The Magical Maze: Seeing the World Through Mathematical Eyes

279 Pages·1998·13.01 MB·English

by  Ian Stewart

SQ"iuq NwA dtvo~dd lq
cdlmI Ohu
l
IAN ST EWA RT
AU T H RO 0 F NATU RE'S N U M BERS
The Magical Maze
SEEING THE WORLD
THROUGH MATHEMATICAL EYES
Ian Stewart
John Wiley & Sons, Inc.
New York * Chichester * Weinheim * Brisbane * Singapore * Toronto
This book is printed on acid-free paper. i
Copyright ' 1997 by Ian Stewart. All rights reserved
First published in the United States in 1998 by John Wiley & Sons, Inc.
Published simultaneously in Canada
First published in Great Britain in 1997 by Weidenfeld and Nicolson
No part of this publication may be reproduced, stored in a retrieval system or
transmitted in any form or by any means, electronic, mechanical, photocopying,
recording, scanning or otherwise, except as permitted under Sections 107 or 108 of
the 1976 United States Copyright Act, without either the prior written
permission of the Publisher, or authorization through payment of the appropriate
per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers,
MA 01923, [phone], fax [phone]. Requests to the Publisher for
permission should be addressed to the Permissions Department, John Wiley &
Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, [phone], fax
[phone], E-Mail: PERMREQ @ WILEY.COM.
This publication is designed to provide accurate and authoritative information in
regard to the subject matter covered. It is sold with the understanding that the
publisher is not engaged in rendering professional services. If professional advice
or other expert assistance is required, the services of a competent professional
person should be sought.
ISBN 0-471-19297-X
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
CONTENTS
ENTRANCE
EXIT
List of illustrations vii
Before You Enter ... ix
Entrance 1
JUNCTION 1 4
PASSAGE 1: The Magic of Numbers 6
JUNCTION 2 35
PASSAGE 2: Panthers Don't Like Porridge 37
JUNCTION 3 67
PASSAGE 3: Marilyn and the Goats 69
JUNCTION 4 99
PASSAGE 4: The Slime Mould Saga 101
JUNCTION 5 134
PASSAGE 5: The Pattern of Tiny Feet 135
JUNCTION 6 157
PASSAGE 6: Turing's Train Set 159
JUNCTION 7 188
PASSAGE 7: Queen Dido's Hide 190
JUNCTION 8 213
PASSAGE 8: Gallery of Monsters 215
Exit 247
Pointers 248
Directory 264
ILLUSTRATIONS
Figure 1 Martin Gardner (ed.), The Annotated Snark, Penguin, Harmondsworth,
1962, p. 79.
Figure 4 Martin Gardner, Mathematics, Magic, and Mystery, Dover Publications,
New York, 1956, Fig. 55, p. 105.
Figure 8 D'Arcy Thompson, On Growth and Form, Cambridge University Press,
Cambridge, 1942, Fig. 448, p. 913.
Figure 10 Przemyslaw Prusinkiewicz and Aristid Lindenmayer, The Algorithmic
Beauty of Plants, Springer-Verlag, New York, 1990, Fig. 4.2, p. 101.
Figure 23 William Feller, An Introduction to ProbabilityT heory and Its Applications,
Vol. 1, John Wiley & Sons, Inc., New York, Fig. 5, p. 84.
Figure 26 Thomas Hofer, 'Modelling Dictyostelium Aggregation', Ph.D. Thesis,
Balliol College, University of Oxford, 1996, Plate 1, p. 5.
Figure 34 Ian Stewart and Martin Golubitsky, Fearful Symmetry, Penguin,
Harmondsworth, 1992, Fig. 9.11, p. 2:39; credited to Mathematical Intelligencer,
Springer-Verlag, New York, Vol. 5, 4 (1983), p. 39.
Figure 36 H.M. Cundy and A.P. Rollett, Mathematical Models, Oxford University
Press, Oxford, pp. 61-62.
Figure 37 Keith Critchlow, Islamic Patterns, Shocken, New York, p. 188.
