Benoit Mandelbrot

Benoît Mandelbrot
Mandelbrot in 2007
Born	Warsaw, Poland
Died	15 October 2010(2010-10-15) (aged 85)
Nationality	French, American
Alma mater	École Polytechnique California Institute of Technology University of Paris
Known for	Mandelbrot set
Awards	Wolf Prize (1993) Japan Prize (2003)
Scientific career
Fields	Mathematician
Institutions	Yale University International Business Machines (IBM) Pacific Northwest National Laboratory
Doctoral students	F. Kenton Musgrave Eugene F. Fama among others

Benoît B. Mandelbrot (born 20 November 1924) is a French and American mathematician, best known as the father of fractal geometry. He is Sterling Professor of Mathematical Sciences, Emeritus at Yale University; IBM Fellow Emeritus at the Thomas J. Watson Research Center; and Battelle Fellow at the Pacific Northwest National Laboratory. Mandelbrot was born in Poland. His family moved to France when he was a child, and he was educated in France. He is a dual French and American citizen. Mandelbrot now lives and works in the United States.

Early years

Mandelbrot was born in Warsaw in a Jewish family from Lithuania. Anticipating the threat posed by Nazi Germany, the family fled from Poland to France in 1936 when he was 11. He remained in France through the war to near the end of his college studies. He was born into a family with a strong academic tradition—his mother was a medical doctor and he was introduced to mathematics by two uncles. His uncle, Szolem Mandelbrojt, was a Parisian mathematician. His father, however, made his living trading clothing. Mandelbrot attended the Lycée Rolin in Paris until the start of World War II, when his family moved to Tulle. He was helped by Rabbi David Feuerwerker, the Rabbi of Brive-la-Gaillarde, to continue his studies. In 1944 he returned to Paris. He studied at the Lycée du Parc in Lyon and in 1945-47 attended the École Polytechnique, where he studied under Gaston Julia and Paul Lévy. From 1947 to 1949 he studied aeronautics at California Institute of Technology. Back in France, he obtained a Ph.D. in Mathematical Sciences at the University of Paris in 1952.

From 1949 to 1958 Mandelbrot was a staff member at the Centre National de la Recherche Scientifique. During this time he spent a year at the Institute for Advanced Study in Princeton, New Jersey where he was sponsored by John von Neumann. In 1955 he married Aliette Kagan and moved to Geneva, Switzerland then Lille, France.

In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York. He remained at IBM for thirty-two years, becoming an IBM Fellow, and later Fellow Emeritus.

Academic career

From 1951 onward, Mandelbrot worked on problems and published papers not only in mathematics but in applied fields such as information theory, economics, and fluid dynamics. He became convinced that two key themes, fat tails and self-similar structure, ran through a multitude of problems encountered in those fields.

Mandelbrot found that price changes in financial markets did not follow a Gaussian distribution, but rather Lévy stable distributions having theoretically infinite variance. He found, for example, that cotton prices followed a Lévy stable distribution with parameter α equal to 1.7 rather than 2 as in a Gaussian distribution. "Stable" distributions have the property that the sum of many instances of a random variable follows the same distribution but with a larger scale parameter.

Mandelbrot also put his ideas to work in cosmology. He offered in 1974 a new explanation of Olbers' Paradox (the "dark night sky" riddle), demonstrating the consequences of fractal theory as a sufficient, but not necessary, resolution of the paradox. He postulated that if the stars in the universe were fractally distributed (for example, like Cantor dust), it would not be necessary to rely on the Big Bang theory to explain the paradox. His model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred.

In 1975, Mandelbrot coined the term fractal to describe these structures, and published his ideas in Les objets fractals, forme, hasard et dimension (1975; an English translation Fractals: Form, Chance and Dimension was published in 1977). Mandelbrot developed here ideas from the article Deux types fondamentaux de distribution statistique (1938; an English translation Two Basic Types of Statistical Distribution) of Czech geographer, demographer and statistician Jaromír Korčák.

The Mandelbrot set and periodicities of orbits.

While on secondment as Visiting Professor of Mathematics at Harvard University in 1979, Mandelbrot began to study fractals called Julia sets that were invariant under certain transformations of the complex plane. Building on previous work by Gaston Julia and Pierre Fatou, Mandelbrot used a computer to plot images of the Julia sets of the formula z² − μ. While investigating how the topology of these Julia sets depended on the complex parameter μ he studied the Mandelbrot set fractal that is now named after him. (Note that the Mandelbrot set is now usually defined in terms of the formula z² + c, so Mandelbrot's early plots in terms of the earlier parameter μ are left–right mirror images of more recent plots in terms of the parameter c.)

In 1982, Mandelbrot expanded and updated his ideas in The Fractal Geometry of Nature. This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "program artifacts".

