Fractal visionary dies: Benoit Mandelbrot, 1924-2010

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Fractal visionary dies: Benoit Mandelbrot, 1924-2010
By Chris Talbot
4 November 2010

School students throughout the world, if they have access to personal computers,
will have probably been given programmes that produce beautiful and complex
pictures called fractals. A simple Internet search will turn up thousands of
computer images, discussion articles and mathematical explanations of fractals.
The most famous of them, the image called the Mandelbrot set, was named after
the man who carried out investigations in the 1960s, 1970s and 1980s at the IBM
laboratories at Yorktown Heights in New York State.

Benoit Mandelbrot coined the word “fractal” in 1975 to describe the
revolutionary approach to geometrical mathematics that he pioneered. From the
time of the ancient Greeks, geometry, associated with the name Euclid, had
studied a restricted range of objects—lines, circles, planes, cylinders and so
on. In the nineteenth century, under the impact of new developments in physics,
especially electricity and magnetism, mathematician scientists like Carl
Friedrich Gauss and Bernhard Riemann extended geometry beyond Euclid into higher
dimensions. They examined properties such as curvature that were often far
beyond our daily experience. This was the type of geometry that Einstein could
apply to his theories of relativity. Nevertheless, even these new geometries
relate to things that are smooth and even “tame” in some sense.

Mandelbrot began looking at many different branches of science—turbulence in
fluid mechanics and meteorology, price fluctuations in economics, the growth of
cells in physiology, the clustering of galaxies, the shape of trees and
plants—and realised that traditional smooth geometry was often at best a crude
approximation. Even the scientific understanding of causal or regular laws of
extension and motion that were combined with or expressed through chance
fluctuations or “noise”, the basis of most branches of science, did not satisfy
Mandelbrot.

Mandelbrot wanted to show that what often appeared to be wild or random
fluctuations that did not obey the laws of statistics could in fact be explained
by simple laws or rules. His fundamental idea was that of “self similarity.” To
take a famous example, when you examine a cauliflower, cut it in pieces and
magnify a piece, you find it has a similar structure to the original
cauliflower. Cut the piece up again and magnify the new tiny pieces, and they
have the same structure again. This approach enables a geometry to be applied to
things where it did not previously seem possible—to clouds, coastlines,
mountains and so on.

Such an unusual and even iconoclastic view of the world can partly be explained
by Mandelbrot’s biography. He was born to a Lithuanian Jewish family, his father
a clothes merchant and mother a dentist. The Mandelbrots moved to Paris in 1936,
fearing the spread of Nazism. During the war they moved south to Tulle in
central France where Benoit had no formal education, living in constant threat
of poverty and capture.

Largely self-taught, but with educated parents and refugee teachers around him,
Mandelbrot developed a highly individual genius for geometrical mathematics.
After the war, he passed the examinations to the elite Ecole Polytechnic in
Paris and, after graduating, visited the California Institute of Technology. He
received a PhD at Paris, and then went to the Institute for Advanced Study at
Princeton under John von Neumann. In 1955 he returned to a post at the Centre
National de la Recherche Scientific in Paris.

Mandelbrot was deeply unsatisfied with the postwar state of mathematics. In
France, mathematics was dominated by Nicolas Bourbaki. This was not a single
person (the name was taken from a French general) but the collective pseudonym
for a group of top mathematicians whose names were kept secret. Bourbaki
produced a series of textbooks, mainly during the 1950s and 1960s that attempted
to give a foundation to the whole of mathematics that was rigorously based on
axioms, and was set out in the driest and most austere manner imaginable,
without a single diagram or picture. The number of mathematicians in the group
was fixed, and they had to retire at the age of 50.

Bourbaki was a reaction against the much more intuitive approach to mathematics,
stemming from Henri Poincare, that predominated in France earlier in the
century. It was very much in tune with a formalist or structuralist zeitgeist
that permeated many academic areas at this time. Not unreasonably, this outlook
was seen to be heavily influenced by Stalinism, even though many of its
mathematical practitioners had no connection with the Communist Party.

