The Reasoning Architect: Mathematics and Science in Design 9780070613911, 0070613915, 9780070613966, 0070613966, 9780071009232, 007100923X

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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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oe

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. \ er

5

p oS -

ed

2 ee

=

5 AY +

7 r A -

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=

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4i

s

Sa

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aS

H|ij

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EH

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.

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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,

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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

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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).

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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,

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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

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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-

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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

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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-

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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,

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]]

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

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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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]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

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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,

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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

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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

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trivial anyhow. I have just one chance of escap-

MATHEMATICS,

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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-

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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-

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’ 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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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