Numbers of the beast

Symmetry and the Monster

December 22, 2006

The measurement of symmetry is one of the great successes of modern mathematics. Symmetry is not like length or time or temperature.

Those can be measured with instruments that yield ordinary numbers.

Symmetry is measured using what are technically known as groups, algebraic systems satisfying certain conditions that were identified in the 19th century. Group theory, the study of these objects, starts at a level simple enough that it can, and does, form part of school mathematics. But research has taken the subject far beyond where even the most sophisticated university course can take undergraduates. It is, therefore, a high ambition to seek to explain some of that research to a non-mathematical public.

This book is about the search for the finite simple groups - the indivisible groups out of which all groups are built. Popular analogy compares them to the atoms from which all chemical compounds are made. In this book, they are called "symmetry atoms". But this usage involves a charming paradox: for although they are atoms according to the Greek etymology (cannot be cut), they are far from being "anything very small", which the dictionary gives as one of the main meanings of the word. The "monster" of the title is a simple group of size greater than 8 x 1053 that measures the symmetry of an object in space of 196,883 dimensions.

Modern mathematics has resolved a number of famous questions. Perhaps best known were the classical geometrical problems - squaring the circle, trisecting an angle, duplicating a cube, now known to be impossible by the methods prescribed by the Greeks. Then came Fermat's last theorem, which was simple to state but involved unimaginably sophisticated ideas when it was finally proved by Andrew Wiles in 1994. The Riemann hypothesis, unresolved but chosen as one of the Clay Institute Millennium Problems for which a prize of $1 million (£510,000) is offered, has led to several popular expositions, such as The Music of the Primes by Marcus du Sautoy.

In sheer weight of numbers and size of monsters, the search for the finite simple groups beats them all. It has involved the concerted efforts of more mathematicians than any previous project. It has produced longer proofs than have ever been seen before. Think of the proof of Pythagoras's theorem: perhaps one page of Euclid's Elements , a leisurely page of exposition by modern standards. Compare that with the theorems about finite simple groups, which have proofs running to several hundred pages of intensely concentrated argument. When these theorems are put together to yield the whole classification, the result is a proof amounting to some 10,000 pages. And it is not just the size of the undertaking that is huge: the area contains some monstrously bold conjectures - known by the felicitous title "monstrous moonshine" conferred on them by John Conway - that are leading to connections between group theory, number theory, geometry and theoretical physics.

Whether Mark Ronan has succeeded in explaining all this to non-mathematicians is something that only non-mathematicians will be able to judge. As a mathematician, I can only certify that he has made a far better fist of it than most of us could have, and that his efforts deserve to succeed.

Peter Neumann is a fellow of Queen's College, Oxford.

Symmetry and the Monster: The story of one of the Greatest Quests of Mathematics

Author - Mark Ronan
Publisher - Oxford University Press
Pages - 255
Price - £14.99
ISBN - 0 19 280722 6

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