# Symmetry in Motion

 TOOLBOX Resize
By Jordan Ellenberg,
an assistant professor of mathematics at the University of Wisconsin and author of the novel "The Grasshopper King"
Wednesday, September 6, 2006

KING OF INFINITE SPACE

Donald Coxeter, the Man Who Saved Geometry

By Siobhan Roberts

Walker. 399 pp. \$27.95

You might not have known geometry needed saving. It's still right there in the high school curriculum where you remember it -- T-shaped proofs, similar triangles and all. But among grown-up mathematicians, geometry is all but ignored; the interesting problems are largely solved, the applications mostly exhausted. For decades now, no high-profile mathematician has worked in classical geometry, with the notable exception of H.S.M. ("Donald") Coxeter, the subject of Siobhan Roberts's thorough and intermittently absorbing biography, "King of Infinite Space."

Every mathematical biographer faces the same vexing problem: Mathematicians spend their lives sitting at a desk and writing in a notebook. Occasionally they switch desks. There are exceptions -- Évariste Galois dying at 20 in a duel, John Nash succumbing to and subsequently emerging from psychosis -- but Donald Coxeter was not an exception. He earned his PhD, married his first girlfriend and settled into a professorship at the University of Toronto, where he stayed from 1936 until his death in 2003, at the age of 96. If there's an interesting story to be drawn from his life, it's a story about the history of ideas -- more precisely, about the history of geometry.

Geometry as we know it starts with Euclid: points, lines and planes. When you've got a point and a line in the plane, there's really just one interesting thing you can do: reflect the point through the line. (Imagine the point as a wet ink spot on a piece of paper; if you fold the paper along the line and press down, the second ink spot you produce is the reflection of the first one through the line.) The grand theme of Coxeter's most important work is reflection, and especially the interplay between different reflections. The latter can produce symmetries and patterns of astonishing richness. Think of a kaleidoscope (unsurprisingly, Coxeter's favorite toy), which is built on exactly this principle: Reflections through several mirrors, set at judiciously chosen angles, produce a beautiful picture.

But Coxeter's work on reflections is good for more than just making complicated toys and beautiful pictures; Roberts takes readers on a wide-ranging tour of contexts in which Coxeter's beloved symmetries have made themselves known, from geodesic domes to the error-correcting codes that make digital recording possible. As always, what is beautiful has ended up being useful. (That, at any rate, is what we mathematicians write in our grant applications.)

Algebraist Robert Moody said of Coxeter: "Modern science is often driven by fads and fashions, and mathematics is no exception. Coxeter's style, I would say, is singularly unfashionable." If Coxeter's work is so important, why is it so out of fashion? Roberts presents an affecting story of a genius left behind by a mathematical profession that has forgotten to value the classical entities of point, line and plane. The truth is more complicated. Geometry is healthier than ever -- but we now recognize that "geometry" covers much more ground than Euclid understood. Not just planes, but curvy high-dimensional spaces are proper subjects for geometry. Indeed, after Einstein, we know that we live in such a space! Even more exotically, the digital codes used to store information on computers can be thought of as points in a "geometry" made of sequences of 0's and 1's. This isn't the kind of geometry where you get to use a protractor, but its underlying essence is the same -- and without it, Coxeter's work would not have proved so terrifically useful.

As a biography of Coxeter, "King of Infinite Space" is exhaustive and definitive. Roberts's painstaking research, documented by 73 pages of endnotes, turns up many gems, like the transcript of a discussion of the exotic carbon molecule Buckminsterfullerene by a befuddled House of Lords. ("Baroness Seear interrupted: 'My Lords, forgive my ignorance, but can the noble Lord say whether this thing is animal, vegetable or mineral?' ") Especially notable is Roberts's access to Coxeter's diaries, which inject the book with anecdotes of rather startling candor -- as when we learn that Coxeter's wife made him wear his tie around the house because she found his neck so unattractive.

This book will be invaluable for historians. For the general reader it may be a harder sell. Roberts covers a huge amount of mathematical ground, as Coxeter himself did -- but her treatment of these topics is that of a conscientious reporter, not an enthusiastic proponent. Her conscientiousness, too, results in a bit too much of whom Coxeter met when and what they said. But for the devotee of geometry, there is no substitute for Coxeter, and no substitute for this long-overdue treatment of his life.

© 2006 The Washington Post Company