Books and Arts

Nature 451, 629 (7 February 2008) | doi:10.1038/451629a; Published online 6 February 2008

Multi-dimensional lives

Mark Ronan1

BOOK REVIEWEDFinding Moonshine: A Mathematician's Journey Through Symmetry

by Marcus du Sautoy

Fourth Estate/Harper: 2008. 400 pp/384 pp. £18.99/$25.95

The mathematics of symmetry emerged from the work of Evariste Galois, the young genius who died aged just 20 in 1832 after being shot in a duel. His tragic story, placed in the context of French history and revolution, is retold here in Marcus du Sautoy's follow-up to The Music of the Primes. Galois's is a dramatic story of strong egos and lost manuscripts, ending with a letter he wrote the night before he died — perhaps the most famous epistle written by a mathematician.

Galois was working on the solution of equations, a problem first tackled some 4,000 years earlier when the Babylonians created a formula for solving quadratic equations. The subsequent theory was developed by several main characters: the twelfth-century Persian mathematician, astronomer and poet Omar Khayyam; four Italians in the early 1500s; Paolo Ruffini in 1799; and Niels Henrik Abel in 1824, the young Norwegian who showed that most fifth-degree equations and higher cannot be solved in terms of square roots, cube roots and so on.

Multi-dimensional lives


Islamic tile patterns display a multitude of symmetries.

Some equations of high degree could be solved in this way, but a method was needed to determine which ones they were. Enter Galois. By examining patterns among the solutions and studying the group of symmetries preserving these patterns, he could solve the problem. If the group of symmetries could be deconstructed into cyclic groups, then the solutions could be expressed in terms of roots. Some groups do not admit any deconstruction — they are 'atoms of symmetry' — and Galois found the first ones.

Atoms of symmetry are the basic building-blocks for all finite groups of symmetry, and some of them have applications in modern technology. For example, they can be used to encode digital data such that small transmission errors can be automatically and efficiently corrected. Most symmetry atoms fit into a 'periodic table' where they belong in one of several families whose members enjoy similar properties. There are 26 exceptions. The largest of these is the 'Monster', a vast group of symmetries requiring at least 196,883 dimensions in which to operate. It exhibits numerical patterns similar to those obtained in an important branch of number theory, a connection dubbed 'Moonshine' by John Conway, who was one of the first to investigate it and marvel at its surprising magic.

Moonshine appears at the end of Finding Moonshine; the main thrust is elsewhere. Twelve chapters, one for each month of the year, include descriptions of the author's own life and work in the months concerned. Du Sautoy describes how he conducts his own research, interacts with other mathematicians, his family and particularly his son. He points out that mathematicians can be strange people, and pokes playful fun at the idiosyncracies of some of those who worked on the Monster. He is admirably self-deprecating, recounting how on a visit to Japan he annoyed local guests by remarking that the sake was only 30 ° proof, which is not a prime number, whereupon his host obligingly produced a 43 °-proof alternative.

Interesting interludes highlight the elementary aspects of symmetry, including its role in the music of Bach. Du Sautoy also discusses the regular solids (tetrahedron, cube, and so on), teaching us that the Romans were so obsessed with dice games that they carried heavy gaming boards on campaigns, and even used 12-sided dice invented by the Etruscans. He says that Pythagoras learned of the dodecahedron from the Romans while in southern Italy. But hold on — when Pythagoras moved to Croton in Italy in the sixth century BC, Rome was an Etruscan kingdom. The Romans came later. Inaccuracies such as these, the lack of references and the projection of unverified feelings onto historical characters, spoil an otherwise delightful account of the early history (particularly of the sixteenth-century Italians) and the subsequent research on equations. Later material leading to the Monster also contains some factual errors.

The appearance of the Monster near the end is where the mathematics gets most interesting and relevant to more applied areas of science. For example, symmetry atoms are used in physics to create the standard model of the quantum forces, and Moonshine finds a home in string theory.

That said, du Sautoy omits the applications to physics, and sticks with simple symmetries that a reader with no mathematical appreciation will understand. I, meanwhile, share Freeman Dyson's sneaking hope that "some time in the twenty-first century, physicists will stumble upon the Monster group, built in some unsuspected way into the structure of the Universe".

  1. Mark Ronan is professor of mathematics at the University of Illinois at Chicago, Illinois 60607, USA, and honorary professor at University College London. He is author of Symmetry and the Monster.