopkins
Hesiod,
University
translated
by
Press A.
for the extract
N.
Athanassaki
(1983); C. Ray Wylie for his poem “Paradox,” from
Science, volume 67, page 63, copyright © 1948 American Association for the Advancement of Sci-
ence; A. D. Editions Ltd for extracts reproduced
from
‘The
Buckminster
Architect as World
Planner”
by R
Fuller, courtesy of Architectural De-
sign Magazine, London; The Bodley Head Ltd and Jonathon Cape Ltd for extracts from The
Buckminster Fuller Reader, edited by J. Meller (1980); Routledge and Kegan Paul for extracts from Paracelsus’ Selected Writings, edited by J. Jacobi (1951); Manchester University Press for extracts
Acknowledgments
Xxi
from the Cooke manuscript, edited by D. Knoop,
Schuster
Ltd
Kunstverlag GmbH for extracts from ‘Late Gothic Structural Design in the ‘Instructions’ of Lorenz Lechler,” by L. R. Shelby and R. Mark, in
Ltd for Haydon, Jennings
extracts by Coleridge, Ruskin, and from Pandaemonium, edited by H. (1985); Harper and Row Publishers Inc
G.
P.
Jones,
and
D.
Hamer
(1938);
Deutscher
Architectura; Princeton University Press for the ex-
tracts by Serlio and Suger in A Documentary History of Art, edited by E. G. Holt (1958); Academy Editions, London, for extracts from L. Alberti’s Ten Books on Architecture, translated by J. Leoni (1955),
from Boulle and Visionary Architecture, edited by
H. Rosenau (1976), from R. Rykwert, The Necessity
of Artifice (1982), and from R. Wittkower, Architectural Principles in the Age of Humanism (1988); Harvard University Press for extracts from Ssu-Yu
Teng
and J. K.
Fairbank,
China's
Response
to the
West (1968), reprinted by permission; Simon and
for
extracts
from
Joseph
Glanvill,
The Vanity of Dogmatising (1970); Andre Deutsch
for extracts from Walter Gropius, The Scope of Total Architecture (1955). Copyright © 1955 by Walter Gropius. Reprinted by permission of Harper and Row Publishers Inc; William Heinemann Ltd for extracts from Aulus Gellius’ Attic
Nights,
translated by H.
Rolfe,
and
from
Plato’s
Philebus and Parmenides, translated by H. N. Fowler; IPC Magazines Ltd for the extract by R. Estling appearing in New Scientist; and the Watchtower Bible and Tract Society of New York Inc for the extract from Life—How Did it Get Here? By Evolution or Creation (1985).
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The Reasoning Architect: Mathematics, Science, and
Art in Architecture
1-1
IDEAS
This is a book about ideas, about ideas in architecture and ideas of architecture. Now this is a very large subject, about which many volumes have been
written, almost all of them concerned with ideas of
the imagination. The ideas discussed in this book
Several themes run through the following ters. At the highest level, there is the notion pervasiveness of mathematics in the western lectual tradition. The extraordinary ubiquity of
chapof the intelmath-
ematics in our culture is not due to its instrumental
efficacy, that is, to the fact that it is useful for solv-
ing practical problems. The peculiar and revered po-
It is the aim of this book to show that although architecture is usually thought to be the product of acts of inspired creation, it is also the product of acts of inspired reason; to demonstrate that science and mathematics are portions of our intellectual culture that cannot be set apart from architecture and
sition of mathematics in the West is a consequence of mathematics’ ancient claim to provide absolutely certain knowledge. From the birth of mathematics as an independent body of knowledge, fathered by the classical Greeks, and for a period of over 2000 years, mathematicians pursued truth. Under the powerful influence of Pythagoras, Plato, and Aristotle, mathematics and philosophy became intertwined, sharing as they did the requirement for ironclad proofs of statements. The mission of philosophy, it was held, was to discover the true knowledge be-
cern of all of us.
and deceptive appearance of this world. In this
are ideas of a different sort: ideas of the intellect, of
reasoning. Such things are commonly believed to be the province of engineering and science and mathematics, and really not of much interest to the
art of architecture. Nothing could be further from the truth.
left to the engineers to worry about, but are the con-
hind the change and illusion, the veil of opinion
4
THE REASONING ARCHITECT
quest, mathematics had a special place, for mathematical knowledge was the outstanding example of knowledge independent of sense experience. It was certain, objective, and eternal. To achieve its marvelous and powerful results,
mathematics relied on a special method, namely, that of deductive proof from self-evident axioms.
Deductive reasoning, by its very nature, guaran-
tees the truth of what is deduced if its axioms are truths. By utilizing this seemingly clear, infallible, and impeccable logic, mathematicians produced seemingly irrefutable conclusions. The accomplishments of the Greek mathematicians were magnificent. They produced a vast body
of theorems about number and space, deduced in
self-obvious steps from self-obvious axioms, offering an almost endless vista of certainty. Although much was lost in the Middle Ages, this notion of mathematics as a peculiar kind of knowledge survived. The indubitability of mathematical knowledge struck the medieval mind as being so different in kind from the piecemeal knowledge we have of the sensory world that they conceived of it as a gift from God (Kline, 1980).
During the Middle Ages, the great churches and
cathedrals were designed on intricate geometrical
lines. If mathematics was a link to the Divinity, then
it was surely obvious that the House of God must be designed according to the mathematical principles that God had vouchsafed to humanity. During the Renaissance this idea was developed in other ways. Mathematics, it was held, was behind all that was beautiful in the world, so beautiful buildings
must therefore be designed with mathematics. An-
other theme of this book, therefore, is that the most
long-lived notion in architectural theory is that math-
ematics provides the key to architectural design. Beyond the game of mathematics itself, mathematical concepts supplied the essence of our un-
derstanding of the world. During the Renaissance it supplied the theory upon which artists could build
a perspective for the realistic depiction of scenes. It
also provided a theory of proportion with which to design buildings. During the scientific revolution individuals such as Kepler and Galileo and Coper-
nicus constructed mathematical theories of optics,
astronomy, and mechanics that were in remarkable
accord with observation. They used to provide a firm grip on the workings understanding that dissolved mystery it by law and order. The American William James expressed this attitude
mathematics of nature, an and replaced philosopher thus:
When the first mathematical, logical and natural uniformities, the first laws, were discovered, men
were so carried away by the clearness, the beauty and simplification that resulted that they believed themselves to have deciphered authentically the eternal thoughts of the Almighty. His mind also thundered and reverberated in syllogisms. He also thought in conic sections, squares and roots and ratios, and geometrized like Euclid. He made Kepler’s laws for the planets to follow; He made velocity increase proportionally to the time of falling bodies, He made the law of sines for light to obey when refracted.... He thought the archetypes of all things and devised their variations; and when we rediscover any of these wondrous institutions, we seize His mind in its very literal intention. (James, 1907, p. 56)
From 1600 to the late 1700s the mathematicians
and scientists were sustained by the belief that God had designed the universe in an orderly way, and
that the task of science and mathematics was to un-
cover that design. In the late Enlightenment this belief gave way to a concern for the attainment of purely mathematical results, unconnected with reallife problems. The idea that there were any indubitable truths at all came under attack, and logical difficulties arose in arithmetic and algebra. The problem of the age became: If God does not guarantee the truth of mathematics,
then what does? Some
answered that truths derive from the basic intui-
tions of the mind and that the real world was essentially unknowable.
Science,
then, was not the
exploration of the world but of our own minds. This idea was developed in the late nineteenth century and our own time into a powerful critique of natural
science, which we shall examine in a later chapter.
THE REASONING
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The assertion of mathematics to provide a very particular and very specific sort of reliable knowledge has been largely accepted since antiquity, and challenged only in this century. Other disciplines have also made this claim. We will therefore be exploring the question of what constitutes valid knowledge and how we can obtain that knowledge. Science is one such discipline. Science staked its own claim quite recently, about 400 years ago. Since that time science has steadily increased the breadth and comprehension of its knowledge claims. One result has been that our present view of the world and of ourselves is shot through and through with scientific ideas, many of which are quite sophisticated and quite subtle, and which form a bedrock, a lowest stratum of thought, pervading our society.
MATHEMATICS,
SCIENCE, AND
ART
IN ARCHITECTURE
5
change the world. What a complex picture of the universe!
Is not all this obvious, is it not common
sense?
It is in fact a picture developed by our own culture, through long centuries. Today’s common sense is yesterday’s radical science. Science and art have allotted places in this particular picture, places very different from those in the pictures created by other cultures. Running through this book is therefore the theme of the history and content of our world picture and the way it structures our experience of the world.
for example, the methods we use to
At each step in the expansion of science’s knowledge claims, individuals have expressed disquiet and concern about the ever-growing ambitions of science. Architectural thinkers have been particularly concerned with this. We will hence discuss the
there are such methods. We do not believe, as was
which it is legitimately employed, particularly in ar-
Consider,
ensure that buildings stay up. First, we assume that
common in the Middle Ages, that the structural in-
tegrity of a building depends in part on the will of
nature of science and its processes and the extent to chitecture.
