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
Belgium
stefaan.vanryssen@hogent.be
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’.