Figure 38 IstvAn Hargittai, Quasicrystals, Networks, and Molecules of Fivefold
Symmetry, VCH, New York, Fig. 14, p. 150.
Figure 40 Jan Stewart, Game, Set & Math, Blackwell, Oxford, Fig. 6.7, p. 83;
credited to Science Photo Library.
Figure 42 G. Nicolis, Introduction to Nonlinear Science, Cambridge University
Press, Cambridge, 1995, Fig. 1l.1b, p. 17.
Figure 43 Hans Meinhardt, The Algorithmic Beauty of Sea Shells, Springer-Verlag,
Berlin, 1995, Fig. 10.17, p. 178.
Figure 44 Shigeru Kondu and Rihito Asai, 'A reaction-diffusion wave on the skin
of the marine angelfish Pomacanthus', Nature, 376 (1995), pp.765-768, Fig. 3
(p. 767).
Figure 46 Hans Meinhardt, The Algorithmic Beauty of Sea Shells, Springer-Verlag,
Berlin, 1995, Fig. 10.18, p. 179.
Figure 48 P.P. Gambaryan, How Mammals Run, John Wiley & Sons, Inc., New
York, 1974.
Figure 77 H.-O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals, Springer-
Verlag, New York, Plate 3, fol. p. 152.
Figure 83 Przemyslaw Prusinkiewicz and Aristid Lindenmayer, The Algorithmic
Beauty of Plants, Springer-Verlag, New York, 1990, Fig. 1.24, p. 25.
Figure 84 H.-O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals, Springer-
Verlag, New York, Fig. 5.48, p. 280.
Figure 85 Michael F. Barnsley and Lyman P. Hurd, Fractal Image Compression,
A.K. Peters, Wellesley, MA, 1993, Plates 13 and 14, p. 53.
Figure 86 Ian Stewart, Game, Set & Math, Blackwell, Oxford, Fig. 9.2, p. 127.
Figure 87 Edward Ott, Chaos in Dynamical Systems, Cambridge University Press,
Cambridge, Fig. 1.13, p. 15.
BEFORE YOU ENTER ...
S
ome scientists talk to the people; most, deplorably, don't. One who
did was Michael Faraday, one of the greatest scientists of the nine-
teenth century. Faraday made enormous advances in the theories of
electricity and magnetism - in particular, he invented the electric motor
and the dynamo. He provided the foundations upon which James Clerk
Maxwell built his masterpiece, the mathematical equations of electro-
magnetic fields. From Maxwell's work it was but a short step to the
discovery of electromagnetic waves, from which - thanks to numerous
mathematicians, physicists, engineers, inventors, and entrepreneurs -
came radio, radar, and television.
The television connection closes a curious historical loop. Faraday's
career is intimately bound up with the Royal Institution of Great Britain,
a building in London that housed its own scientific library, laboratories,
and lecture theatre. He first became interested in electricity while he was
working as an apprentice bookbinder, and read an article on the topic in
the third edition of the Encyclopaedia Britannica. He was given a ticket to
attend a lecture at the Royal Institution, given by the great chemist Sir
Humphry Davy. The young Faraday was spellbound. He wrote to Davy
asking for a job, and when one of the great man's assistants was sacked
for getting into a fight, Faraday became Davy's laboratory assistant. By
1820 he knew as much chemistry as anyone alive - but his attention was
turning to electricity once more.
In 1821 Faraday married Sarah Barnard, and settled permanently at the
Royal Institution. For the next thirty years he carried out his epic work on
electricity and magnetism, along with much else. And he did not neglect
the general public. In 1826 the Royal Institution began a tradition that
continues to this day: the Christmas Lectures for young people. They
have been held every year since, with one short break during the Second
World War. Faraday gave nineteen of the lectures between 1827 and 1860.
And here is where the circle closes, for in recent years the British
Broadcasting Corporation has televised the Christmas Lectures.