In the rest of this chapter we discuss mathemat-
God. This implies that, second, there is a causal connection between what we do, the materials we use,
ics, science, and the humanities and arts in general
main upright. It is not just an accident that a con-
mixture in architecture.
and their sizes, and the ability of a structure to recrete beam of such-and-such a size works. There is a reason for it. Third, this reason involves proper-
terms (Box 1-1). We examine the differences between them,
their place in society, and their ad-
ties of the structure, the beam, and the forces act-
1-2 SCIENCE AND ART
ies are helping it, or because the beam itself wants
1-2-1 Achievement and Arts
ing on it. The beam does not do its job because fairto work.
Beams, we hold, do not think for them-
in the Sciences
It is possible for us to find out things about the beam.
The International Mathematics Olympiad is a sort of Olympic Games for budding mathematicians. In 1986 a 10-year-old boy became the youngest com-
it will do if the loads are changed. Sixth, we can use
studying second-year university mathematics, first-
selves. Fourth, the reasons behind the structure’s
behavior are discoverable by engineers or scientists.
Fifth, we can predict the behavior of the beam, what
petitor to enter the Olympiad. At the time, he was
all this knowledge to alter the structure, to manip-
ulate it to suit our purposes.
Now consider what all these assumptions add
up to: that events that occur in the world fall into patterns or regularities that are the product of rules
and natural laws, that an invocation of these rules is the only legitimate explanation of these events, and that these rules can be discovered and understood by humanity, which can then use them to
' The ways in which knowledge is classified by a culture provide a window into the intellectual life of that society. Thus we
usually distinguish between the sciences, humanities, and the
arts. Sometimes we also refer to the crafts. The sciences are often divided into natural (physics, chemistry, biology, geology) and social (psychology, sociology, economics, anthropology). The arts are usually divided into visual (architecture, sculpture,
painting), music, and performing (theater, dance, cinema). The humanities are usually taken to comprise philosophy, literature, religion, classics, and the languages.
6
THE REASONING ARCHITECT
BOX 1-1 ISMS: HUMANISM AND POSITIVISM Two words we shall meet a great deal in these pages are humanism and positivism. Humanism, we shall see later, emerged in the Renaissance as a certain way of looking on the world. At that time it reterred to a respect for the human as opposed to the divine affairs that had preoccupied the Middle Ages. Humanists were then those scholars interested in politics and history. These studies were revived in the Renaissance as an adjunct to their enthusiastic studies of the classical civilizations of Greece and Rome, so humanists were also interested in the classics and the languages of these civilizations. Thus, today, the humanities as an academic discipline generally refer to the study of the classics, history, language and literature, and philosophy.
During the Enlightenment (late eighteenth century), the idea was
extended in a further way. The distinction from religion was emphasized, so that humanism came to reter to a nonreligious way of life that had more pleasant connotations than the word atheism. A second development in the nineteenth century broadened the concept, as the aspects of humanism as the study of humanity were emphasized. This saw human beings as capable of moral development, of pertectibility, and referred to a kind of learning that lauded music, literature, painting, theater, film, and sculpture, and sought to
encourage some sort of intellectual, spiritual, and aesthetic progress
in society. Atthe same time science had reached a point of maturity in which it seemed to many people that its methods could solve many of humanity’s ills. The enthusiastic embracing of science as a powerful means, perhaps the very best means, of understanding the world and everything in it, including human beings, was given the name Positivism. Previous means of inquiry, it held, were inferior to science. The proper way to do things was to discover facts and ascertain relationships between these facts. From these relationships general laws could be deduced. The process was completely objective. In the general critique of science that has developed since then, positivism has come to refer to the idea that the methods of the natural or physical sciences are applicable to all the endeavors of humanity. The term has acquired largely negative connotations, and the term scientism is now often used as a synonym (Williams, 1985;
Barnes, 1985).
THE REASONING
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year university physics and computing science, final-year high school year Latin and geography, and intermediate high school English. Now this order is very interesting. He was a full eight years ahead of his age in mathematics, but only two years ahead in English. Clearly he was a prodigy, but while he was merely advanced in English, he was
positively brilliant in mathematics. This pattern of
youthful genius in mathematics is in fact quite common, and most universities throughout the world
have a handful of mathematical child prodigies selfconsciously attending lectures. Precocity in the arts and humanities, however, is quite rare. How many
10-year-olds are there in the world’s architecture
schools? I am willing to bet that there is not a one. Why should this be? Consider some of the mathematicians whose work I present in later chapters: Gaspard Monge,
inventor of the descriptive geometry that is the fundamental means of communication in the building industry, was a professor of physics at 16. Karl Friedrich Gauss,
by common
consent one of the
three greatest mathematicians of all time, was recognized as a prodigy at 14 and did his most important work between 18 and 21. Augustin-Louis Cauchy, who invented group theory, had become one of the foremost mathematicians of his time by the age of 27. William Hamilton, who codified the
mathematics of complex numbers, was elected professor of astronomy at 22. Arthur Cayley had published 25 papers by the same age. Janos Bolyai devised non-Euclidean geometry at 29. Christopher Wren was a professor of astronomy at 25, well before he thought of architecture as a career. Isaac Newton invented modern physics and mathematics at 24. Charles Darwin had conceived of the the-
ory of evolution by the age of 29. Albert Einstein had completed his major work on relativity by 26. Richard Feynman had finished the work in physics
that won him a Nobel Prize by the time he was 30. James Watson had revealed the structure of DNA to the world at 25.
Most of the truly great mathematicians showed their genius before they were 20, and almost every
significant mathematical discovery was made be-
MATHEMATICS,
SCIENCE, AND
ART
IN ARCHITECTURE
7
fore its discoverer was 25 (Hardy, 1969). Now compare this to the careers of architects. Le Corbusier designed Notre-Dame-du-Haut at Ronchamp, often considered the most important building of the century, at 63. Frank Lloyd Wright designed Falling Water at 68. Louis Kahn became famous when he did the Yale University Art Gallery at 50. The genius of Mies van der Rohe was first revealed in his German pavilion at Barcelona when he was 43. Christopher Wren, having turned from science to architecture, designed his masterpiece, St. Paul’s Cathedral, also at 43. As Adolf Placzek said in the
preface to the monumental Macmillan Encyclopedia of Architects:
Unlike so many of the great poets and musicians, great architects are a peculiarly long-lived lot. They are tough. Dealing with material, structure and society’s demands, they have always had to be. Among the encyclopedia’s twenty most outstanding architects, only two—Raphael and H. H. Richardson—died before they were fifty, and several did some of their finest work after they were seventy. (Placzek, 1982, p. xii)
This picture of youthful achievement in the sciences and mathematics and elderly achievement in architecture and art is the conventionally accepted portrait. The truth is just the opposite. Figure 1-1 shows the proportion of a lifetime’s work accomplished each decade of life for a variety of disciplines. From Figure 1-1c you can see that in general productivity rises quickly to peak in the 40s, then declines slowly to old age. Mathematicians (Figure 1-1b) are not in fact especially precocious, but they do have an unusually flat productivity profile (very
similar to that of humanists). That is, each decade after the age of 29 they produce about the same amount of work. Architects (and other artists), con-
trary to Placzek, in fact, produce very little in old
age. Scientists in their seventies are only slightly
less productive than those in their exuberant thir-
ties, but an elderly architect is much less than half
as productive.
However, this overall picture of productivity says
8
THE REASONING ARCHITECT
5
30
30
25
25
3=
= 20
20
=2 15
2 = 15
5
§ 10
@
°
a
a
°
8
5
@
0
20s
30s
40s
Age
50s
60s
5 0
70s
20s
30s
(a) Architects
30 x
:A
£
&
25
40s
Age
50s
60s
70s
(b) Mathematicians
O Sciences @
Humanities
B Arts
20
15
6
= 10 @Q
8
@
5 0
20s
30s
40s
50s
60s
70s
Age (c) The sciences, arts, and humanities in general
Figure 1-1 The proportion of a lifetime’s work accomplished each decade of life for a variety of disciplines. (a) Architects. (b) Mathematicians. (c) The arts, sciences and humanities in general. (Sources: Architects, Stevens, 1988a; Others, Dennis,
1966)
nothing about the quality and development of one’s
Could the reason for this difference in produc-
life’s work. The anecdotal evidence mentioned above is, I suggest, being misinterpreted. Mathe-
tivity lie in differing psychologies? We have known
seminal ideas, when very young. Thereafter, they
different modes of thought (Schmitt and Worden,
maticians may well do their best work, or have their
produce at a more or less steady rate. Architects
probably take rather longer to mature and do not reach their qualitative peak until middle age. Thereafter their output diminishes rapidly.
for some time that the human brain is lateralized,
that is, the two cerebral hemispheres specialize in 1975). As far as we know, we are the only such an-
imals in which this occurs, although there is some
weak evidence for its existence in the other pri-
mates. The left hemisphere seems to be good at lan-
THE
REASONING
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guage, analytical thought, and numbers. The right hemisphere
is better at three-dimensional vision,
pattern recognition, and music. It is almost a linguistic idiot (Campbell, 1984). Most people pass their lives with one hemisphere
MATHEMATICS,
SCIENCE, AND ART
IN ARCHITECTURE
Q
ics must come mostly from the activities of the left,
just as the development of literature must come more from the right even though linguistic skills are left-based. It would seem that the abilities of the left hemisphere are at their zenith from childhood,
brainers, and artists and humanists right-brainers,
decaying slowly from early adulthood, but those of the right hemisphere mature slowly. Our capacity for analytic thought is prewired into our brains, but the richness of imagination characteristic of artistic genius is not so programmed and must be absorbed from a cultural milieu.
the proclaimed dominion of the artists. Or, rather,
gifted people seem to have more in common with other gifted people in any field than they have with their mediocre colleagues in the same discipline.
dominant: some seem almost entirely intuitive, oth-
ers entirely rational. It would be a mistake, how-
ever, to conclude that this dichotomy is inevitable
or even desirable. It would likewise be false to conclude that scientists and mathematicians are leftand never the brains shall meet. Creativity, the production of fresh insights, the recognition of unseen patterns, is the forte of the right hemisphere and the visual artists, since the right is geometrically and visually sophisticated as well as intuitive, whereas literary artists—poets and novelists—must use the left hemisphere since the right hemisphere cannot speak or write. The classic example of a fusion of the left and right functions may, perhaps, be seen in Leonardo da Vinci, who was as brilliant an artist
as he was a technologist and investigator of the natural world (Aaron and Clouse, 1982).
