*Review of*King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry

*by Siobhan Roberts; review was in the*Toronto Star

*in fall 2006.*

*King of Infinite Space*is a book about geometry, so it’s fitting that it’s a little oddly shaped itself. Nominally a biography of a famous Toronto mathematician, the book is equally concerned with geometry’s larger role in math, science, and art. Ostensibly aimed at a general readership, it’s back-loaded with 150 pages of appendices and endnotes. And ultimately it’s structured as not a life story but a series of interlocking subjects and episodes – evoking nothing so much as one of the complex theoretical figures that come up so often in the book’s pages.

Siobhan Roberts, the author, is a freelance journalist, and the book grew from a profile she wrote for

*Toronto Life*in 2003. The subject was Donald Coxeter, a well-known University of Toronto geometry professor who died that year at the age of 96. Coxeter was born in the U.K. and studied geometry at Cambridge University; in 1936 he immigrated to Toronto to accept a teaching post at U of T, and he remained there ever after, publishing books and papers, travelling to conferences, inspiring acolytes, and becoming a legend in the field.

The Coxeter that we glimpse in Roberts’ book is a mildly eccentric and intriguing figure, but his was not an especially dramatic life and this is far from an in-depth character study. Coxeter’s politics and social values are touched on a little, his relationships with his wife and two children somewhat less. Whatever personal problems or crises he may have had are all but ignored. (One comic high point is a throwaway list of the Coxeters’ grievances with successive maids in the 1930s and ’40s.) Despite its subtitle,

*King of Infinite Space*is too academic, cautious, and respectful in tone to really function as a biography.

Which is fine, because it’s clear that Roberts would rather talk about Coxeter’s work anyway. She writes with enthusiasm about his intellectual and aesthetic interest in symmetry and shapes and in diagrams and models, in a time when much of the mathematical establishment was hostile to visual aids. And she discusses at length a couple of his major legacies: Coxeter diagrams, which are a kind of shorthand for describing complex shapes using points, lines, and numbers, and Coxeter groups, which are groups of symmetrical shapes generated by reflection.

At least, I think I have those descriptions right. Enthusiasm or no, Roberts’ book is rather heavy going for those without much geometry background (i.e. me, admittedly, but presumably many other general readers too). Her passion for the subject is obvious, but at times I wished she had a little more of, say, Malcolm Gladwell’s gift for breaking down complicated insider concepts into graspable and enlightening outsider lingo. Roberts herself seems to tacitly address this shortcoming by liberally stacking the book with rather mushy testimonials from Coxeter’s colleagues and admirers. “Coxeter’s perspective and ideas are in the air we breathe,” says one younger geometer, Ravi Vakil. “It’s not that his ideas are used to solve problems, it’s that the fundamental problems grow out of his ideas. He’s the soil, part of the substrate, part of the building in which we work, in which we live.” This and similar passages seem designed to reassure readers that Coxeter is a towering giant even if we can’t get our heads around exactly why.

Still, Roberts pursues some tangents that will intrigue even the uninitiated. In the 1950s, Coxeter formed a friendship with the Dutch artist M.C. Escher, who was no math expert (“[Coxeter’s] hocus-pocus text is no use to me at all,” Escher complained) but managed to apply complex geometric principles to his drawings through sheer work and will. And although Coxeter was a pure mathematician, mainly concerned with investigation for its own sake, some of the most interesting parts of the book cover the way geometry intersects with other fields. The familiar problem of how best to stack spheroids, for example, came in handy for early efforts at electronic information transmittal. And the shapes of different proteins are relevant in designing drugs to combat disease.

Some important larger themes emerge, too: the declining position of geometry within the mathematical cosmos, and the declining interest in the visual within geometry. Bourbaki, a group of French mathematicians, was openly hostile to classical Euclidean geometry – the group was associated with the battle cry “death to triangles” – and mistrustful of Coxeter’s beloved visual teaching and learning, considering it inferior to pure logical reasoning. And Roberts notes that geometry’s struggle to hold the interest of the academy could have long-term consequences, as future scholars are forced to rediscover lost knowledge that their forebears already had.

*King of Infinite Space*rarely hits heights of urgency and approachability, but at times it’s quietly invigorating as it looks at the joys and rewards of the pursuit of knowledge.

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