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Balancing Act

The Enormous Theorem and the jigsaw of numbers.

Mar 19, 2007, Vol. 12, No. 26 • By DAVID GUASPARI
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Symmetry and the Monster

The Story of One of the Greatest Quests of Mathematics

by Mark Ronan

Oxford, 272 pp., $27

The earliest example of "modern" and "abstract" mathematics--and one of the most beautiful--is the theory of groups. It grew out of the study of equations, but can be understood as a theory of symmetry.

You might cryptically define a "symmetry" as a transformation or rearrangement of something that leaves it unchanged. For example, a square looks unchanged if you rotate it 90 degrees, or examine its reflection in a suitably positioned mirror. Collectively, the rotations and reflections that leave a square looking the same make up "the symmetry group of the square." Symmetries abound not only in mathematics, but also in physics. The theory of relativity says that the transformation from one observer's point of view to another's may alter the values of some observations but will leave the laws relating those observations unchanged.

The principle that any fundamental physical theory must possess certain kinds of symmetry has become a scientific axiom, and a tool of discovery.

In Symmetry and the Monster, Mark Ronan guides the lay reader through one of the major mathematical achievements of the 20th century--officially known as "the classification of finite simple groups," but often called "the enormous theorem," because its proof occupies more than 10,000 pages of dense reasoning spread through several hundred journal articles (and never-published manuscripts), with significant contributions from perhaps a hundred authors. For posterity's sake, a project is under way to simplify and unify this body of argument and present it in full detail. Estimated completion date: 2010. Anticipated length: 12 volumes.

The enormous theorem identifies and characterizes the infinitely many finite simple groups, which Ronan calls "atoms of symmetry" because their role in group theory is something like that of the elements in chemistry or the prime numbers in arithmetic: Building blocks from which all the groups can be constructed. Eventually, a kind of periodic table emerged into which almost all the known simple groups could be fit. Immense labor then proved that there are precisely 26 exceptions--simple groups not in the table and fitting no known pattern.

The existence of the largest and strangest of these, the eponymous Monster, was conjectured in the early 1970s and confirmed 10 years later. Here are some of its Babe Ruthian statistics: The Monster can be represented as a collection of rotations in a space of 196,883 dimensions. The size of that collection, which has been calculated exactly, is comparable to the number of elementary particles in the planet Jupiter. To write it in decimal form takes 54 digits. One is amazed, if not aghast, that the human mind can deal with such complexity.

The Monster also has spooky connections with seemingly remote branches of mathematics and with string theory, an ongoing attempt to unify all the fundamental forces of physics in a single theory. Glimpses of these mysterious affinities can make the world slip out of focus, as the boundary between sober scientific fact and numerological superstition seems to dissolve before our eyes.

It is an understatement to say that Ronan has set himself a formidable task, but to a reader willing to meet him part way, he offers an absorbing tale of discovery. It begins in 1830 with the great variste Galois, and chronicles both spectacular intellectual virtuosity and outsize personalities. Galois himself--killed in a duel at age 20--was a combination of Mozart and James Dean who introduced whole new fields of mathematical enquiry in a testament hastily composed the night before his trip to the field of honor.

Ronan's presentation naturally skews toward things that are more easily explained. So we get a great deal about the eccentricities of Sophus Lie, but only the most glancing account of the deep subject called (in his honor) Lie Theory, from which the periodic table of simple groups derives. Lacking a single theme, the narrative loses some momentum halfway through, but revs up when it reaches the 1960s and remains in high gear as "The Classification" evolves from a pipe dream to thinkability and then inevitability, and the quest to complete it culminates in a kind of international treasure hunt for the mysterious exceptions.

Many of the principals are still working mathematicians, and Ronan makes good use of their own words to describe the circumstances and excitement surrounding key discoveries. A reader may glimpse what it feels like to do mathematics at a high level, the sense of exploring a terrain that is mysterious, surprising, and completely unforgiving.