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King of Infinite Space. Donald Coxeter, the Man Who Saved Geometry

by Siobhan Roberts
Walker & Co, New York, 2006
416 pp., illus. b/w, Trade, $27.95
ISBN: 0-8027-1499-4.

Reviewed by Stefaan Van Ryssen
Hogeschool Gent


Anyone who has dabbled in group theory, even as an amateur, will have dealt with Coxeter diagrams, concise and very abstract symbols summarising the symmetry properties of groups in any number of dimensions. Few people will know how these nice little drawings, consisting of nothing but a series of dots connected by numbered lines, developed as shorthand for the structure of kaleidoscopes. And even fewer will know anything about the man who developed them: H.M.S. Coxeter.

Coxeter’s life spanned practically the entire Twentieth Century (1907 — 2003). Starting as a precocious middle-class British boy with an inflated imagination — inventing his own language and mythology long before Tolkien started dreaming about Ents and Orks — and an interest in the fourth dimension, he went on to read mathematics at Cambridge, become a Fellow of Trinity and moved to Toronto, where he would spend most of his long life as a professor in and a staunch defender of the most unfashionable branch of maths: pure geometry. Against the tide of formalisation and holding his position in the surf of growing interest in algebra, group and number theory, topology and analytic geometry, Coxeter safeguarded the Euclidean tradition. When practically every mathematician was under the spell of the extreme abstractions of the Bourbaki group, he stuck to visualisation, diagrams, lines and planes, regular solids and n-dimensional polytopes, gradually finding deeper and deeper insights in the basic structure of spaces and shapes. Of course, in the long run, his knowledge of symmetry in a purely geometrical sense turned out to be closely related to symmetries in any other branch of maths, but that was only recognised when most of his work had already been done and when ‘Coxeter’ had become a household name for certain classes of mathematical objects.

Siobhan Roberts chose not to talk too much about maths when writing this biography of a man whose life was devoted to mathematics. Instead of really delving rather too deeply into his contributions to geometry, she sticks to the unharmful eccentricities, the momentous events and a few personal anecdotes. Mathematician might prefer to read more of a systematic exploration of Coxeter’s contributions to their field, and avid readers of biography might want a bit more excitement, sensation and life, but neither of these is here. For the math, one should simply read Coxeter’s books, and for entertainment, the man’s life probably was not spectacular enough. Yes, he did meet Einstein and Von Neumann, Wittgenstein and Buckminster Fuller (he disliked his brashness but liked his domes). Yes, Coxeter was a lifelong vegetarian and pacifist with a tendency to believe in the Platonic equation between beauty and truth, but. . . . A biography of this great mathematician was certainly justifiable, and I will be the last to claim that it hasn’t been done well, but there is only one conclusion possible and that is that Coxeter’s life was more platonic than applied. If the man would have had a choice, he probably would have projected himself in four, or five, dimensions outside our own. But the book is a good read for a quiet evening or two, and it wets the appetite for more triangles, diagrams and ‘kissing circles’.




Updated 1st February 2007

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