Mathematician’s life detailed, but ideas remain opaque

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This article was published 19/11/2006 (7183 days ago), so information in it may no longer be current.

The Man Who Saved Geometry

By Siobhan Roberts

House of Anansi, 399 pages, $37

Reviewed by Neil Schipper

DONALD Coxeter was a mathematician who spent most of his working life at the University of Toronto. He has been called the 20th-century’s greatest geometer.

Born in London in 1907, and fascinated from a young age by shapes and patterns, he devoted his life to their study, and made discoveries that offered insights to seemingly unconnected areas both within and beyond pure mathematics.

His highly influential books and papers are praised for their unusual elegance, economy of expression, and intellectual beauty. These books have become the 20th-century analog to Euclid’s Elements.

The biography of this remarkable man, King of Infinite Space, comes to us by Canadian journalist Siobhan Roberts in her first full-length book.

Coxeter’s parents were a well-to-do surgical equipment manufacturer, also a hobby sculptor, and his artist wife. His youth, however, was not without distress. The breakup of his parents’ marriage and several years at a boarding school with a cruel headmaster and bullying boys were agonizing for the slender, sensitive boy.

Coxeter’s schooling left him so unprepared for college admissions exams in mathematics that he required two years of tutoring. His advanced study took place at both Cambridge and Princeton. In 1936, he accepted an offer by the U of T, considered at the time something of a disappointment.

Coxeter had a stunningly long career. He died in 2003 at the age of 96, remaining mathematically active until very near the end, having delivered an original paper in 2002.

Modern geometers examine two-dimensional polygons (triangles, squares and such), three-dimensional polyhedra (like the tetrahedron and cube), and polytopes, which exist in four dimensions and higher.

Polytopes are not only not constructible in the world, they aren’t even visualizable in the normal sense. However, with no small amount of deep reflection, like imagining how a “slice” of a polytope should “project down” to a three- dimensional figure, geometers have discovered many polytopes, and enumerated and classified their properties.

Roberts devotes considerable space to the role Coxeter played in restoring the dignity of classical geometry. This is a branch of mathematics that by the early- and mid-20th century had come to be considered as something suitable for weekend amusement, and not an area where a mathematician might expect to find something newly true and truly new.

The very approach of reasoning through visualization had become suspect as fraught with unreliability. Analytic approaches based on algebraic manipulation had become dominant.

Coxeter restored balance to the relationship between these two perspectives. Publishing a string of masterful works, he devised ways of describing symmetries that helped translate between algebraic and geometric treatments of certain problems.

Over time, they came to be revered for their beauty and their utility. Contributions to non-academic popular mathematics books further enhanced his reputation.

Among those who credit Coxeter for having a significant impact on their own development are Benoit Mandelbrot, Buckminister Fuller, Freeman Dyson and Douglas Hofstadter. The latter provides an especially enjoyable foreword to Roberts’ book.

Roberts deserves credit for attempting to bring Coxeter’s story to a wider public. Her writing maintains a crisp, journalistic gait as she weaves brief math-historical vignettes into the biographical narrative.

However, she offers little genuine mathematical explication, and it is not altogether compensated for by the many quotations from accomplished mathematicians and scientists, sincerely worshipful though they may be, nor by material that’s only vaguely relevant (such as the origin of Rockefeller Fellowships, or the search for Carbon-60 with which Coxeter had no clear involvement).

The voluminous footnoting (70 pages of endnotes for some 250 pages of text) suggests a certain misplaced thoroughness. The story of the man’s life is told in fine detail, but the ideas of the man remain largely opaque.

Winnipegger Neil Schipper has worked as an R & D engineer and as a math and science teacher.