Mathematics and science are also the result of a fruitful interaction between the two modes of thought. The great scientific discoveries were all made through imaginative leaps by the right hemisphere, which is so much better at perceiving the whole and sensing patterns than the left. ‘Discovery,” said Szent-Gyé6rgi, “consists of seeing what everyone else has seen, and thinking what no one
else has thought” (quoted in Pottage, 1983, p. 71). But these patterns must be subjected to the critical and skeptical faculties of the left hemisphere, for there is no other way to determine if the right has found a real pattern, or simply imagined one. Newton’s achievement in synthesizing the calculus and the laws of motion was intuited by the right hemisphere, but the arduous effort he exerted in veri-
fying and refining them were labors of the left hemisphere. Sparked as they may have been by the right hemisphere, the development of science and mathemat-
Against these differences, we should note that
They tend to be introverted, dominant, radical, self-
sufficient, and emotionally sensitive. They also tend to score highly on tests of neurosis but have high ego strengths, that is, they can control their neurotic urges. Other common traits are divergent thinking, intelligence, and preference for complexity (Beloff, 1970; Albert, 1983; Nicholls, 1972). The most intriguing difference between scientists and architects is that the former score highly on tests of masculinity, while the latter score much
closer to
the feminine end of the scale (Broadbent, 1973).
1-2-2
The Content of Our Culture
Let us try to get a feel for the magnitude of the various components of our intellectual life that exist in our society.” One very broad indicator is the number of works translated each year, divided by Dewey or UDC classification (UNESCO, 1985) (Figure 1-2). Literature is by far the largest section, accounting
for 46 percent of all translated books. Then the social sciences, technology, and natural sciences. An-
other indicator is the number of prominent individ? The reliability and validity of the data I use here might be, of
course, open to question. However, it is quite an impression of the state of affairs, which is do. The concepts are inherently fuzzy, and any to invoke could be challenged on the basis of
adequate to convey all one can hope to authority one cared some bias or other.
10
THE REASONING ARCHITECT
General
Table
Philosophy
1-1
Production of Books and Graduates in
the USA and UK, 1981, and Periodicals in the USA, 1980
Geography and history
Field
Books
Architecture Arts
Gy
Literature
Natural sciences
|
and technology Social sciences
Technology
Mathematics and computing
Arts Figure
1-2
Translations by UDC class for the world,
(Source: UNESCO, 1985)
uals who
1980.
Source:
have contributed
to modern
thought
(Bullock and Woodings, 1983), shown in Figure 1-
3. By far the greatest number are writers, followed
by natural scientists. Architects hold a respectable place, accounting for about 2.5 percent of the total, about equal in number to influential psychologists or technologists, and a
Humanities Other
little less than the number
of influential literary critics or anthropologists.
Since the British tradition assigns a rather dif-
ferent place to architects than other societies, we will now confine ourselves to the English-speaking nations, in particular the two major Anglophone
nations. Table 1-1 and Figure 1-4 show data for the
UNESCO,
1.1% 8.5 23.3 16.0 3.2
37.9 10.0
Periodicals
Graduates
0.1% 34 30.6 93 0.2
40.2 16.3
0.8% 3.3 23.7 29.8 2.8
18.4 21.1
1985.
production of graduates, books, and periodicals in the United States and the United Kingdom.
Just over half (54 percent) of all graduates have
degrees in the sciences or technology, and about
half as many have humanities degrees. Against this mass, architecture is very tiny indeed (0.8 percent),
and it is even swamped puterists. The annual graduates is about the the service trades. This
by mathematicians and comproduction of architecture same as domestic science or could be interpreted in many
ways. We may not need many architects, or we may not want many architects. Society does seem to need
alot more scientists, and a lot more humanists, than Figure
1-3
Prominent contributors to modern thought.
(Source: Bullock and Woodings,
1983)
Architecture
Literature
Arts
architects. Perhaps we get value for money from our architects, and hence require only a few of them,
or perhaps architects are just not very important, or perhaps they are too expensive, or society’s needs may be met by a very small number.
Lasting information is transmitted through the printed word. Even architects communicate through print, although the paper may hold more graphic material than text. The quantity of printed material
existing in a discipline should be a reasonable, if very rough, indicator of the amount of activity in that discipline. In both books and periodicals the humanities lead the way. This is as it should be, since the end product of scientists and technolo-
Humanities
gists is often an artifact, of architects, buildings, and Natural sciences
of artists, artworks,
but the end product of a hu-
THE
REASONING
ARCHITECT:
manist is usually a written document itself. The number of architecture books published annually is a little higher than the number of folklore and ethnography books, or books on military matters. If we compare books and periodicals across disciplines,
architecture,
the
arts,
and
mathematics/
computing show a heavy emphasis on books over
MATHEMATICS,
SCIENCE, AND
ART
IN ARCHITECTURE
]]
recent,’ about 200 years old, but as long as we do not take the boundary as hard-and-fast, and admit
into each parts of the other, they are useful distinctions if only because scientists and artists do see themselves as carrying out quite different sorts of activities.
periodicals. The sciences, as we shall see later, rely
Though they may be different, it does not necessarily lead to the conclusion that they are op-
munication and information transmission. The arts
or pursuit. Here, for example, the astronomer Carl
crucially on periodicals as a means of regular com-
do not seem to have a need for such a communication system, nor for a method of rapidly dissem-
inating information. Neither, it would seem, do the
fields of mathematics and computing. What emerges from these statistics is the marginality of architecture in the intellectual life of society. Notwithstanding the obvious importance of buildings, it does not seem that many people are
concerned with problems of their design and history, in the way that a vast number of people are concerned with the problems of the humanities or the sciences. Likewise, for that matter, mathemat-
ics does not figure prominently in the life of intellectuals.
posed. The two can be unified in the one individual
Sagan discusses why a
record containing sounds
and pictures of earth was launched with the spacecraft Voyager II in 1977, and in so doing draws from
the humanist achievement:
culture
to illuminate
a scientific
Why? Because the future would be very different from the present. Because those in the future
would want to know about our time, as we are curious about our antecedents’.
Because there
was something graceful and very human in the gesture, hands across the centuries, an embrace
of our descendants and our posterity. There have been many time capsules.... Esarhaddon, son of Sennacherib, was a mighty gen-
Science and Art in Society
eral and an able administrator, but he also had a
The proper place of science, mathematics, and reason in our society has concerned thinkers since the
itary glory but his entire civilization to the future, burying cuneiform inscriptions in the foundation stones of monuments and other buildings. Esarhaddon was king of Assyria, Babylonia and Egypt. His military campaigns extended from the mountains of Armenia to the deserts of Arabia.
1-2-3
fifth century B.c. The place of art, too, has been one
of the perennial questions. Since the industrial revolution the question of the former has received more attention, as science has grown into an enormous social institution, commanding great resources, in a way that art has not and does not. Art as commodity may be common enough, and artists and their critics plentiful, but not in the way that science absorbs tremendous quantities of resources in society. For each art gallery displaying the works of hopeful young painters, there are a dozen, a hundred, firms and institutions packed with scientists. When Bismarck said that politics was an art, not
a science, he made a common distinction between the two. A distinction is often made also between
art and craft. These dichotomies are in fact quite
conscious interest in presenting not just his mil-
For all that, his name is hardly a household word
today, but his works have made a significant contribution to our knowledge of the Middle East in the seventh century B.c. His son and successor, > The Greeks distinguished between areas in which we could have true knowledge (episteme), and those which were mere collections of techniques or crafts (techne). The first does not really match any modern area of endeavor, and the second is roughly what we would mean by “art and craft.” Our modern terms “‘science” and ‘art’ (in the most common sense of “fine art’) reached their present meanings only during the Enlightenment.
