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Monday, February 12, 2007

Reading List: King of Infinite Space

Roberts, Siobhan. King of Infinite Space. New York: Walker and Company, 2006. ISBN 0-8027-1499-4.
Mathematics is often said to be a game for the young. The Fields Medal, the most prestigious prize in mathematics, is restricted to candidates 40 years or younger. While many older mathematicians continue to make important contributions in writing books, teaching, administration, and organising and systematising topics, most work on the cutting edge is done by those in their twenties and thirties. The life and career of Donald Coxeter (1907–2003), the subject of this superb biography, is a stunning and inspiring counter-example. Coxeter's publications (all of which are listed in an appendix to this book) span a period of eighty years, with the last, a novel proof of Beecroft's theorem, completed just a few days before his death.

Coxeter was one of the last generation to be trained in classical geometry, and he continued to do original work and make striking discoveries in that field for decades after most other mathematicians had abandoned it as mined out or insufficiently rigorous, and it had disappeared from the curriculum not only at the university level but, to a great extent, in secondary schools as well. Coxeter worked in an intuitive, visual style, frequently making models, kaleidoscopes, and enriching his publications with numerous diagrams. Over the many decades his career spanned, mathematical research (at least in the West) seemed to be climbing an endless stairway toward ever greater abstraction and formalism, epitomised in the work of the Bourbaki group. (When the unthinkable happened and a diagram was included in a Bourbaki book, fittingly it was a Coxeter diagram.) Coxeter inspired an increasingly fervent group of followers who preferred to discover new structures and symmetry using the mind's powers of visualisation. Some, including Douglas Hofstadter (who contributed the foreword to this work) and John Horton Conway (who figures prominently in the text) were inspired by Coxeter to carry on his legacy. Coxeter's interactions with M. C. Escher and Buckminster Fuller are explored in two chapters, and illustrate how the purest of mathematics can both inspire and be enriched by art and architecture (or whatever it was that Fuller did, which Coxeter himself wasn't too sure about—on one occasion he walked out of a new-agey Fuller lecture, noting in his diary “Out, disgusted, after ¾ hour” [p. 178]).

When the “new math” craze took hold in the 1960s, Coxeter immediately saw it for the disaster it was to be become and involved himself in efforts to preserve the intuitive and visual in mathematics education. Unfortunately, the power of a fad promoted by purists is difficult to counter, and a generation and more paid the price of which Coxeter warned. There is an excellent discussion at the end of chapter 9 of the interplay between the intuitive and formalist approaches to mathematics. Many modern mathematicians seem to have forgotten that one proves theorems in order to demonstrate that the insights obtained by intuition are correct. Intuition without rigour can lead to error, but rigour without intuition can blind one to beautiful discoveries in the mathematical objects which stand behind the austere symbols on paper.

The main text of this 400 page book is only 257 pages. Eight appendices expand upon technical topics ranging from phyllotaxis to the quilting of toilet paper and include a complete bibliography of Coxeter's publications. (If you're intrigued by “Morley's Miracle”, a novel discovery in the plane geometry of triangles made as late as 1899, check out this page and Java applet which lets you play with it interactively. Curiously, a diagram of Morley's theorem appears on the cover of Coxeter's and Greitzer's Geometry Revisited, but is misdrawn—the trisectors are inexact and the inner triangle is therefore not equilateral.) Almost 90 pages of endnotes provide both source citations (including Web links to MathWorld for technical terms and the University of St. Andrews biographical archive for mathematicians named in the text) and detailed amplification of numerous details. There are a few typos and factual errors (for example, on p. 101 the planets Uranus and Pluto are said to have been discovered in the nineteenth century when, in fact, neither was: Herschel discovered Uranus in 1781 and Tombaugh Pluto in 1930), but none are central to the topic nor detract from this rewarding biography of an admirable and important mathematician.

Posted at February 12, 2007 22:57