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Meet Mr. Polytope

Tony Rothman

King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry. Siobhan Roberts. xvi + 399 pp. Walker and Company, 2006. $27.95.

Less than the time needed to open a book is required for a prospective reader to understand that the publishers of King of Infinite Space, Siobhan Roberts's new biography of geometer Donald Coxeter, have cheerfully absorbed the precepts of Hollywood marketeers: Packaged with enthusiastic blurbs from physicist Freeman Dyson, mathematician John Conway, writers Martin Gardner and James Gleick, and historian Peter Galison, not to mention a foreword by Gödel, Escher, Bach author Douglas Hofstadter, the whole is designed, much like full-page movie ads in the New York Times, to render harmless slings and arrows hurled by errant reviewers. In this case, though, one can sympathize with the tactic: It's a safe bet that few people outside of narrow mathematical circles have ever heard the name Donald Coxeter, despite the fact that many mathematicians regard him as the greatest geometer of the 20th century.

This photograph of geometer Donald Coxeter...Click to Enlarge Image

Coxeter's lack of name recognition is only the first challenge Roberts faces. Another is that . . . well, let's 'fess up right away, we academics tend not to lead the most exciting lives, by romance-novel standards anyway, and in this Coxeter was no exception. Born in London in 1907, Donald would have led an unblemished childhood but for the trauma of his parents' divorce. The boy showed an early gift for mathematics and music, becoming an accomplished pianist by the age of 10 and also, for a time, a hopeful composer. He was "incarcerated in a boarding school," he reported later in life, and there he fell in love with classical geometry. A few years afterward he received a scholarship to Trinity College, Cambridge, where he led an existence a Trappist would appreciate, studying relentlessly and graduating in 1928 as Senior Wrangler (which is to say, high scorer) in the legendarily difficult Tripos examination. He stayed on, completing his Ph.D. in 1931, at which point he became a fellow at Trinity and was allowed to cut across the courtyard grass.

Afterward, Coxeter spent a few years shuttling back and forth across the Atlantic between fellowships at Princeton and Cambridge until he accepted a permanent position at the University of Toronto in 1934. He remained there until 2003, when he died with his cat on his stomach at the ripe old age of 96, by which time many of his younger colleagues imagined he had departed this world a generation earlier. In the interim, he married, had children, taught, wrote papers and at least two classic texts, managed to stave off middle-aged ossification of the mathematical brain and once ducked out of some conference talks to visit an art museum.

No, the excitement in Coxeter's life was internal. His métier was classical geometry—the study of those solids we encounter in high school, such as dodecahedra and icosahedra. But he found three dimensions confining. With little but his mind's eye to illuminate the terrain, Coxeter climbed into the fourth, fifth, sixth, seventh and eighth dimensions to explore, construct and classify geometric objects according to their symmetry properties. Decades later, Coxeter's work is turning up in arcane fields such as string theory and microbiology, where "Coxeter groups" may help to describe the shape of n-dimensional space or viruses.

Here, though, the third challenge to a Coxeter biographer plants its feet astride the path. Aside from his spectacularly genteel existence—and despite his acknowledged influence on artists such as M. C. Escher and architect Buckminster Fuller—Coxeter was a pure mathematician whose work was far removed from the tangibles of daily life. How, then, is one to impart the essence of a discrete symmetry group of order 120 to readers who by college may well have spurned Plato's warning, "None but those who know geometry may enter here"?

This last challenge is easier to deal with. Roberts has evidently heeded the famous dictum that contributed to Stephen Hawking's success, that every equation in a popular book halves the readership. She goes light on the mathematics and, what's more, is able to exploit geometry's visual aspect to give an idea of the higher-dimensional analogies. Generally her straightforward approach succeeds, but at times one does wish that a CD-ROM and 3-D glasses had been provided to allow a better view of Coxeter's constructions. And occasionally I wonder what the average reader will make of a statement such as: "Take for example an 8-dimensional polytope, which has 711,244,800 symmetries, 240 vertices, 6,720 edges, finishing with its 7-dimensional cells, namely 17,280 simplexes and 2,160 orthoplexes."

Presenting Coxeter himself, however, is perhaps the most daunting of the challenges—the man doesn't seem even to have indulged in an extramarital affair with which to titillate today's readers. Roberts wisely chooses to generate interest by threading Coxeter's story through the tapestry of 20th-century mathematics. The most engaging aspect of the book is its chronicle of the war between geometry and algebra, which pits Coxeter, geometry's David, against Nicolas Bourbaki, algebra's Goliath.

Bourbaki was said to frequent the Café Capoulade in Paris, where he worked on the Éléments de mathématique, a treatise intended to encompass all mathematics. The odd thing about Bourbaki was that no one ever saw him. His name, which may have been taken from a general in the Franco-Prussian War or an omnivorous hedgehog in a story by anarchist Octave Mirbeau, was in fact the nom de plumefor a secret society of French mathematicians who intended to determine the direction of mathematics worldwide—and nearly succeeded.

The Bourbaki were proponents of formal, algebraic mathematics devoid of geometrical reasoning, and they made it a point of honor to publish no diagrams in their books and papers. "Death to triangles!" shouted Jean Dieudonné at a conference, a declaration that became their rallying cry. The influence of the Bourbaki was enormous. Before reading King of Infinite Space, I had not realized how enormous, for it was they who along with Sputnik ushered in the New Math of the 1960s, with the consequence that a generation of students failed to learn arithmetic and geometry was almost stamped out as a discipline. As one who was in grade school at the time, I can testify that students were subjected to liberal doses of set theory, perhaps to the detriment of multiplication.