Natural sciences Figure
1-4
Production of books and graduates in the United States and United Kingdom,
United States, 1980. (Source: UNESCO,
1985)
1981, and periodicals in the
THE
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Assurbanipal, perhaps influenced by the timecapsule tradition of his father, accumulated a mas-
sive library on stone tablets comprising the knowledge of all that was known in that remote
epoch. The remains of Assurbanipal’s library are a remarkable resource for scholars of today.... For those who have done something they consider worthwhile, communication to the future is an almost irresistible temptation, and it has
been attempted in virtually every human culture. In the best of cases it is an optimistic and farseeing act; it expresses great hope for the future; it time-binds the human community; it gives us a perspective on the significance of our own ac-
MATHEMATICS,
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IN ARCHITECTURE
]3
impact of Snow's lecture would probably have faded in time. What kept it alive, and placed it permanently on the agenda of western intellectuals, was
the cranky reply of the famous English literary critic F. R. Leavis (1963). Leavis was a critic in the es-
teemed British tradition of radical conservatives,
such as Jonathan Swift, William Blake, and John
Milton, and he yearned passionately for a return to what he took to be the golden age of sixteenthcentury England. He seemed to confirm Snow’s worst fears of the humanist who combined ignorance with hostility to science. The differences between the two cultures have
tions at this moment in the long historical jour-
been institutionalized in universities for over a century. The conceptions that each culture has of their
Billions of years from now our sun, then a dis-
mutually unintelligible jargon and unrelated met-
ney of our species.
tended red giant star, will have reduced Earth to a charred cinder. But the Voyager record will still
be largely intact, in some other remote region of the Milky Way galaxy, preserving a murmur of an ancient civilization that once flourished—perhaps before moving on to greater deeds and other worlds—on the distant planet Earth. (Sagan,
1978, pp. 3-4, 42)
It is, I believe, enormously harmful to our soci-
ety that the two have indeed come to be seen as antagonistic. In 1959 a man who was both a scien-
tist and a novelist gave a lecture at Cambridge University that crystallized the disquiet felt in modern
times about this issue. In the lecture and book The
Two Cultures (1963), Charles Percy Snow argued that
the intellectual leadership of the West, the elites re-
sponsible for the important social and political decisions in our society, had divided into two camps,
one of scientists and one of humanists (and artists). The educations of the two were so radically differ-
tasks, their place in society, their divergent values,
aphors, symbols, and analogies have created a tension between them, conflicts of values, misunder-
standings, and incomprehensions (Olson, 1982). Each culture has had its eloquent and erudite proponents. Snow himself had no doubt that science was the proper basis for all our culture. He was disheartened to think that scientists had been forced out of the mainstream of our present humanist culture, shut up in academe and industry, leaving the rest of us bumbling along as best we could (Barnes,
1985). He felt that scientists and engineers should
take over from the literati, to create a wonderfully
rational and benign technocracy in which all we had to do was be happy. In response to this somewhat chilling idea, critics such as Herbert Marcuse (1964), Jurgen Habermas (1971), and Theodore Roszak (1976) expressed their fears that science threatened to destroy all other
competing value systems and establish a monolithic totalitarian society that would dominate individuals. Political decision making, they held, had already
ent that they shared almost no basic goals and val-
been removed from the people and given to tech-
values in the face of competition from the Commu-
and knowledge to make decisions about the course
ues. Snow was concerned about the loss of western
nist nations. He feared that an overemphasis on values of the humanist culture could inhibit the full potential of science that the West had to harness if
it was to ensure the victory of the “free” world. The
nocrats, who were the only ones with the expertise
of society. The great mass of people could not take part in decision making because they were ignorant. The technocrats, though knowledgeable, have
a grievous fault in the way they approach decision
14
THE REASONING ARCHITECT
making, for they transform issues that are rightly social and political into mere technical problems. Thus, for example, they are more interested in find-
ing the most economical way to generate more electricity, or the most efficient method of making nuclear power safe, than in asking whether we need
more generation capacity, or whether there are so-
cial
considerations
that
outweigh
the
cost-
effectiveness of nuclear power. In vitro fertilization, an extremely expensive process that allows infertile affluent westerners to have children, could seem misplaced in a world where a major problem is over-
population. We will return to this critique of science in a later chapter. My own view is that there is little danger
little danger of scientific culture annihilating humanist culture. The table also brings out a distinction I will discuss later, that between high culture and populist culture (Gans, 1974). When we refer to Snow’s
two cultures, we really are talking about a very small proportion of humanity, the participants of high culture. Most people are interested in rather different
parts of our culture, the parts represented by Walt Disney, Barbara Cartland, and Enid Blyton.
That said, I believe that the greater menace lies in the chasm between the two cultures. There is also, I think, more bridging to be done from the humanist side than the scientific. Antiscience movements have a long history in the West, but there has never been an antiart movement or an
of one culture annihilating the other.* Table 1-2 shows the 20 most translated authors of 1980 (the
antihumanities movement. Those who reject or are
consists entirely of humanists and writers, except
ture. When people dislike art, they reject only one
positions of the Marxist authors are partially due to state-supported industries of translations). The list
hostile to science dislike the whole enterprise and wish to see it dismantled. It is impossible to find anyone who rejects the entire idea of art or litera-
for the science populist Isaac Asimov. There seems
style or school in favor of another. Quite often, those
hostile to science readily lend their support to in-
dividuals and movements that propose alternatives
to mainstream science. Scientists who dislike the humanist culture, on the other hand, do not sup‘ There was and is considerable disagreement as to which of the cultures was under threat, depending on which was perceived as containing the essence of our civilization. Sir Kenneth Clark’s landmark television series Civilisation, for example, was entirely a history of art. This so incensed Jacob Bronowski that it inspired him to produce the equally seminal series The Ascent
of Man, as a history of science.
port countermovements to literature or art (McCauley, 1978). Even
the violent cultural disturbances this
century in Germany, Russia, and China did not reject
art per se, although the new regimes attempted to suppress particular schools and styles. What, exactly, would such a countermovement look like, anyway?
While the artifacts of science, the products of tech-
Table
1-2.
The Twenty Most Translated Authors in
the World in 1980 1. V. I. Lenin
11. L. 1 Brezhnev
4. A. Christie 5. J. Verne 6. E. Blyton
14. J. London 15. H. C. Andersen 16. M. Twain
2. Walt Disney Productions 3. The Bible
7. K. Marx 8. B. Cartland
17. F. M. Dostoevsky 18. I. Asimov
9. F. Engels
19, G. Simenon
10. W. Shakespeare Source:
UNESCO,
12. J. Grimm 13. R. Goscinny
20. L. N. Tolstoy 1985.
nology, are everywhere about us, there is little un-
derstanding of the nature and processes of science. A look through any quality newspaper will find critical articles on literature, theater and cinema, his-
tory, and biography. By its nature, professional criticism reveals the process of humanist culture in a way that a report of a medical breakthrough does not reveal the process of science. Humanist culture has developed an entire body of individuals, crit-
ics, whose task is to mediate between the produc-
ers and consumers of humanist culture. We all assume that we have the right to comment on the humanist culture in a way we do not presume for
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science. “I don’t know anything about music, but I know what I like,” said a character in Max Beerbohm’s novel Zuleika Dobson, and the phrase has
passed into common use. Science lacks these me-
diators and is thereby rendered less accessible. Intelligent members of the humanist culture will in general know much less about science than members of the scientific culture will know about humanist pursuits.
The difficulty of science may in part be due to
the right-left brain lateralism. The left brain, it would seem, is good for science and mathematics only for
a few years; the right brain can handle the arts and humanities until senility. Once the scientific boat is gone, it is gone forever, but the humanist ship can be boarded at any time. A clear example is the tendency for scientists to drift into the humanities, particularly philosophy, as they age. In the field of architectural science, the archetypal example is Henry Cowan. Since virtually inventing the field in the 1950s, he has devoted more and more of his time to
the history of his discipline (such as Cowan, 1977). Typical examples in the physical sciences are Ziman (1984) and Medawar (1984).
