Symmetry is more than a mathematician's conceit.
May 12, 2008, Vol. 13, No. 33 • By DAVID GUASPARI
This gracefully written book is both a leisurely introduction to the mathematics of symmetry and its author's apologia: Why dedicate one's life to mathematical problems? What is such a life like?
Roughly speaking, a symmetry is a transformation of some object that leaves it looking unchanged. Marcus du Sautoy illustrates on a visit to the Alhambra, whose interior is covered with a spectacular profusion of intricate repetitive patterns: the world's most gorgeous display of wallpaper. Imagine two copies of a wallpaper pattern, one atop the other in perfect alignment. A symmetry of the pattern is any way of moving the top copy--sliding, spinning, flipping, etc.--that leaves it aligned with the bottom. If the pattern is a simple grid of squares, for example, sliding the top copy sideways by the length of one square is a symmetry. So is rotating a quarter turn around the center of any square, or flipping the top sheet upside-down by spinning it around a square's diagonal.
The collection of all an object's symmetries is a group, the object's symmetry group. (Its essence is a table showing how symmetries combine: Two quarter-turns around the same point are the same as a half-turn around that point; two quarter-turns around different points are the same as a half-turn followed by a slide.) The endless variety of wallpaper patterns gives rise to precisely 17 different groups, and objects more exotic than wallpaper give rise to infinitely many others.
Du Sautoy's visit is a mathematical life in miniature. The patterns are beautiful and the pleasures of deeper insight are compelling--of seeing, for example, how the endlessly various designs manifest a few basic forms; or the pleasure of surprise in seeing that the same form underlies designs that look quite different.
Like all mathematics, the visit is a quest, in this case a modest attempt to verify that the Alhambra's architects had (without benefit of theory) created every possible kind of pattern; that is, exemplified each of the "wallpaper groups." It's dogged by repeated failures--thinking a new group has been found but having to recant--and graced by moments of illumination, such as realizing that an elusive group has been literally underfoot all day, in the symmetries of the pavement's brickwork. And the audience for his achievement will be small, his young son growing bored long before the trip ends.
Groups can be studied in themselves, apart from what they're the symmetry groups of, and some of them play a role like that of the chemical atoms. Every group can, in a technical sense, be decomposed into atoms, which cannot be further decomposed. (Atomic groups are officially called "simple." One of my few complaints is that du Sautoy never precisely defines "group," which can be done in a paragraph, or "simple," which takes a page or two.) One of the most remarkable projects in mathematics has been the "classification of finite simple groups"--analogous to filling in the periodic table of elements.
Symmetry covers much of the same ground as Mark Ronan's Symmetry and the Monster (reviewed in the March 19, 2007, issue of THE WEEKLY STANDARD), surveying the wide role of symmetry in nature and human nature, and giving central place to the classification project. Du Sautoy's telling introduces work at Cambridge on the remarkable Atlas of Finite Groups and adds, in counterpoint, a contemporary story about his own work.
A chemical formula listing numbers and kinds of atoms does not determine the chemical uniquely because the atoms can bond in different ways. So it is with groups. Du Sautoy has spent more than a decade on the problem of how many different groups can be built using two (or three, or four . . .) copies of the same atom. It is astonishingly difficult, and solutions for particular numbers of atoms have not suggested a pattern. For even longer, he's puzzled about a pattern that leapt out of a related calculation, which seemed always to produce a palindrome: a sequence of numbers, such as 2,4,7,3,7,4,2, that reads the same forward and backward. Could that always be true?