Benoît Mandelbrot

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Benoît Mandelbrot in 2006

Benoît B. Mandelbrot (born November 20, 1924) is a mathematician, best known as the "father of fractal geometry".

Benoît Mandelbrot was born in Poland, but his family moved to France when he was a child; he is a French citizen and was educated in France. Mandelbrot now lives and works in the United States. He is a 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.

[edit] Early years

Mandelbrot was born in Warsaw, Poland to a Lithuanian-Jewish family. Anticipating political developments, the family fled Poland to France in 1936 when he was 11. He remained there 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. One, Szolem Mandelbrojt, was a famous 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. In 1944 he returned to Paris and in 1945-47 attended the École Polytechnique, where he studied under Gaston Julia and Paul Lévy. Then he spent two years at the California Institute of Technology where he studied aeronautics. Back in France, he obtained a Ph.D. in Mathematical Sciences at the University of Paris in 1952.

From 1949 to 1957 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 then Lille.

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.

[edit] Later years

Mandelbrot and the Mandelbrot set

From 1951 onwards Mandelbrot worked on problems and published papers not only in mathematics but also in real-world fields including information theory, economics and fluid dynamics. He became convinced that two key themes, fat tails and self-similar structure, ran through a multitude of these problems.

Mandelbrot found that price changes in financial markets did not follow a Gaussian distribution, but rather other 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^[1].

In 1975 Mandelbrot coined the term fractal to describe these structures, and published his ideas in Les objets fractals, forme, hasard et dimension (translated into English as Fractals: Form, chance and dimension^[2]) in 1977.

In 1979, while on secondment as Visiting Professor of Mathematics at Harvard University, 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 discovered 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^[3]. This influential work brought fractals into the mainstream of both professional and popular mathematics.

On 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 asteroid 27500 Mandelbrot was named in his honour.

In December 2005, Mandelbrot was appointed to the position of Battelle Fellow at the Pacific Northwest National Laboratory^[4].

On November 23, 1990, he was made a knight in the French Legion of Honor and promoted to officer on January 1, 2006.^[5]

[edit] Mandelbrot, fractals, and the new theme of regular roughness

Benoît Mandelbrot was the first to use a computer to plot the Mandelbrot set and to phrase key conjectures concerning its very complex structure. The most basic one remains unproven.

Another view of the Mandelbrot set.

Although Mandelbrot invented the word fractal, some objects featured in The Fractal Geometry of Nature had been previously described by other mathematicians (the Mandelbrot set being a notable exception). However, 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 around into essential tools for the long-stalled effort of extending the scope of science to non-smooth parts of the real world. He highlighted their common properties, such as self-similarity (linear, non-linear, or statistical), scale invariance and (usually) non-integer Hausdorff dimension.

He also emphasized the use of fractals as realistic and useful models of many phenomena in the real world that can be viewed as rough. Natural fractals include the shapes of mountains, coastlines and river basins; the structure of plants, blood vessels and lungs; the clustering of galaxies; Brownian motion. Man-made fractals include stock market prices but also music, painting and architecture. Far from being unnatural, Mandelbrot held the view that fractals were, in many ways, more intuitive and natural than the artificially smooth objects of traditional Euclidean geometry. As he says in the Introduction to The Fractal Geometry of Nature:

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 has been called "a living legend" and "a visionary" ^{[citation needed]}. 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. It sparked a widespread popular interest in fractals as well as contributing to chaos theory and other fields of science and mathematics.

[edit] Pronunciation

Benoît is read as "ben-wa" [bənwa]. The pronunciation of the name "Mandelbrot", which is a Yiddish and German word meaning "almond bread", is given variously in dictionaries. The Merriam-Webster Collegiate Dictionary and the Longman Pronouncing Dictionary give [ˈmæn.dəlˌbɹoʊt] (first syllable sounds like "man"; last syllable rhymes with "boat"); the Bollard Pronouncing Dictionary of Proper Names gives the quasi-French pronunciation [ˈmæn.dəlˌbɹɔː] (last syllable rhymes with "draw"); the American Heritage Dictionary gives [ˈmɑːn.dəlˌbɹɑt] (first syllable has the vowel sound of the 'a' in "father"; last syllable rhymes with "pot").

Mandelbrot himself, as most Frenchmen do, pronounces his name as [mɑ̃dɛlbʀot] when speaking in French.^[6]

[edit] See also

• How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
• Mandelbrot Competition

[edit] External links

Wikiquote has a collection of quotations related to:

Benoît Mandelbrot

• Mandelbrot's page at Yale
• Yale Economic Review - Review of The (mis)Behavior of Markets
• Interview of the École Polytechnique site
• Biography of Mandelbrot
• Video stream of Mandelbrot lecturing at MIT
• O'Connor, John J., and Edmund F. Robertson. "Benoît Mandelbrot". MacTutor History of Mathematics archive.
• Benoît Mandelbrot at the Mathematics Genealogy Project

[edit] References

1. ^ New Scientist, 19 April, 1997
2. ^ Fractals: Form, Chance and Dimension, by Benoît Mandelbrot; W H Freeman and Co, 1977; ISBN 0-716-70473-0
3. ^ The Fractal Geometry of Nature, by Benoît Mandelbrot; W H Freeman & Co, 1982; ISBN 0-716-71186-9
4. ^ PNNL press release: Mandelbrot joins Pacific Northwest National Laboratory
5. ^ Légion d'honneur announcement of promotion of Mandelbrot to officier
6. ^ Recording of the September 11, 2006, ceremony during which senator Pierre Laffitte presented the Officer of the Legion of Honor insignia to Mandelbrot

[edit] 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
• Nel Mondo dei Frattali, Di Renzo Editore, Roma, 2005
• A focus on the exceptions that prove the rule, by Benoît Mandelbrot and Nassim Taleb; Financial Times, March 23, 2006. [1]

[edit] External links

• Video of Benoît Mandelbrot at the Peoples Archive