A second factor rendering science difficult to come to terms with is its accumulative nature. One mathematical proof has been in the making for 40 years and now runs to 15,000 pages, representing
the combined efforts of over 100 mathematicians
from across the world (Gorenstein, 1985). Science
is built piece by piece, and for this to happen with-
MATHEMATICS,
SCIENCE, AND
ART IN ARCHITECTURE
]5
has an infinity of unknown worlds. Had Coperni-
cus
and
Kepler
never
lived,
then
someone
else
would have formulated the heliocentric hypothesis of the solar system and the laws of planetary motion, simply because these are facts of the universe waiting to be found. Perhaps much later, but it would have happened. Had Bach never lived, or Beethoven, music would be a
different thing, and
there is no assurance that it would be anything like
the music we have today (Deutsch, 1958). This is
not to deny that artists are embedded in their own
times, nor that these times constrain in certain ways
what artists can do. Yet Kepler was constrained in a different way, in that he could not possibly have (correctly) found that the planets move in circles. Fourth, much of science, and especially mathe-
matics, is counterintuitive. The sciences with the longest modern history, and that are also the most developed, are physics and chemistry. Beyond an elementary level, their theories are inexplicable with-
out sophisticated mathematics. Even simple theories of physics, such as the laws of motion, are directly contrary to what common experience would suggest to be the case, as we shall see later. For these reasons, understanding the sciences is harder than understanding the humanities and arts,
and crossing from the humanities to the sciences is more arduous and less often undertaken than the return journey.
out wasted effort, scientists must have efficient ways
1-2-4 Science and Art in Architecture
task, must talk to each other. A sophisticated com-
In no other discipline is the tension between the
ferences and meetings has therefore developed at
relationship between science and architecture is a complex one that we shall explore in the rest of this book. A good case could be made that science and technology have been the most important influences
of communicating. Scientists, by the nature of their munication system of scholarly journals and con-
the center of the social institution of science. It is scarcely possible to do science without being part of this system. Novelists or painters, though, could easily carry out their work in grand isolation, and many have done so in the past and do so now (Price,
two cultures more evident than in architecture. The
on architecture since at least 1750 and that the his-
tory of architecture since that time should be read
1961).
as its attempts to understand the new technologies
that the world of the humanities is not. A scientist
revolution. The split between the two cultures has been in-
Third, the world of science is bounded in a way
has but one world to discover; an author or artist
and social institutions generated by the industrial
16
THE REASONING ARCHITECT
stitutionalized in architecture, and for 200 years the
two strands of culture have struggled to fashion architecture in their own image. The scientific strand was the orthodoxy in the second and third quarters
of this century, embodied at first in the Modern Movement,
and in what is now known
as the In-
ternational Style. The aesthetic they fashioned was
called ‘‘functionalism,” and it held that buildings
should be the simplest, cheapest, and most direct
answer to a design problem (Broadbent, 1979). In
its milder form, functionalism held that buildings
should in some way explain themselves, that they should make in their design a rational statement
about their function and construction. In its stronger form,
it maintained
that functional consider-
ations could define the form of the building in a necessary and deterministic way.
The result was being criticized extensively by the 1970s, the most damning charge being that it had
“led to the wholesale destruction of small-scale, hu-
mane environments and their replacement by bleak, windswept piazzas and urban motorways, lined with
the
dimmest,
greyest,
and
most
soul-
destroying of high-rise apartment slabs” (Broad-
bent, 1979, p. 103). Its practitioners were not from the scientific culture, and they took its icons without understanding its values and nature, in a sort
of Cargo Cult mentality that the benefits of science that they saw about them could be imported into architecture simply by adopting the images of sci-
tules, into a tool of an exclusively technological character. Its main concern becomes how to build in an efficient and economical manner, while avoiding questions as to why one builds and
whether such activity is justified in an existential context... Because positivistic thought has made it a point to exclude mystery and poetry, contemporary man lives with the illusion of the infinite
power of reason. He has forgotten his fragility and his capacity for wonder, generally assuming that all the phenomena of his world, from water
to fire to perception or human behavior, have been “explained.” For many architects, myth and poetry
are generally
considered
synonymous
with dreams and lunacy, while reality is deemed equivalent to prosaic scientific theories. In other words, mathematical logic has been substituted
for metaphor as a model of thought. (PérezGomez, 1983, pp. 3-6)
Such criticism, while it is important in keeping the role of science on our debating agenda, is enormously dangerous,
I believe, because it builds an
iron curtain between the two cultures and because
it rests on fundamental misunderstandings about science. The contempt it encourages of science can only be harmful, for the reasons given by Jacob Bronowski:
ence. The criticism has since expanded into a wide-
The scholar who dismisses science may speak in
such as Joseph Rykwert (19822) and Alberto Pérez-
To think of science as a set of special tricks, to
ranging critique of science in architecture by critics Gomez (1983). Like Marcuse and Roszak they see science as having turned the crucial problems of hu-
man existence into illegitimate questions and of replacing a world rich in meaning and value by a dessicated husk:
[The] malaise from which architecture suffers to-
fun, but his fun is not quite a laughing matter.
see the scientist as the manipulator of outlandish skills—this is the root of the poison that flourishes in the comic strips. There is no more threatening and no more degrading doctrine than the
fancy that somehow we may shelve the respon-
sibility for making the decisions of our society by
passing it to a few scientists armoured with a spe-
day can be traced to the collusion between ar-
cial magic. This is another dream. ..in which the
as it developed in the early modern period.... This functionalisation of architectural theory implies its transformation into a set of operational
manity which has no business except to be happy....But in fact it is the picture of a slave society, and should make us shiver whenever we
chitecture and its use of geometry and number
engineers rule, with perfect benevolence, a hu-
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MATHEMATICS,
SCIENCE,
AND
ART
IN ARCHITECTURE
]7
hear a man of sensibility dismiss science as some-
one else’s concern. The world today is made, it
is powered by science; and for any man to abdicate an interest in science is to walk with open eyes towards slavery. (Bronowski, 1962, p. 16)
The fundamental fallacy committed by such critics, as the British architect Sir Leslie Martin (Martin and March, 1972) saw, is in regarding
creativity and reasoning as two watertight compartments of the human
intellectual makeup.
Since architecture is clearly a creative activity, it follows that architecture cannot be about reasoning, and from this it is a straightforward step to conclude that it must not be about reasoning. The critique perpetuates the wholly wrong idea that
creativity in architecture is the domain of design and design alone and that all the other components of architectural knowledge are just so many dry facts that are sometimes handy to the architect but preferably left to a consultant. The result
of such attitudes, among other consequences, is
that architects are doing less and less in the construction process, as the masters of all these dry facts chip away slowly but steadily at the architects’ role.
Figure 1-5 The Vitruvian architect in the late twentieth century. While architects have often shown unbounding enthusiasm for mathematical systems of proportioning. they become quite hostile when mathematics is used for
nonaesthetic purposes in architecture.
Modern architecture’s attitude to mathemat-
ics reflects the tensions of the art-science divide. On the one hand, architects have been enraptured by mathematical systems of proportioning, as the reception given to Le Corbusier’s Modulor shows. Such systems are often adopted with un-
critical enthusiasm, as though the secrets of the
universe have been revealed. On the other hand,
the intrusion of mathematics into nonaesthetic
areas of architecture is greeted with horror and derision (Figure 1-5). Philip Steadman, author of Architectural Morphology, to which this book is indebted, recounted this story:
In 1975 William Mitchell, Robin Liggett and I de-
veloped a computer program which generated architectural plans of a certain type automatically. ... These were plans of rectangular rooms, set together to form arrangements with a rectangular
shape overall—the sorts of plans typical of many small houses and flats....This work provoked
some strange reactions. Mitchell outlined the sys-
tem to an architect acquaintance in Los Angeles, and was told flatly ‘That's impossible.” Later, Mitchell and I submitted a paper describing the
work to the British Architects’ Journal. The article
was refused by the then editor in a letter of scarcely concealed hysteria: ‘This work is strictly non-architectural, . . .it has nothing to do with architecture.” (Steadman, 1983, p. 1)
]8
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Architecture has confined mathematics to one
ficed an ox to the gods in joy when he discovered
tenance its use in others. Why should this be so? I
A painting by Leonardo, a statue by Rodin, a sym-
particular role, the aesthetic, and refused to coun-
believe that this is the wrong question, and that what needs to be explained is why mathematics is present at all in architectural thought. The answer has been presented concisely by Hillier (1985b). The central problem of architectural theory is to determine the abstract principles underlying architectural form.
Once discovered,
it is believed, these prin-
ciples will then enable architects to design good architecture. Since at least Greek times, it has seemed
this theorem.
phony by Mozart—all are flawed because they rely on the experience of the senses for their appreciation. But a mathematical proof is a creation of the intellect pure and simple, and appreciated by the
mind directly, freed of the petty shackles hampering our five puny senses. The mathematician Henri Poincare spoke thus of mathematics and science:
self-evident that the fundamental principles of architectural form must be mathematical. Sometimes the principles have been numerical ones, sometimes geometrical. The former assert that order in architecture arises from the regularities in which numerical ratios can be combined. They lead to propor-
If nature were not beautiful, it would not be worth knowing, and if nature were not worth
that they provide ways of generating forms. The latter assert that architecture must emulate the un-
I mean that profounder beauty which comes from the harmonious order of the parts and which a pure intelligence can grasp. This it is which gives body, a structure so to speak, to the iridescent appearances which flatter our senses, and without this support the beauty of these fugitive dreams would only be imperfect, because it would be vague and always fleeting. On the con-
tional and modular systems which are synthetic in
derlying geometrical order of nature, and tend to
produce schemes for the analysis of finished forms.
Mathematics has always seemed the only hope for
constructing a theory of order.