Upon his retirement from IBM in 1987, Mandelbrot joined the Yale Department of Mathematics. At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences. His awards include the Wolf Prize for Physics in 1993, the Lewis Fry Richardson Prize of the European Geophysical Society in 2000, the Japan Prize in 2003, and the Einstein Lectureship of the American Mathematical Society in 2006. The small asteroid 27500 Mandelbrot was named in his honour. In November 1990, he was made a Knight in the French Legion of Honour. In December 2005, Mandelbrot was appointed to the position of Battelle Fellow at the Pacific Northwest National Laboratory. Mandelbrot was promoted to Officer of the French Legion of Honour in January 2006. An honorary degree from Johns Hopkins University was bestowed on Mandelbrot in the May 2010 commencement exercises.

Fractals and regular roughness

Although Mandelbrot coined the term fractal, some of the mathematical objects he presented in The Fractal Geometry of Nature had been described by other mathematicians. Before Mandelbrot, they had been regarded as isolated curiosities with unnatural and non-intuitive properties. Mandelbrot brought these objects together for the first time and turned them into essential tools for the long-stalled effort to extend the scope of science to non-smooth objects in the real world. He highlighted their common properties, such as self-similarity (linear, non-linear, or statistical), scale invariance, and a (usually) non-integer Hausdorff dimension.

He also emphasized the use of fractals as realistic and useful models of many "rough" phenomena in the real world. Natural fractals include the shapes of mountains, coastlines and river basins; the structures of plants, blood vessels and lungs; the clustering of galaxies; and Brownian motion. Fractals are found in human pursuits, such as music, painting, architecture, and stock market prices. Mandelbrot believed that fractals, far from being unnatural, were in many ways more intuitive and natural than the artificially smooth objects of traditional Euclidean geometry:

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
  —Mandelbrot, in his introduction to The Fractal Geometry of Nature

Fractal geometry is useful to accurately describe the development and resulting shape of many growth processes evident in nature, both organic and inorganic. Mandelbrot's work has changed the way researchers in many fields both perceive and characterize the phenomenon of natural growth.

File:Maple Tree Fractal Branch 1.jpg

A limb of a maple tree, illustrating organic fractal branching.

File:Frost Water crystal on Mercury 20Feb2010 cu2.jpg

Natural water frost crystal growth on cold glass, showing fractal branching growth in a purely physical system.

Mandelbrot has been called a visionary and a maverick. His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made The Fractal Geometry of Nature accessible to non-specialists. The book sparked widespread popular interest in fractals and contributed to chaos theory and other fields of science and mathematics.

Views of the Mandelbrot set. Each successive image is a magnification of the previous image.

Honours and awards

A partial list of awards received by Mandelbrot:

2004 Best Business Book of the Year Award AMS Einstein Lectureship Barnard Medal Caltech Service Casimir Frank Natural Sciences Award Charles Proteus Steinmetz Medal Franklin Medal

Harvey Prize Honda Prize Humboldt Preis IBM Fellowship Japan Prize John Scott Award Lewis Fry Richardson Medal

Medaglia della Presidenza della Repubblica Italiana Médaille de Vermeil de la Ville de Paris Nevada Prize Science for Art Sven Berggren-Priset Władysław Orlicz Prize Wolf Foundation Prize for Physics

• Member of the Norwegian Academy of Science and Letters.

See also

• "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension", a 1967 paper by Mandelbrot
• Mandelbrot Competition
• Skewness risk and Kurtosis risk

Further reading

• The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward, by Benoît Mandelbrot and Richard L. Hudson; Basic Books, 2004; ISBN 0-465-04355-0
• A focus on the exceptions that prove the rule, by Benoît Mandelbrot and Nassim Taleb; Financial Times, 23 March 2006.
• Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe and Cornelia Zahlten: Fractals: An Animated Discussion (63 min video film, interviews with Benoît Mandelbrot and Edward Lorenz, computer animations), W.H. Freeman and Company, 1990. ISBN 0-716-72213-5 (re-published by Films for the Humanities & Sciences, ISBN 978-0-7365-0520-8)
• A Multifractal Walk down Wall Steet by Benoit Mandelbrot, Scientific American, February 1999 [1]
• "Hunting the Hidden Dimension: mysteriously beautiful fractals are shaking up the world of mathematics and deepening our understanding of nature", NOVA, WGBH TV, PBS, October 28, 2008.
• The Fractal Geometry of Nature, by Benoît Mandelbrot; W. H. Freeman, 1983; ISBN 0716711869

External links

• Mandelbrot's page at Yale
• Interview of the École Polytechnique site
• Biography of Mandelbrot
• Video stream of Mandelbrot lecturing at MIT
• "Mandelbrot Set", a song by Jonathan Coulton (video)
• O'Connor, John J.; Robertson, Edmund F., "Benoit Mandelbrot", MacTutor History of Mathematics Archive, University of St Andrews
• Benoit Mandelbrot at the Mathematics Genealogy Project
• Template:Peoples Archive
• Who Discovered the Mandelbrot Set?
• Finance & Mandelbrot (Facebook Group)
• Benoit Mandelbrot video interview on Big Think
• "Benoit Mandelbrot: Fractals and the art of roughness", at TED2010