The Bourbaki indeed had the upper hand for much of the century's remainder. The algebraic mode of thinking may be as different from the geometric as Baroque music is from modern, and for the past generation at least, the training of mathematical scientists has tended to be along the algebraic route. Even such "geometric" subjects as Einstein's general relativity have been largely reduced to algebraic computation.

Jakob Steiner, the great 19th-century geometer, reportedly disliked algebra, believing that calculation replaces thinking, whereas geometry stimulates the imagination. There is much to be said for Steiner's view. Having spent the past year writing a book on Japanese geometry, I was amazed at how difficult it was, after an adulthood of algebraic calculation, to relearn how to see. It is comforting to learn that Coxeter himself carried around large kaleidoscope mirrors as a crutch to help him visualize the geometries he intended to explore.

The conflict between algebra and geometry is as eternal as the battle of the sexes. Roberts's book bears the subtitle The Man Who Saved Geometry. As a rule, perjury in subtitles should be forgiven. The author herself points out that during this period geometry thrived in Russia, Italy and the Netherlands. Nor did Coxeter openly battle the Bourbaki; he merely continued to practice geometry and by force of results did much to keep the flame alive. In any case, Roberts has done a real service in revealing the struggle for supremacy between algebra and geometry. A full-scale study of the Bourbaki's influence on mathematics education would seem well worth some enterprising scholar's effort.

At the same time, all the above does raise the more general question of whether the history of science is well served by biographies. That Coxeter was admired, even adored, by his colleagues is not in doubt. At a recent symposium at Princeton University organized in conjunction with the book's publication, a pleasantly fruity bunch gathered at the astrophysics department to pay homage to the master. It was a 1960s day transplanted into the 2000s and the sort of occasion Princeton-like institutions should host more frequently. John Conway spoke of his "love" for Coxeter; others chronicled Coxeter's legions of fans, who sent him geometric models from around the world. Artists displayed unbelievably complex geometric sculptures and demonstrated crocheting as an effective representation of hyperbolic manifolds. Coxeter's students confirmed that he was gracious and generous with his time and ideas.

Nevertheless, in the book Coxeter comes across as a somewhat reserved character whose relationship with his children was severely strained. Throughout he is upstaged by cameo appearances of the supporting cast: by John von Neumann, who "always wore a suit one size too small" and who engaged in an ongoing contest with another mathematician in which each tried to catch the other working; by the inventor of "surreal" numbers, John Conway himself, who once built a periscope to see four dimensions, yet who amiably snapped, "None of your business! That's personal!" when asked how one indeed imagines four dimensions; by Ludwig Wittgenstein, who Coxeter notes "talked very interestingly about blindness and deafness, and why you get seasick on a camel but not on a horse," but who engaged Coxeter in "painful" discussions and gave 150-minute lectures that the mathematician found unintelligible; and by Bucky Fuller, whom Coxeter regarded as something of a crank, labeling Fuller's book Synergetics "a lot of nonsense," despite the fact that Fuller dedicated it to him.

Scientific biographies should perhaps be group portraits, for science is a collective endeavor, whereas a traditional biography invariably illuminates the contributions of the subject while leaving everyone else in shadows. Several pages in King of Infinite Space are devoted to Coxeter's television demonstration of non-Euclidean geometry in which he draws triangles on a globe. This is quite a standard exercise, and Riemann, who pioneered spherical geometry a century earlier, goes unmentioned. János Bolyai, who developed non-Euclidean geometry, receives due credit, whereas Nikolai Lobachevsky, who independently developed such a system and published it first, escapes notice. Much discussion is given over to Frederick Soddy, who in 1936 became the third or fourth person to rediscover the Descartes Circle Theorem, and who published his results as a poem that included a generalization to spheres, although the same result and an essentially identical proof had been produced in Japan more than a century earlier. And although Bucky Fuller surfaces prominently, the fact is that the first geodesic dome was built by the Zeiss Optical Company shortly after World War I to house their new planetarium projector.

In science, surely, the ideas and currents of thought should be the protagonists. What is quite interesting here, for example, is Coxeter's interaction with artists, in particular M. C. Escher, whose art was sometimes inspired by Coxeter's work (although Escher confessed he couldn't understand anything of Coxeter's letters) and who reciprocally served as the inspiration for several of Coxeter's mathematical papers. We often hear of artists who are inspired by science—the artists in attendance at the Princeton symposium are living examples—but less often do we hear of scientists inspired by art. Twenty years ago Linda Henderson in her bookThe Fourth Dimension and Non-Euclidean Geometry in Modern Art chronicled how the 19th-century craze of speculating about higher dimensions influenced literature (for example, Edwin Abbott's Flatland) and painting (the works of Marcel Duchamp, among others). Here Roberts provides a complementary demonstration of how Charles Howard Hinton, the prophet of four dimensions, and H. G. Wells's novel The Time Machine indisputably helped turn the young Coxeter toward his lifetime pursuits.

There is a lesson here. Even if Roberts's title character is a slightly wan hero, King of Infinite Space supplies ample grist for the mills of anyone pondering the state of modern mathematics, its sociology and its interaction with other pursuits.

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