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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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.