Mandelbrot, who always had a highly visual and experimental approach to
mathematics, was very unhappy in France in the 1950s. He complained that he
“loathed and feared” Bourbaki. He had a close personal connection with Bourbaki
since his uncle, Szolem, was one of the founders of the group when it was a club
of like-minded mathematicians in the 1930s. Szolem fled Paris during the war but
returned as a professor at the University of Paris after the war and worked on
the Bourbaki volumes. He told his nephew that “geometry has been dead for nearly
a century” [1] and suggested he could work instead on a largely forgotten area
of mathematical analysis in a paper by Gaston Julia, one of Poincare’s students.
Benoit was not impressed, though the knowledge he gained about iterative
sequences of complex numbers turned out to be central to fractal theory, and the
student of fractal pictures will recognise the name Julia.

Escaping the stifling atmosphere of Bourbaki, Mandelbrot moved permanently to
the US, taking a post at IBM’s research centre. This gave him the freedom to
research into many areas—linguistics, aeronautics, engineering, physiology,
economics, and so on, including also the history of science and the study of
recondite areas of mathematics. Even here he did not escape the prevailing
hostility to a materialist philosophy that had permeated mathematics and led
mathematicians to see their formalistic world as somehow above and apart from
reality. Mandelbrot takes a quote from Marshall Stone, then top mathematics
professor at Harvard:

“While several important changes have taken place since 1900 in our conception
of mathematics and in our points of view concerning it, the one which truly
involves a revolution in ideas is the discovery that mathematics is entirely
independent of the physical world….”

Mandelbrot strongly disagreed, insisting that the vitality of mathematics
depended on applications to the real world. At IBM, he was able to investigate
mathematics applied to many areas, “to retain permanently the ‘infantile
sensibility’ I enjoyed as an adolescent” he wrote. The remark was provoked by
the response of a mathematician who described his “mathematical sensibility” as
being “rather infantile and somewhat dull.” This reaction, explains Mandelbrot,
“echoes my uncle’s opinion.”

At IBM, Mandelbrot was able to use computers in a new way that has since become
central to many branches of science. “I made them [the computer and computer
graphics], not a tool to be called on only if needed, but a constant and
integral part of my process of thinking,” he wrote.

Many scientists reacted against Mandelbrot’s intrusion into the fields in which
they were experts. As James Gleick points out, “He was always an outsider,
taking an unorthodox approach to an unfashionable corner of mathematics,
exploring disciplines in which he was rarely welcomed, hiding his grandest ideas
in efforts to get his papers published, surviving mainly on the confidence of
his employers at Yorktown Heights.” [2] His defensive approach made other
scientists regard him as “prickly,” explains Gleick.

Over the decades, Mandelbrot’s ideas have become incorporated into a new
mathematical area called chaos theory and are now widely respected. It has to be
said, however, that so far fractals and chaos theory have relatively rarely been
directly applied to science or engineering, even when chaotic motion is observed
or fractal dimensions calculated. The mathematical rules used by Mandelbrot and
even those further developed in chaos theory are often too simple to be of
immediate application to real world situations. But computer modelling has
entered the mainstream of many branches of science, and consequently fractal or
chaotic phenomena can often be seen in computer graphics output. Their most
ubiquitous application is in computer games programming and film making.

Philosophically, Mandelbrot’s ideas, and the chaos theory that has developed
from them, help to challenge the widespread bias against determinism among
intellectuals. Determinism as applied to history is far richer and more complex
than determinism in physics. But even in physics, Newton’s laws of motion can be
applied to phenomenon such as the asteroid belt of our solar system and produce
chaotic results of enormous complexity. The equations that were developed by the
nineteenth century physicists Navier and Stokes, which are completely
deterministic, can now be seen to give rise to the chaotic motion seen in
turbulent fluid flow. Mandelbrot’s life and work has helped to widen and enrich
our understanding of determinism in ways that are only just beginning to be
realised.

[1] Mandelbrot’s Introduction to Fractals for the Classroom: Part One, by
Heinz-Otto Pietgen, Harmut Jürgens, Dietmar Saupe, Springer, 1992.

[2] Chaos, James Glieck, Penguin,1988.

http://wsws.org/articles/2010/nov2010/mand-n04.shtml

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