The Magazine

Balancing Act

Symmetry is more than a mathematician's conceit.

May 12, 2008, Vol. 13, No. 33 • By DAVID GUASPARI
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Symmetry

A Journey into the Patterns of Nature

by Marcus du Sautoy

Harper, 384 pp., $25.95

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?

By the middle of the 20th century, the known simple groups could be organized into a few general families, but for five unclassifiable ("sporadic") exceptions discovered in the mid-19th century. In 1962 the monumental Feit-Thompson Theorem--255 densely written journal pages--provided tools that seemed powerful enough to complete the classification, perhaps to show that all simple groups had been discovered. Then appeared a one-page paper with another (175,560 symmetries of a 7‑dimensional object) and the rush was on to make a name by finding more. Some were depressed by the prospect that this might go on without end; others, who regarded these unclassifiable oddities as gems, were exhilarated.

The Atlas of Finite Groups began in the mid-1970s as a huge blank ledger in the Cambridge office of John Horton Conway, who had found a spectacular sporadic group himself
(4,157,771,806,543,630,000 symmetries of a 24-dimensional object). He'd done that when he was financially strapped, with three small children, after a deal with his wife allowed him to work at specified times on a much-thought-about hunch. In the very first session, a marathon of more than 12 hours, he found what he was looking for.

"I knew I was a good mathematician," he later said, "but I hadn't done the work to prove it. I'd been feeling really black for several years. .  .  . The discovery of this group wiped out that guilt. It removed the black feeling."

Conway--full of jokes and puzzles and mathematical games, always "on"--became a cult figure. He enlisted four other eccentrics to help systematize everything known about the finite simple groups, to create the Atlas. Of those four, du Sautoy gives the fullest picture of Simon Norton, whose favorite nonmathematical topic of conversation is bus and railway timetables. (Mathematics departments have a huge tolerance for oddity, but I can testify that his fellow graduate students, of whom I was one, found him extreme.) No one knew whether the Atlas could have a last page, or whether sporadics would go on forever. There turned out to be precisely 26 of them--no one knows quite why. The largest, called the Monster, has more symmetries than there are atoms in the sun and lives in 196,883 dimensions.

By the mid-1980s the classification was complete; but for the makers of the Atlas, this remarkable collective triumph of, perhaps, 100 mathematicians was bittersweet. The book was closed. Group theory was no longer hot. Conway accepted a chair at -Princeton, got divorced, became depressed, and attempted suicide. He survived, but is melancholy about growing old and the prospect of losing his powers. With Conway gone, the Atlas group dispersed and Norton became (in du Sautoy's telling) a tragic figure: "Without the political and social skills to survive the cut and thrust of the academic world, [he] was rather abandoned by everyone."

Symmetry begins with du Sautoy's thoughts on his 40th birthday--an event that makes him ineligible for the most prestigious mathematics prize, the Fields Medal--and it returns periodically to the middle-aged question, "What's it all for?" Symmetry ends at a conference in Edinburgh on the work of Richard Borcherds, the 1998 Fields medalist for, among other things, solving a problem of Conway and Norton. He proved their Moonshine Conjecture, which seemed as much numerology as science, asserting that certain patterns in the Monster group's Atlas entry had important connections with number theory. (The key insight came, after eight years of work, when he was stuck on a bus in Kashmir.)

The conference is buzzing about a paper on Asperger's Syndrome--a variant of autism consistent with high intelligence, whose signs include "impaired social interaction" and "all-absorbing narrow interests." Among mathematicians, its incidence is high. Borcherds noted that he had five of the six symptoms on one diagnostic list, Norton seems the classic type and, suggests du Sautoy, Conway (despite his one-way bonhomie) is a plausible candidate. Du Sautoy seems to be silently mulling the tradeoffs involved and is glad that we don't get to choose.

"Before the award," Borcherds later says, "I used to think it was terribly important, but now I realize that it's meaningless. However, I was over the moon when I proved the Moonshine Conjecture." Du Sautoy has learned that the conjecture he hoped to prove is false: The palindromes were charming patterns that proved deceptive. But he savors a moment of discovery like Borcherds's moment, an important advance he made while stuck on the phone, unable to reach his wife.

"Borcherds is right," he concludes. "In mathematics the real prize is not a medal, .  .  . but making the breakthrough on the problem you've dedicated your life to. The prize might be claimed at any time and any place: on a broken-down bus in Kashmir, on a Saturday in Cambridge at twenty past midnight, or while listening to an engaged signal on the end of a telephone line in Bonn."
If you want to know what being a mathematician is like, read this book.

David Guaspari is a writer in Ithaca,
New York.