Symmetry is something that is easy for us to appreciate. It might be that we have an evolutionary taste for symmetric creatures; we suspect there is something wrong if a horse has an uneven gait, and it has been shown that we prefer symmetric faces. Of course symmetry is part of our art and architecture. So it is an inherently interesting subject for everyone, but mathematicians have taken the study of symmetry to heights that the rest of us can barely imagine. One of those mathematicians is Marcus du Sautoy, who has shown in his previous _The Music of the Primes_ that he has the capability of descending from the mathematical summits enough to have readers understand a bit of what mathematicians do. Now in _Symmetry: A Journey into the Patterns of Nature_ (Harper), du Sautoy has told the story of a mathematical quest that has gone on for centuries and which, it seems, was essentially completed in the 1980s. There are lots of different symmetries, some of which have complicated ways of being manipulated in dimensions higher than anyone will ever be able to depict. To prove that every single symmetry has been mathematically classified was a real triumph of a branch of mathematics known as Group Theory. The scale of the triumph only mathematicians will come close to fully understanding, but the rest of us can get an idea of how monumental a victory this was from du Sautoy's engaging look at how the job was done.
Imagine an equilateral triangle. You can leave it where it is, or you can rotate it around by a third, or by two thirds, and it looks just the same. You can flip it around three different axes, and it looks the same. Those are its six symmetries. The Greeks were fascinated with the symmetry of solid figures, the Muslims with that of tiles and plane figures. But shapes and tilings are not all there is to symmetry; different ways of shuffling a pack of cards have symmetry, as does the number lock on a piece of luggage. The change ringers who team up to ring five bells in the exactly 120 different orders in which five bells can be rung are performing a symmetrical operation. There are symmetries of many properties of matter and physics, and it might be that they will help explain string theory. Indeed, the great story here is that like prime numbers, symmetric groups are at the heart of the mathematics of our universe, but unlike prime numbers, there is only a finite number of symmetries, and they have all now been found. It was thought in the 1920s that group theory had reached a dead end and there were no further families of symmetry to be found. But in 1965, a new group was found that didn't fit into any of the previous families, and like the athletes who found themselves able to do a four minute mile once one person had done so, mathematicians were inspired by this discovery into finding further such groups. The largest of the groups is called the Monster. "The Monster is like some huge, great symmetrical snowflake that you can see only when you get to 196,883-dimensional space," says John Horton Conway, one of the heroes of the quest described here. Forget the six symmetries of that equilateral triangle; there are more symmetries in the Monster than there are atoms in the Sun. Even more wonderful is that there are mathematicians like Conway who somehow can picture such an object in their minds, and have included it and all the other possible symmetries in the massive, inclusive _Atlas of Finite Groups_. If the math (even at the level of this popularization) gets dense, there are always funny stories about mathematicians, who are often just as strange as you would expect such eggheads to be.
Symmetries is not a textbook on symmetry, and the concepts here require not only textbooks but mathematical textbooks, textbooks only a few people, even a few mathematicians, are ever going to read. What du Sautoy has done is to give a feeling of the importance of the hunt and the excitement of it. He has also done readers the service of explaining just what a mathematician does. "Many of my friends have the impression that I'm sitting in my office doing long division to a lot of decimal places, and wonder why a computer hasn't put me out of a job by now." It is hunting for patterns and discovering them that keeps him going. The twelve chapters here not only go through the history of group theory but each tells of a month in his life, the joys and frustrations of mathematical research, of mathematical teamwork and competition, or of the push to publish one's ideas. Maybe you will never master group theory, but if you want to know what mathematicians do, and why it is important, this is a wonderful guide.