1-3 THE LIFE OF MATHEMATICS 1-3-1
Mathematics as Art
If there is any subject that could be labeled “dry,” itis high school mathematics. It is presented in text-
books with all the excitement of a dramatization of the phone book. This is not how mathematicians see it. To most, mathematics is an aesthetic discipline, and mathematicians are the discoverers of the
only things that can truly be called beautiful. Take
the theorem of Pythagoras: The sum of the squares on the sides of a right-angled triangle is equal to the square on the hypotenuse. What an extraordinary thing to discover about a triangle! Who could have
suspected that such a beautifully simple relationship should hold between the three sides? It is certainly not obvious. It is said that Pythagoras sacri-
knowing, life would not be worth living. Of course I do not here speak of that beauty which strikes the senses, the beauty of qualities and of appearance; not that I undervalue such beauty, far from it, but it has nothing to do with science;
trary, intellectual beauty is sufficient unto itself,
and it is for its sake, more perhaps than the future good of humanity, that the scientist devotes
himself to long and difficult labours. (Poincare,
1958, p. 8)
The pure mathematician regards his work as a
form of avant-garde art. Of what use is it? None at
all! Does anyone ask of what use is the Mona Lisa, or of what use is Michelangelo’s David? Here the mathematician Godfrey Hardy speaks: Thave never done anything “‘useful.”” No discov-
ery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least dif-
ference to the amenity of the world. Judged by
all practical standards, the value of my mathematical life is nil; and outside mathematics it is
THE
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trivial anyhow. I have just one chance of escap-
MATHEMATICS,
SCIENCE, AND
ART
IN ARCHITECTURE
]9
sary groundwork for more advanced (and interest-
ing a verdict of complete triviality, that 1 may be
ing) work in mathematics. Perhaps 20 of the high
ing. And that I have created something is un-
without which the rest of mathematics is largely un-
judged to have created something worth creatdeniable: the question is about its value.
school topics could be considered an essential core,
value which differs in degree only, and not in
intelligible. The remainder were selected largely to direct students along one specific path in mathematical thought, that leading to a working knowledge of calculus. Calculus is of immense importance in all technical fields—except architecture. Within
ematicians, or of any of the other artists, great or
cept to establish some results in the environmental
The case for my life, then, is this: That I have
added something to knowledge, and helped others to add more,
and that these things have a
kind, from that of the creations of the great math-
small, who have left some kind of memorial be-
hind them. (Hardy, 1969, pp. 150-151) 1-3-2
The Discipline of Mathematics
We saw earlier that the production of mathematical work is only a small component of the total production of intellectual material of our culture. One guess puts the total mass of mathematics at about
100,000 fat volumes, far smaller than the content of
many intellectual fields (Davis and Hersh, 1981). Most of this has been produced in the past generation. When some of the older mathematicians now working were very young, it was thought that all of mathematics was within the grasp of a single individual. This is not possible today. Mathematics differs from many other fields in the great degree of serial dependence involved. To understand calculus, you have to understand coordinate geometry, and to understand that, you must know some algebra and so on back to number theory. It is quite impossible to grasp calculus oth-
our own discipline there is little use for calculus exsciences. There are indeed vast areas of mathemat-
ics of immediate use in architecture, alas, very few
of which you have studied in high school. Figure 1-6 is my impression of the structure of mathematics. The four central disciplines are analysis, topology, number theory, and algebra. Analysis includes calculus and can be thought of as the study of infinite processes. Topology started life as a generalization of geometry and is concerned with the relationships between spaces as they form connected areas. It is concerned with the continuity,
connectedness, and adjacency of spaces, regardless of their shape and size. Algebra is the study of vari-
Figure 1-6
My impression of the structure of mathematics.
Analysis
erwise. This is quite different from architecture, say,
where you can study modern architecture in depth
while knowing very little about baroque or classical
buildings. The student may not have a full appre-
ciation of modern style, but his ignorance of pre-
ceding history does not render his studies impossible. This serial dependence
is one reason why
mathematics is hard, and one reason why it is offered as an advanced subject in high school. In order to get anywhere useful, students have to undertake a long apprenticeship. Many of the things you studied were the neces-
Algorithmics Operations research
20
THE REASONING ARCHITECT
ables and their permissible manipulation. Hover-
ing around these are geometry, combinatorics (how things may combine with each other), algorithmics
(the creation of procedures for carrying out math-
ematical tasks), and statistics and operations research (a way of looking at certain sorts of prob-
lems). Gluing all these together is the theory of mathematical logic. Figure 1-7 shows roughly the sort of mathemat-
ics required in a conventional technical discipline
like engineering. Your high school course was in-
tended to lay the foundations for just this sort of discipline. The main thing an engineer needs to know about is great wads of calculus, since it is the main tool involved in the analysis of engineering
structures. He or she will also need some topology for the analysis of frames and some algebra for mak-
ing the theory easier. Most engineering structures
are analyzed with computers these days, so algorithmics is needed.
Figure 1-8 shows the sort of mathematics used in
Algorithmics Operations research
Figure 1-8 My impression of the mathematics used in architecture.
to engineering,
ory is concerned with ways of deriving combinations of things. Odd as it may sound, combinato-
theory of combinatorics plays a large role. This the-
of some of the greatest architects of the past, among
architectural
studies.
Compared
very little calculus is used. On the other hand, the
Figure 1-7. My impression of the mathematics used in engineering.
rics helps in understanding the architectural style other things. Geometry is, of course, of great importance in the design of buildings, and in topology also.
Mathematics is often divided into two parts, pure and applied. Pure mathematics is a grand game, played for the joy of it all. As Hardy said, a pure
mathematician has no interest at all in potential ap-
plications. An example of mathematics at its purest is provided by a very simple problem, known as Fermat's last theorem after its creator, Pierre de Fermat (Delong, 1971; Stephens, 1984). It is a problem
in number theory, which deals with the properties of integers and things like primes and perfect numbers. This theory was, like many other things, started by the Greeks, in particular by a mathematician called Diophantus. Fermat owned a copy of
his works, and in the margin of one page jotted this Operations research
down:
It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in gen-
THE
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eral any power higher than the second into powers of like degree. In modern notation what Fermat is saying is this:
We all know about Pythagorean triples, which are triplets of integers which obey the Pythagorean formula:
Caeth One such triple is 3, 4, 5. There is an infinite num-
ber of such triples. Fermat now considers the gen-
MATHEMATICS,
SCIENCE, AND
ART
IN ARCHITECTURE
2]
simple to state, it has turned out to be fiendishly difficult to solve. Its simplicity has attracted crackpots for hundreds of years, and most universities receive a few letters each year from people who claim to have solved it. Such are the amusements of pure mathematicians. Applied mathematicians have quite a differ-
ent job. Their role is to discover solutions to real-
life problems from the great mass of existing mathematics. From the many games played by their purist counterparts, applied mathematicians must
where a, b, c, and n are all integers. He says that there are no solutions at all for this equation when
find some games that correspond to bits of reality and use them to solve problems in those bits. This can be a very difficult and subtle job. My newsdealer sells a notepad for $1.60. How much do two notepads sell for? $3.20? No. At my newsdealer you can get two for $3.00, so that’s the
For example, there are no integers that satisfy c = a> + Bb. Here n has the value of 3. In spite of its name, this is certainly not a theorem because Fermat does not supply a proof. It is really a con-
dealer has notepads reply that notepads
but the name of the problem has been hallowed by history. The quality that has tantalized mathematicians so is that Fermat also wrote in the margin:
happens that in this case those rules do not apply. If the newsdealer chooses not to use the rules of
eral formula:
a+ B=" n > 2, apart from the uninteresting triplet 1, 0, 1.
right answer. But some people will say the news-
jecture. Further, it wasn’t Fermat's last anything,
of real arithmetic to calculate your answer, but it so
Thave discovered a truly remarkable proof which this margin is too small to contain.
If he did have a proof, he never wrote it down. The galling thing is that to this day we do not know whether the theorem is true or not. Although 300 years have elapsed since Fermat wrote this theo-
rem, no one has ever been able to discover a proof, and on the other hand, no one has been able to
applied a discount for quantity, should really sell for $3.20. To it is quite impossible for you to should sell for. You have used
and two which I say what the rules
real arithmetic, that is his business, not yours.
To give another example. Let us say the insurance value of the Mona Lisa is $10 million. How much would two Mona Lisas be worth? In this case, only
a complete lamebrain would insist that the answer
was $20 million. The point to this is that even in
quite simple cases, blind application of bits of math-
ematics to real-world problems can be utterly mis-
leading in their results. The major concern of applied mathematicians is in ensuring that the bits of maths they use really do match the problem at hand. We
will look at this task, which is called ‘“model
matter whatsoever. It is a concern for pure math-
building,” in a later chapter. Another job for applied mathematicians is to find efficient ways of solving problems (Kolata, 1974). There are some problems for which we have efficient, fast methods of solution. We can call them
for all values of n less than 125,000, but there is still
vided into two sorts, depending on how many op-
disprove it.
