- 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.
February 2007 