Symmetry and the monster: one of the greatest quests of mathematics

264 Pages·2006·0.831 MB·English

by  Mark Ronan

Symmetry and the
Monster
One of the greatest quests of mathematics
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Symmetry and the
Monster
One of the greatest quests of mathematics
Mark Ronan
1
3
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© Mark Ronan 2006
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First published 2006
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ISBN 0–19–280722–6 978–0–19–280722–9
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Preface
In recent years several books on mathematics have been
published, presenting intriguing pieces of the subject. This book
also presents some interesting gems, but in the service of explain-
ing one of the big quests of mathematics: the discovery and
classification of all the basic building blocks for symmetry. Some
mathematicians were sceptical of explaining it in a non-technical
way, but others were very encouraging, and I would like to thank
them. In particular I owe thanks to those mathematicians who
read all, or large parts, of the manuscript: Jon Alperin, John
Conway, Bernd Fischer, Bill Kantor, and Richard Weiss. I also
thank my son and daughter who were always positive about the
outcome, and finally my editor Latha Menon who made very
helpful criticisms.
Mark Ronan,
February 2006
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Contents
Prologue 1
1 Theaetetus’s Icosahedron 5
2 Galois: Death of a Genius 11
3 Irrational Solutions 27
4 Groups 42
5 Sophus Lie 53
6 Lie Groups and Physics 71
7 Going Finite 79
8 After the War 88
9 The Man from Uccle 97
10 The Big Theorem 113
11 Pandora’s Box 127
12 The Leech Lattice 142
13 Fischer’s Monsters 157
14 The Atlas 171
15 A Monstrous Mystery 190
16 Construction 205
17 Moonshine 216
Notes 230
Appendix1:TheGoldenSection 238
Appendix2:TheWittDesign 240
Appendix3:TheLeechLattice 242
Appendix4:The26Exceptions 244
Glossary 248
Index 251
vii
To Grace Varndell,
my headmistress from primary school,
who still remembers exactly where I sat in class.
Prologue
What we know is not much. What we do not know is immense.
Pierre-Simon Laplace (1749–1827), said to be his last words
In November 1978 an English mathematician named John McKay
was reading a research paper at his home in Montreal. He worked
in a branch of mathematics called group theory, which deals with
the study of symmetry. It was an area that had recently produced
some exceptional objects in many dimensions, but McKay was
taking a break by reading a paper in number theory, the part of
mathematics that deals with the whole numbers. There was no
connection– or so he thought.
The largest of the exceptional symmetry objects had been called
the Monster. It had not yet been constructed, but a careful exam-
ination of the data showed that the Monster – if it existed – could
probably be viewed in 196,883 dimensions. Now McKay was read-
ing about an object in number theory, and out popped the number
196,884. He was astonished. Any relationship with the Monster
seemed absurd – they came from completely different parts of
mathematics – but he felt he should tell someone, so he wrote a
letter to John Thompson, the great guru in group theory.
Another person receiving the letter might have waved aside the
coincidence as too speculative and beyond understanding, but not
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