The Geometer as Hero

Review of the book King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry, by Siobhan Roberts


Jeffrey S. Rosenthal

(Published in the Globe and Mail, October 14, 2006)

The latest candidate for Canadian hero comes from an unlikely source: abstract mathematical research.

Donald Coxeter is widely regarded as the 20th century's leading expert on geometry. Thin and frail-looking despite his 50 push-ups a day, he inspired a curious affection within the normally staid academic community. By the 1980s, he was mostly retired, but we university math students whispered about occasional Coxeter sightings and lectures the way other youths stalked actors and rock stars.

Now Siobhan Roberts, a Toronto journalist, has taken on the considerable challenge of turning this insider story -- a mathematician admired by other mathematicians -- into a tale for the masses. The result is part biography, part scientific history and part epic.

King of Infinite Space describes Coxeter's humble beginnings in England, his rigorous studies and social awkwardness at Cambridge University in the 1920s, and his gradually increasing mathematical fame during his many years teaching at the University of Toronto. He emerges as a gentle, diffident, humble person with just one true passion: geometry.

Coxeter worked on "classical" geometry, studying the intricacies of shapes and angles, lines and vertices, polygons (triangles, squares and so on) and their higher-dimensional analogues, polytopes. He had a childlike fascination with everyday images: the cross-section of an apple core, flowers with five petals, homemade cardboard models, reflections through angled mirrors. To Coxeter, a picture really was worth a thousand words.

The book does a good job of hinting at Coxeter's most important mathematical discoveries in understandable ways. The famous "Coxeter groups" are nicely described as the many different reflections -- like a kaleidoscope -- that you would see through multiple intersecting mirrors. And the widely used "Coxeter diagrams" are presented as a form of shorthand, able to describe complicated high-dimensional polytopes with just a few lines and dots.

Roberts also nicely presents Coxeter's wide range of contacts and influence. He assisted an engineer with the design of an early computer modem, despite his distaste for computers. He exchanged friendly correspondence with the artist M. C. Escher, whose later drawings took inspiration from the diagrams in Coxeter's work (though Escher was baffled by Coxeter's "hocus-pocus" equations). He had a love-hate relationship with the architect and visionary Buckminster Fuller, after staring in awe at Fuller's geodesic dome at the 1967 Montreal Expo.

And his student days at Cambridge were enlivened by Alicia Boole Stott, the 68-year-old "housewife geometer" who had introduced the word polytope many years earlier, and who affectionately mentored the young Coxeter in his burgeoning geometric pursuits.

The book is at its best in discussing the surprising tensions among mathematical researchers about how mathematics should be studied.

At one extreme was Bourbaki, a shadowy group of French mathematicians dedicated to reducing all of mathematics to symbols and equations and logical deductions, with no room for the pictures and models and intuition that Coxeter held dear. Bourbaki inspired the formal New Math curriculum taught in elementary and high schools in the 1960s and 1970s.

Coxeter opposed this movement, declared that learning mathematics from a purely formal perspective was "not only regrettable, but unreasonable."

By bucking this trend in favour of more intuitive and concrete geometric constructions, Coxeter helped to save classical geometry from virtual extinction as a research subject. My own university was an unexpected casualty of Coxeter's strong feelings: He had planned to bequeath his Rosedale mansion to the University of Toronto, but when the institution reduced its emphasis on classical geometry, his will was quickly rewritten.

Mathematical debates don't excite everyone. As Coxeter's daughter Susan put it, while accompanying her aging father to a conference: "To think, we've come all this way to talk about circles touching circles when there are so many more important things going on in the world."

Fortunately, the book also occasionally offers poignant looks into Coxeter's soul. Coxeter confesses that he loved his childhood governess, who taught him division and quadratic equations, more than his own mother. As a shy young man, he was heartbroken on learning that a beautiful dance partner, despite her ringless fingers, was actually married (his mathematical reaction: "99.9 per cent of American married women wear a ring").

He later endures a sad marriage ceremony just days after his father's sudden, unexpected death. As an old man, he admits that he was so consumed by his scholarship that he "was not able to love Rien [his wife] as fully and completely as one should." These emotional moments touch us all, and I found myself wishing there were more of them.

Roberts also hints at Coxeter's social values. A lifelong vegetarian (for health reasons) and pacifist, he declined to serve as a code-breaker during the Second World War. In 1997, he petitioned against his university granting an honorary degree to U.S. president George H. W. Bush, later declaring that he thought the first president Bush was the worst thing in the world until the second President Bush came along.

The book is not without flaws. In Roberts's zeal for asides and background and context, the text is sometimes superfluous or repetitive. That Coxeter accepted a Rockefeller Foundation grant to spend a year at Princeton University is an important development in his life story, but a two-page aside on why John Rockefeller created this granting program in the first place is not. And, several events (such as Princeton mathematician Solomon Lefschetz dismissing Coxeter's research as "trivial") are each needlessly presented in two separate places.

The order of topics is occasionally awkward. For example, Bourbaki is presented as an actual person on page 13, and only revealed to be a clandestine collective on page 120. And a step-by-step account of Coxeter's final conference appearance, in Budapest in 2002, would have made a touching finale to a long life of highs and lows -- but as the book's first chapter, it will appeal only to those who are already fans.

The book is very scholarly, containing 74 pages of endnotes and eight appendices with such titles as "Schläfli Symbols of the 3-D and 4-D Regular Polytopes". It is based on a tremendous number of interviews with leading mathematicians and scholars, plus two years observing and conversing with Coxeter himself (the basis for her 2003 National Magazine Award-winning article in Toronto Life). The depth of detail is at times overwhelming, but is also oddly persuasive: If Roberts loved Coxeter enough to research him so meticulously, then perhaps we should love him too.

Coxeter the man died in 2003, at the age of 96. Thanks to Roberts's passionate writing, Coxeter the legend lives on.

Jeffrey S. Rosenthal is a professor in the department of statistics at the University of Toronto, and the author of Struck by Lightning: The Curious World of Probabilities. He receives his e-mail on a computer server named after Donald Coxeter.