Mathematics

# Mirror games

Jan 31st 2008

From *The Economist* print edition

SYMMETRY has been a source of fascination since ancient times. In Plato's “Symposium”, Aristophanes argues that the origins of love lie in the search for symmetry, while Pythagoras and Theaetetus—who discovered the icosahedron, the geometric shape with 20 triangular faces—explored the symmetries of geometric figures.

But it was not until the 19th century that mathematicians such as Evariste Galois, an unhappy French genius, discovered symmetries hidden in the solutions to mathematical equations. In his new book, Marcus du Sautoy, a professor of mathematics at Oxford University, gives a fascinating account of the long quest to unearth the mathematics of symmetry.

Geometric symmetry can be understood as a property of shapes that remain unchanged despite being twisted or flipped—the letter H looks the same after being given a half turn; Y looks the same in a mirror. Other, more abstract objects—the shuffle of a pack of cards, the forces that govern our universe—can likewise be thought of as symmetrical if they remain unchanged after mathematical operations analogous to twisting and flipping. But studying such symmetries required a whole new language. A crucial step here was made by Arthur Cayley, a Victorian mathematician who showed that the symmetries of any object could be described by a mathematical structure known as a symmetry group.

This was the beginning of an important mathematical quest: to understand and classify all possible types of symmetry. Much as every integer can be broken down into a product of prime numbers, the symmetries of any object can be constructed from a collection of basic building blocks known as simple groups. The challenge for mathematicians became how to classify the complete set of these simple groups. For a while it looked as if almost all of them would fall into a few straightforward families, leading Leonard Dickson, an American mathematician, to declare in the 1920s that group theory was dead. This turned out to be premature.

In 1965 a new simple group (with 175,560 symmetries) was discovered which did not fit into any of the standard families. This kicked off a mathematical gold rush. Over the next ten years 26 new simple groups were found. These did not seem to belong to large families, but were isolated examples, and so became known as sporadic groups. The largest is called the “Monster” and is impressively huge: a number with 54 digits, describing the symmetries of a 196,883-dimensional “object”.

In the 1980s the quest was finished. All the sporadic groups had been found, and the classification was shown to be complete. This huge accomplishment resulted in a proof running to some 10,000 pages across 500 journals. It is still being checked.

Alongside the mathematical story is an equally fascinating personal one. Mr du Sautoy's own work involves the study of symmetry, and he gives an illuminating account of the life of a mathematician. Each of the 12 chapters charts a month in his life, detailing the joys of mathematical discovery and the frustrations of mathematical research. He describes how he became a mathematician, and talks about the challenge of interesting his nine-year-old son in the beauties of mathematics.

“Finding Moonshine” is full of insight into the nature of symmetry and the people who study it. It makes for a fascinating and absorbing read.