Cached from
http://www.burymenot.com/2006/10/random-shapes-tangents.html
The Mathematician
As Shape Shifter:
How Donald Coxeter reinvigorated geometry
By ROBERT OSSERMAN*
Wall Street Journal, November 25, 2006; Page P10
King of Infinite Space
By Siobhan Roberts
Walker, 399 pages, $27.95
In 1923, at age 16, a high-school student in England named Donald
(H.S.M.) Coxeter seemed well on his way to becoming a composer. He had
written a variety of musical pieces, including songs, a string quartet
and something called "Devil Music" -- incidental music for "Magic," a
play by G.K. Chesterton. His mother thought to obtain an independent
judgment, however. She managed to arrange a visit to the composer
Gustav Holst, who was less than encouraging.
Fortunately, Donald
had another interest -- mathematics. He had written an essay for a math
class titled "Dimensional Analogy," in which he extended the familiar
elements of high-school geometry to a fourth dimension. The kind of
question he asked was this: If we were four-dimensional creatures
wanting to build a four-dimensional house, what would four-dimensional
bricks look like and how would they fit together?
Many a young
teenager with a flair for mathematics has felt the pull of the
mystical-sounding fourth dimension and has stumbled toward some rough
idea of the forms there. The reasoning usually goes this way:
Two-dimensional "bricks" are simply flat rectangles with which one can
tile a patio. Each rectangle, or "brick," has four edges. A
three-dimensional brick has six faces, each of them a rectangle. And
the fourth-dimensional version? A four-dimensional "brick" should have
eight "faces," each of which is a normal three-dimensional brick. What
distinguished Donald Coxeter's youthful effort was its astonishing
sophistication.
Not surprisingly, what began as a teenage
passion became a life's work. Over several decades, Coxeter (who moved
to Princeton, N.J., and then to Toronto after graduating from
Cambridge) investigated the higher-dimensional "bricks" and other
shapes that evolve when you start with various multi-sided polygons
instead of rectangles. Seven of his first eight research papers in the
1920s and '30s described these higher-dimensional shapes, or
"polytopes." Two of his best-known books did, too.
The title of
Siobhan Roberts's book, "King of Infinite Space," comes from a passage
in "Hamlet" cited in one of his books by Coxeter himself: "I could be
bounded in a nutshell and count myself a king of infinite space."
Neither Coxeter nor Ms. Roberts cares to complete Hamlet's thought: "
-- were it not that I have bad dreams." Among Coxeter's bad dreams --
no doubt disturbing the peace of his cerebral kingdom -- was the
indifference of his peers, even their scorn. Solomon Lefschetz, a
leading mathematician at Princeton in the 1930s (and '40s), was
referring to Coxeter's polytopes when he said: "It's good to talk about
trivial things occasionally."
But Coxeter won out in the end.
Somewhat in the manner of Barbara McClintock, whose basic work on the
genetics of maize was done in the 1930s but only recognized properly
decades later (she won the Nobel Prize in 1983), Coxeter worked
steadily, well into his 90s, and became more and more famous every
year. He died in 2003, at 96.
The study of polytopes is now
known to be useful in a number of applications, in particular in
business management and economics. The principles behind their
formation extend into other fields as well, including physics, where
"group theory" -- to which Coxeter was led through his study of the
simple kaleidoscope -- plays an important part. Advanced computer
graphics in the last two decades of the 20th century have allowed all
to see Coxeter-ian visions that until then had existed only in the
mind's eye.
Among much else, "King of Infinite Space" shows how
style and fashion occasionally govern the study of mathematics.
Coxeter's own style was highly visual and geometry-based. That tended
to make him a maverick in the mid-20th century, when he was doing his
major work. At that time, the mainstream was best exemplified by a
group of mathematicians who called themselves, collectively, Bourbaki.
Their ambitious goal was to develop a unified approach to all the
principal areas of modern mathematics through a series of interlocked
volumes, starting with elementary set theory and branching into
algebra, topology, geometry and so forth. Their principal spokesman,
Jean Dieudonné, was in many ways Coxeter's opposite. He wrote that
"current mathematics has inevitably become the study of very general
abstract structures." Coxeter's approach was, by contrast, often
enumerative, in the sense of working out specific examples in great
detail.
Ms. Roberts seeks to portray the David of Coxeter
slaying the Goliath of Bourbaki -- or, if not exactly slaying him, then
at least putting up a good fight and emerging with his head held high.
There is a certain amount of truth to this, but it oversimplifies
mathematics into a war of dualisms: geometry vs. algebra, or concrete
vs. abstract. In fact, mathematics is far more varied, including
categories that amount to "none of the above" or "both." Geometry
itself contains many dimensions, of which the Coxeter variety is only
one.
That said, geometry did almost disappear from the
curriculum and from the faculties of the leading research universities
during the heyday of Bourbaki. Coxeter kept the flame alive. He was an
excellent expositor -- his "Introduction to Geometry" is a classic of
clarity and rigor and full of fascinating sidelights. By the late
1970s, geometry was making a major comeback, and Coxeter clearly
deserves some of the credit. Ms. Roberts reproduces the cover of the
January 1980 issue of a mathematical journal depicting a skeleton
clothed in black, holding a sheet with a geometric drawing and
accompanied by the caption: "Is Geometry Dead?" She does not mention
that an issue of the same journal, the following year, printed a
coverline reading: "Geometry Lives!"
That issue included an
article by none other than Jean Dieudonné singing the praises of
geometry as well as another article noting that a new generation of
brilliant, young, un-Bourbakian geometers was then emerging. This group
included future Fields Medalists William Thurston and S.T. Yau, both of
whom continue to do important work today. Both played a role in the
recent solution of the 100-year-old Poincaré conjecture.
"King
of Infinite Space" can be enjoyed even without a specialized knowledge
of geometry or math. (Ms. Roberts's own exposition is admirably clear
and conscientiously footnoted.) And the book's narrative is heartening.
Too often -- think of "A Beautiful Mind" or "Proof" -- mathematicians
are portrayed these days as seriously disturbed or weirdly obsessed or
burnt out at an early age. Here, by contrast, is the true story of an
eminent mathematician, active, alert, acute and ever alive to new ideas
over a period of 80 years.
* Mr. Osserman, special projects
director of the Mathematical Sciences Research Institute in Berkeley,
Calif., is the author of "Poetry of the Universe: A Mathematical
Exploration of the Cosmos."