Fermat's last theorem is the grand unsolved prob-
lem of pure mathematics. Its solution would, as far
as anyone can tell, have no effect on any practical ematicians alone. Partial proofs have been found
no proof that the theorem is true for all n. Although
“easy” problems. Solution procedures can be di-
erations are needed to solve them. Naturally, the
22
THE REASONING ARCHITECT
number of operations depends on the size of the problem,
and
the
more
operations
needed,
the
longer the time to solve the problem. For easy problems, the number of operations required is a poly-
nomial function of the size. For example, if the problem is of size n, it may require n* operations to solve. An example is that of constructing colored computer graphics images. The time required of the computer
to produce a realistic (but not too fancy) image is proportional to the number of faces on the objects
to be display. There are some problems for which no fast method of solution is known. These we may refer to as “hard” problems. A famous example is the problem called ‘‘the traveling salesperson’s problem.” A salesperson must visit every city and town in his territory. He wants to do this as speedily as possible, so he wants to minimize the total distance he has to travel. What is the best way to do this? Another example is what Mitchell (1977) calls ‘the
Palladian problem.” We start with a dimensionless
plan (Figure 1-9). That is, the location of each room
and wall is decided but not its dimensions. Given that the proportions of each room are decided upon, there will be many possible plans that satisfy the proportioning requirements. Listing all the possi-
ble plans is a hard problem. In such a problem, the number of operations is an exponential function of its size, requiring, for example, 2” operations to solve a problem of size n. Now let’s consider a procedure of each sort, of size 500. The first sort of procedure could be done in 500? ~ 10° operations. Modern computers -ap-
proach the ability to do 10° operations per second and could therefore carry out the solution procedure in 1 ten-thousandth of a second. A procedure of the second sort would require 2° ~ 10'™ op-
erations. This could not be done ina reasonable time
by a modern computer, so let us create a supercomputer to tackle it. We will make this computer very large. In fact, we will make it as large as the entire universe. There
are perhaps 10® protons and neutrons in the uni-
verse. Let us construct our computer so that it can
carry out an operation with but a single proton or
neutron. It can therefore carry out 10” operations at once. We will also make this computer extremely fast. The smallest unit of time is about 10 ~ “s, below which the laws of quantum mechanics break down. It is much smaller than the time a photon would take to cross a distance equal to the size of a proton. Call this a chronon. We will have our com-
puter accomplish each operation in a chronon. It
can therefore carry out 10° operations each chroFigure
1-9
A dimensionless plan.
non or 10 x 10*° = 10’? operations per second. The problem will require 10°10" = 107” s tocom-
plete its task. The universe is, by best guess, about 10°° years old, equal to 10's (Rowan-Robinson, 1985).
Had the computer started its calculations at the creation of the universe, it would have completed only 1 ten-billionth (1017/10?” = 10 ~ '°) of the operations required to solve it. What a difference!
1-3-3
Mathematical
Issues
Most of the difficulties arising from understanding mathematics come from its highly abstract nature,
its concern with generalization, and its insistence
on formalization. Much mathematics starts from an examination of practical problems, quickly becomes
dissociated from the original context, and is elab-
THE
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ARCHITECT:
’ orated into an intricate theory barely betraying its origins. Geometry starts, perhaps, from pondering a stretched string. This is abstracted into the notion of a line without thickness, not something one is likely to bump into in the real world. Everyone knows what a circle is, but how often are perfect circles drawn? Any image of a circle will have minute imperfections.5 This abstracting of qualities is one reason why mathematics has proved of such tremendous use in the world. The further and further removed it becomes from concrete considerations, the more powerful it becomes. It allows us to concentrate on one particular aspect of reality, to extract this characteristic of interest and examine it closely, to pull and prod it, to develop a general theory and to see if it has applications elsewhere.
MATHEMATICS,
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ART
IN ARCHITECTURE
23
Here, for example, is the plan of the very first
MacDonald’s, in Fresno (Figure 1-10). Consider the
relationships between the rooms. To describe these relationships, and these relationships only, the plan contains an overabundance of information. We could abstract out the information by drawing a diagram in which each room was represented by a dot, and the dots connected if there was a door be-
tween the corresponding rooms (Figure 1-11). Such a diagram is called a graph. Each dot is called a vertex, and each connecting line an edge. Gone is all the geometric information. The sizes and shapes of each room are immaterial if we are simply concerned with the relationships between rooms. Now a more complex example (see Hillier, Hanson,
and Graham,
1987, for the complete dis-
cussion). Figure 1-12 shows two houses from rural
France. On the right of each is its graph, drawn in
such a way that the outside, symbolized here by a circle with a 5A story is told about the father of modern painting, Giotto. Around 1300 the pope wanted some frescoes done, and was looking for the right man. Giotto was asked to provide a sample of his work for viewing. With a single stroke of his brush, it is said, he drew a
perfect circle, which was duly presented to the
pope. Giotto got the job. Figure 1-10
cross in it, is on the bottom, and the
rooms arranged in order of the number of spaces or separating rooms it is from outside. The plan of the top house (1) seems at first to be a simple linear layout. Its graph shows that there is in fact considerable complexity in the way that the spaces access
The MacDonald's restaurant in Fresno, 1955. Figure
1-11
A graph of the Fresno MacDonald's.
Workroom
Office
O Toilet Storage
O Preparation area
Serving area
Toilet
24
THE REASONING ARCHITECT
(= i |
Figure
1-12
—— +. br]
*
trl co
s ba =
led v2
a
ede
L AL
Two houses from France.
each other. The graph also reveals that there are
most stimulating examination of mathematics as a
one connecting v2, de, I, la, sc, and the outside; and one connecting sc, co, v1, and the outside. The
Broadbent (1973) has a good review of the research
vation room), | (dairy), la (washing room); the sec-
comprehensively in Kline (1972). For the history of ideas I suggest you start with Burke (1985) and
two major rings, or circuits, through the building—
first ring consists of work spaces: de (food preser-
ond consists of living spaces. The room sc (salle com-
mune) is the main living space, and it separates and connects the work from the living portions of the house. The second (lower) house shows a similar form.
The main living space, m (maison), lies on two rings,
one for working and one for living. Thus the graphs can be used to elucidate relationships that are not clear by simple inspection of the plans.
discipline is without doubt Davis and Hersh (1981).
on the psychology of architects. The history of mathematics is succinctly covered in Kline (1980) and
Boorstin (1984). Williams (1985) is excellent on the
history of changing concepts. The literature on art and science is large, but for a start try Alfert (1986), Ross
(1967),
Brown
(1958),
Cooper
(1980),
Kuhn
(1969), Malina (1974), Medawar and Shelley (1980), Meyer (1974), Moravzcsik (1974), Pollock (1981), and Root-Bernstein (1984b). Bailin (1988) is an interest-
ing introduction to creativity. D. K. Simonton has written extensively on creative productivity, much
READING
of
For culture, art, and science, the journals Leonardo,
lection of articles on all aspects of mathematics.
Daedalus, Minerva, and the Journal of the History of Ideas and Critical Inquiry are worth investigating. The
which
is
summarized
in
Simonton
(1984).
Campbell and Higgins (1984) is a very varied colHillier’s application of graphs to architecture is presented in Hillier and Hanson (1984).
Primitive Notions: Mathematics in Preclassical Civilization
2-1 2-1-1
BEFORE
HISTORY
The Prehistory of Culture
Our species is young, considering our time and place. Mammalian species since the demise of the dinosaurs have lasted about 1 to 2 million years. Genetic dating makes Homo sapiens only about 600,000 years old, so we
are in our adolescence
(Raup and Stanley, 1978). Toolmaking is the oldest
part of our cultural history (Figure 2-1). Other spe-
cies, notably our cousin primates, also use tools but only occasionally. Our ancestors the Australopithecines were the first to use tools as a matter of course about 2 million years ago, and they were the first to go out of their way to make tools, notably flint axe heads. Building,
too, is a most ancient craft. The
earliest evidence is a hut of mammoth bones around
a hearth found in Terra Amata in France, dated to
about 420,000 years ago (Kostoff, 1985). Aesthetic
concerns seem also present at very early stages in the history of our species. A hand axe formed
out of a fossil sea urchin has been dated to 200,000 years ago, although whether this reveals artistic pur-
pose is moot and could perhaps be equally taken to indicate an interest in paleontology. The first indis-
putable evidence of artistic concern, or of at least a desire for decoration, comes from an amulet, found
in Hungary, that is about 100,000 years old. At about the same time as the amulet was being crafted, some
people in Iraq were thinking enough about others to bury their dead. Another 43,000 years pass, and
we find bear skulls set in niches in caves throughout Austria and France, presumably indicating a cult of the cave bear, and the beginnings of religion. About 2000 years later a mutation occurs in a population of humans living on the Iranian plateau. This mutation effects a subtle change in the wiring of their brains, and language proper—the sentence, the song, the poem, the story—is born. The pace of change accelerates tenfold. These humans take to the sea, populate Europe, Africa, America, and Australia, overwhelming
the Neanderthals.
Tools im-
28
THE REASONING ARCHITECT
: Huts; France —
——
Amulet; Hungary
Burial. Iraq
==
100,000 —————__ 8
3
Th
=]
7
Herbal remedies; Iraq
I
50,000
Language proper; Iranian Plateau
42,000
44,000
40,000
|
28,000
20,000
9,000
8,000 |
—+~——————._
22,000
26,000
—+————_
24,000
| Floodwater gardening; Egypt
16,000 aa Agricultural clearance; Thailand
I
12,000 ————
Symbolic tokens, accountancy; Iran Domestication of wheat; Palestine
Urbanization; Towns, brick houses; Palestine Mesopotamia Current interglacial phase begins Z Irrigation. Mesopotamia Cities; Anatolia __
Metalworking; Anatolia
Figure 2-1
1984)
34,000 ceramics, sculpture; Czechoslovakia
Painting; 18,000 ———_,_.
