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The Mathematician
As Shape Shifter:
How Donald Coxeter reinvigorated geometry

By ROBERT OSSERMAN*
Wall Street Journal, November 25, 2006; Page P10

King of Infinite Space
By Siobhan Roberts
Walker, 399 pages, $27.95

In 1923, at age 16, a high-school student in England named Donald (H.S.M.) Coxeter seemed well on his way to becoming a composer. He had written a variety of musical pieces, including songs, a string quartet and something called "Devil Music" -- incidental music for "Magic," a play by G.K. Chesterton. His mother thought to obtain an independent judgment, however. She managed to arrange a visit to the composer Gustav Holst, who was less than encouraging.

Fortunately, Donald had another interest -- mathematics. He had written an essay for a math class titled "Dimensional Analogy," in which he extended the familiar elements of high-school geometry to a fourth dimension. The kind of question he asked was this: If we were four-dimensional creatures wanting to build a four-dimensional house, what would four-dimensional bricks look like and how would they fit together?

Many a young teenager with a flair for mathematics has felt the pull of the mystical-sounding fourth dimension and has stumbled toward some rough idea of the forms there. The reasoning usually goes this way: Two-dimensional "bricks" are simply flat rectangles with which one can tile a patio. Each rectangle, or "brick," has four edges. A three-dimensional brick has six faces, each of them a rectangle. And the fourth-dimensional version? A four-dimensional "brick" should have eight "faces," each of which is a normal three-dimensional brick. What distinguished Donald Coxeter's youthful effort was its astonishing sophistication.

Not surprisingly, what began as a teenage passion became a life's work. Over several decades, Coxeter (who moved to Princeton, N.J., and then to Toronto after graduating from Cambridge) investigated the higher-dimensional "bricks" and other shapes that evolve when you start with various multi-sided polygons instead of rectangles. Seven of his first eight research papers in the 1920s and '30s described these higher-dimensional shapes, or "polytopes." Two of his best-known books did, too.

The title of Siobhan Roberts's book, "King of Infinite Space," comes from a passage in "Hamlet" cited in one of his books by Coxeter himself: "I could be bounded in a nutshell and count myself a king of infinite space." Neither Coxeter nor Ms. Roberts cares to complete Hamlet's thought: " -- were it not that I have bad dreams." Among Coxeter's bad dreams -- no doubt disturbing the peace of his cerebral kingdom -- was the indifference of his peers, even their scorn. Solomon Lefschetz, a leading mathematician at Princeton in the 1930s (and '40s), was referring to Coxeter's polytopes when he said: "It's good to talk about trivial things occasionally."

But Coxeter won out in the end. Somewhat in the manner of Barbara McClintock, whose basic work on the genetics of maize was done in the 1930s but only recognized properly decades later (she won the Nobel Prize in 1983), Coxeter worked steadily, well into his 90s, and became more and more famous every year. He died in 2003, at 96.

The study of polytopes is now known to be useful in a number of applications, in particular in business management and economics. The principles behind their formation extend into other fields as well, including physics, where "group theory" -- to which Coxeter was led through his study of the simple kaleidoscope -- plays an important part. Advanced computer graphics in the last two decades of the 20th century have allowed all to see Coxeter-ian visions that until then had existed only in the mind's eye.

Among much else, "King of Infinite Space" shows how style and fashion occasionally govern the study of mathematics. Coxeter's own style was highly visual and geometry-based. That tended to make him a maverick in the mid-20th century, when he was doing his major work. At that time, the mainstream was best exemplified by a group of mathematicians who called themselves, collectively, Bourbaki. Their ambitious goal was to develop a unified approach to all the principal areas of modern mathematics through a series of interlocked volumes, starting with elementary set theory and branching into algebra, topology, geometry and so forth. Their principal spokesman, Jean Dieudonné, was in many ways Coxeter's opposite. He wrote that "current mathematics has inevitably become the study of very general abstract structures." Coxeter's approach was, by contrast, often enumerative, in the sense of working out specific examples in great detail.

Ms. Roberts seeks to portray the David of Coxeter slaying the Goliath of Bourbaki -- or, if not exactly slaying him, then at least putting up a good fight and emerging with his head held high. There is a certain amount of truth to this, but it oversimplifies mathematics into a war of dualisms: geometry vs. algebra, or concrete vs. abstract. In fact, mathematics is far more varied, including categories that amount to "none of the above" or "both." Geometry itself contains many dimensions, of which the Coxeter variety is only one.

That said, geometry did almost disappear from the curriculum and from the faculties of the leading research universities during the heyday of Bourbaki. Coxeter kept the flame alive. He was an excellent expositor -- his "Introduction to Geometry" is a classic of clarity and rigor and full of fascinating sidelights. By the late 1970s, geometry was making a major comeback, and Coxeter clearly deserves some of the credit. Ms. Roberts reproduces the cover of the January 1980 issue of a mathematical journal depicting a skeleton clothed in black, holding a sheet with a geometric drawing and accompanied by the caption: "Is Geometry Dead?" She does not mention that an issue of the same journal, the following year, printed a coverline reading: "Geometry Lives!"

That issue included an article by none other than Jean Dieudonné singing the praises of geometry as well as another article noting that a new generation of brilliant, young, un-Bourbakian geometers was then emerging. This group included future Fields Medalists William Thurston and S.T. Yau, both of whom continue to do important work today. Both played a role in the recent solution of the 100-year-old Poincaré conjecture.

"King of Infinite Space" can be enjoyed even without a specialized knowledge of geometry or math. (Ms. Roberts's own exposition is admirably clear and conscientiously footnoted.) And the book's narrative is heartening. Too often -- think of "A Beautiful Mind" or "Proof" -- mathematicians are portrayed these days as seriously disturbed or weirdly obsessed or burnt out at an early age. Here, by contrast, is the true story of an eminent mathematician, active, alert, acute and ever alive to new ideas over a period of 80 years.

* Mr. Osserman, special projects director of the Mathematical Sciences Research Institute in Berkeley, Calif., is the author of "Poetry of the Universe: A Mathematical Exploration of the Cosmos."