The Magical Maze came into being because Professor Peter Day, Director
of the Royal Institution, invited me to give the 1997 Christmas Lectures
- the 168th lectures in the series, and only the second time (shame!) that
they have focused on mathematics. The opportunity to produce a book,
based around the lectures and published simultaneously, proved irresist-
ible. Of course the book had to be written before the lectures were given;
moreover, the style of a book differs somewhat from the style of a lecture.
So The Magical Maze treats the material in a rather different manner, with
descriptions and pictures replacing demonstrations, apparatus, and inter-
active sessions with members of the audience. It is aimed at anyone who
is interested in mathematics, not just 'young people'. The lectures them-
selves are drawn from roughly half of the book: the other half is a bonus.
(We haven't, at the time of writing, decided exactly which topics will
appear in the lectures, so I can't tell you which half. Sorry.)
When an author first conceives of a book, it is a ghostly, shimmering
wraith - a virtual object. It doesn't yet exist. So many things could go in,
so many words could be placed on the page ... As the writing progresses,
the ghost becomes more solid, the virtual becomes more real. Words,
paragraphs, chapters come into being.
There are many choices, many decisions. What to include? What to
exclude? Each decision carries consequences: early material cannot be
omitted if later material depends on it. The possible structures for the
book are an interconnected network.
And yet every book must tell a story. A story has a beginning, a middle,
and an end. It is read as a linear sequence.
When a book is being written, it is a maze of possibilities, most of
which are never realised. Reading the resulting book, once all decisions
have been taken, is like tracing one particular path through that maze.
The writer's job is to choose that path, define it clearly, and make it as
smooth as possible for those who follow.
Mathematics is much the same. Mathematical ideas form a network.
The interconnections between ideas are logical deductions. If we assume
this, then that must follow - a logical path from this to that. When
mathematicians try to understand a problem, they have to thread a maze
of logic. The body of knowledge that we call mathematics is a catalogue
of interesting excursions through the logical maze.
This is why I have chosen to use the metaphor of a maze to describe
mathematics. As for its magical nature ... well, we'll get round to that
shortly. It is also why I have structured The Magical Maze as one particular
journey through the verbal maze of books that might have been instead.
Instead of a preface or prologue, I've started with an Entrance. Instead of
reading chapters, we wander through Passages. Passages meet at Junctions.
In place of a conclusion, climax, or epilogue, we finish at the Exit.
And on at least one occasion, we run smack bang into a Dead-end.
You don't need to thread the maze yourself: my job is to guide you
through that one selected path that crystallised out when vague thoughts
transmuted into words on paper.
I must admit that I nearly didn't have a 'Before You Enter' section - my
term for a foreword - at all. However, I needed to explain about mazes
before I propelled you into one. And I also needed to do various things
that most authors do in forewords, such as telling you why the book ever
got written, which I've now done, and thanking the people who helped
make the book possible, which I'm just getting round to. Foremost
among those people is Peter Day, without whose interest I would never
have had the opportunity to put together five television programmes on
mathematics. Ravi Mirchandani at Orion Publishing deserves consider-
able credit for his enthusiasm and guidance, as does Benjamin Buchan.
My debt to Caroline Van den Brul and Martin Mortimore of the BBC is so
vast that I scarcely dare acknowledge it in print. Professor Sir Brian Follett,
Vice Chancellor of the University of Warwick, graciously redefined my
duties to make time for me to engage with the public. Alan Newell,
Chairman of the Mathematics Department, was an enthusiastic supporter
of this development, which now takes tangible form as MAC@W - the
Mathematics Awareness Centre at Warwick.
Finally, I owe a long-standing debt to Professor Sir Christopher Zeeman,
founding father of Warwick's Mathematics Institute and for many years
my boss. Christopher was the first mathematician to give the Christmas
Lectures; he also influenced my career in many ways, in particular by
encouraging me to indulge in non-academic activities aimed at bringing
new mathematics to the people.
And we are all indebted to Michael Faraday, without whom neither the
Christmas lectures, nor television, would have been invented.
I.N.S.
Singapore, Australia, New Zealand, Hawaii, Sweden, and Coventry
January-May 1997
The Magical Maze