10,000
36,000
Tailored clothing; Ukraine
) c =velocity of sound, m/s (LT~') 4n = constant (1)
Po = reference pressure of 2 x 107° pascals (ML 'T~?)
r = distance of the hearer from the source (meters) (L)
The dimensions of Lp can then be calculated as:
{L,] _ (ML?T~9) (ML~3][LT~1}
IML TP [Lp
[M2L?L3LT~3T-4]
[M?7L?L-2T—4] _ (MT~4]
~ [MT] = [1]
Decibels have the same dimensions as pure numbers. They are not units of something. Decibels are actually a measurement of ratios between sound pressures. The reference pressure is taken to be at the threshold of hearing.
42
PRIMITIVE
whee PeeD
50
lon
fifa
\
75
5]
my
I
150
Fine
ee
|!
|
eT ee Ht
1
!
|
|
t
=
'
TTT !
cas
wT
'
10 2
60
l
bacea baad ne
g
i! 1
100
300-mm module
'
at
ane
NOTIONS:
TTT '
aca)
10
250-mm module Figure 2-9 mm
Subunits of two possible modules, one 300
and one 250 mm
long, for the building industry.
MATHEMATICS
IN PRECLASSICAL
CIVILIZATION
43
nominator. The integers can then be seen as a subset of a much larger set of numbers that also includes fractions, the rationals, symbolized by Q (Figure 2-10). A rational can always be written as a decimal expansion, in one of two ways. The simpler sort is a terminating expansion, with n digits before the decimal point and m after it, like this: G = Ay Any
Ay Ay A_y A_7 A_3.. Ay
such as 694.76534, in which n = 3 and m = 2. The
The Babylonians were smarter than most in recognizing the deficiencies of the decimal system. They were even smarter in adopting 60 as their number base, thereby utilizing the nice properties of the number 12 while recognizing the biological fact of our 10 fingers, for 60 is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, 30. To this system we owe the fact that there are 60 s in a minute, 60 min in a degree, and 360 degrees in a circle, as well as 12 constellations
in the zodiac. They also employed a positional num-
ber system, something the Greeks or Romans never
did, in which they could write numbers with the
same ease that we can in the decimal system. That the Greeks could not was to influence their architecture, as we shall see later.
The Babylonians might have come across the fractions in a second way by discovering that adding, subtracting, and multiplying integers always gives
another integer, but division does not. This is a
purely mathematical reason, quite different to the everyday problem of sharing one amongst many. This way of thinking about the fractions is clearly shown in our notation for them, in which we write
2¥46 as just another way of saying 21 divided by 46.
This gives us two sorts of number,
fractions and
integers, with different methods of manipulation. Adding the fractions % and % requires a much more elaborate algorithm than adding the integers 23 and 57. Having two completely different sorts of numbers is most inconvenient. A more aesthetically pleasing situation is to look on the integers as special sorts of fractions: fractions with a 1 in the de-
other sort of rational does not terminate but contains a repeating block of digits, symbolized by the digits c, to c,,:
Z = Ay Agys+ Ay Ay Ay Ay A_3.. Ay Cy Cy C36 Cees such as 0.692307692307...(%3), which has a single
repeating block of the digits 692307. Thus the decimal expansions have one of two characteristics. Either they are of some definite length, or they have a repetitive block within them (shown by the c terms above). The rationals have decimal expansions with patterns.
By about 350 B.c. the Babylonians (Achaemenid Persians, really) had developed a sophisticated al-
gebra and arithmetic. They could solve equations
with one unknown, such as x? — ax = b; and with two unknowns, such asx — y=aorxr+y=b, They knew the formula for the expansion of (a + b)? and the sums of geometric and arithmetic series. They constructed tables of the lengths of chords of a circle of a given radius, corresponding to modern tables of sines and cosines. The Babylonians had a passion for numbers and astronomy, equaled only by the Mayans. By the Hellenistic period they had a complex armory of numerical techniques, which they had successfully applied to the motions of the planets. What they lacked was a model of the heavens—they had no idea why the planets moved as they did, although they could predict their positions with accuracy. Likewise, their arithmetic comes to us as a collection of intricate
44
THE REASONING ARCHITECT
50
Reals
Transcendental irrationals
Algebraic irrationals
Algebraics
|
Rationals s
ls there a number between these two integers? Figure 2-11 Ifthe integers are pictured as a number line, it may occur to someone that there might be numbers between the integers. Then the question arises, how many
numbers are there between integers?
Q is only one rational between any two adjacent inOther fractions
[|
Negative integers
tegers. But it is easy to calculate the average of this Integers
Z
Natural numbers (counting numbers)
Figure 2-10
The sorts of numbers.
procedures, lacking a sense of fitting into a grander scheme of things. It would be left to the Greeks to make the fundamental step that transformed this into modern science and modern mathematics. 2-2-3
Greek Irrationals
We have seen two ways in which the rationals might be discovered. A third way is to do some geometric
rational and the lesser integer, which must be an-
other rational number. Therefore there are at least two rationals between integers. Continuing in the same vein, one concludes that if there is at least one
rational, there must be an infinite number of rationals between any two integers. It is obvious that the number line is full up to the brim with rationals.
All tainly there which
the numbers have been discovered. They cerpack the number line. The only niggle is that seem to be as many integers as rationals, is a bit odd. One of the first discoveries of
the classical Greeks was that there were other numbers. They were the irrationals.
The first proof that there must be numbers that
are not rationals (hence the name irrationals) was due to the Pythagoreans. The proof they gave is a
classic use of the method called reductio ad absurdum.
This technique consists of assuming some statement
thinking. The obvious way to think about integers
true and then showing that such an assumption leads to a contradiction. The existence of this con-
ever. A quite different way is to imagine a line marked out at regular intervals (the familiar number line) so each mark stands for an integer.
be false.
is as collections of things, heaps of pebbles or what-
This is a significant change of perspective. Immediately it conjures up in the quizzical mind the possibility that there might be numbers between the
tradiction shows that the original assumption must We now investigate the innocent-looking V2.
This is clearly a number on the number line and can
be easily constructed geometrically (Figure 2-12). As-
sume that V2is rational. Then it can be represented
integers (Figure 2-11). Perhaps some ancient archi-
by some fraction, say, a/b. Make this the simplest
day at his neatly marked measuring rod. How many numbers are there between every integer? Say there
example, 4 is simpler than %), so that a and b have
tect was the first to ponder this, as he looked one
fraction for V2 by canceling any common terms (for no common divisor. Then:
PRIMITIVE
NOTIONS:
v2
—e— Figure 2-12
Construction of V2 using the diago-
nal of a square. Other square roots can be simi-
larly constructed. ‘a\2
a= (;
CIVILIZATION
45
Let us straighten out the terminology first. The notion of infinity is very elusive, so I will provide a definition of it as follows: Infinity is the cardinality of the set of counting numbers. Forget any other property that this number may have, and just consider it as describing the number of counting numbers. To distance this number from any thoughts you may have about ©, I will use the symbol that mathematicians use for it, which is Xp, pronounced
aleph-null. At first sight it would seem obvious that there is an infinite number of reals. Here we encounter our first peculiarity about the infinite. There there must also be an &) number of reals. So there
Since the left-hand side is even, the right-hand side
must be as well. The only way that a“ can be even is if a is even. Therefore a is even. It can be written as a multiple of 2, say, as 2n. Hence:
2h = (2n)* = 4n?
BP = 2n? Therefore b must also be divisible by 2. This contradicts the assumption that a and b had no common factor. Therefore the original assumption that V2 is rational must be false. How Many
IN PRECLASSICAL
are an Xp) number of counting numbers, and it seems
“2b = @
2-2-4
MATHEMATICS
Reals?
Mathematical beauty involves notions of simplicity, elegance, generality, and depth. A notion of unexpectedness, too, of having derived far-reaching
results with simple tools. As an example of a beautiful proof, I offer a strange result first obtained by Gregor Cantor in the late nineteenth century. Cantor asked a very odd question, a question that could
occur only to a mathematician. We all know that
there are an infinite number of counting numbers
(1, 2, 3,...%). Cantor asked: How many real num-
bers are there? The real numbers consist of the rationals and the irrationals.
are as many counting numbers as reals. Yet the counting numbers are a subset of the set of reals. Does this not seem odd? Yet the only other possibility is that there are more reals than counting numbers, in which case we are left pondering how the
number of reals can be larger than Xp. What is larger than infinity? Cantor showed that this second possibility was in fact the case and that there are more reals than counting numbers. This struck many mathematicians of the time as a ridiculous result, as it may so strike you. Bear with me, though, and see what you make of the proof. We use the same method presented earlier to compare our collection of hides. If there are as many reals as there are counting numbers, then we will be able to match every real with a counting number. If there are more reals than counting numbers, then after we have matched every real to a counting number, we will have some reals left over. Let’s try out this matching
process.
We
keep
things
uncomplicated by just looking at the reals between 0 and 1 to start off with. On the left is the integer 1, matched with a single real on the right represented by an infinite decimal expansion. 1