Tribute to Benoît B. Mandelbrot

French American mathematician Benoît B. Mandelbrot was born 20 November 1924. Born in Poland, he moved to France with his family when he was a child. Mandelbrot spent much of his life living and working in the United States, and he acquired dual French and American citizenship. Mandelbrot worked on a wide range of mathematical problems, including mathematical physics and quantitative finance, but is best known as the popularizer of fractal geometry. He coined the term fractal and described the Mandelbrot set. Mandelbrot also wrote books and gave lectures aimed at the general public. Mandelbrot spent most of his career at IBM’s Thomas J. Watson Research Center, and was appointed as an IBM Fellow. He later became a Sterling Professor of Mathematical Sciences at Yale University, where he was the oldest professor in Yale’s history to receive tenure. Mandelbrot also held positions at the Pacific Northwest National Laboratory, Université Lille Nord de France, Institute for Advanced Study and Centre National de la Recherche Scientifique.

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 situation 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 Fractals: Form, Chance and Dimension.

While 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”.

Mandelbrot left IBM in 1987, after 35 years and 12 days, when IBM decided to end pure research in his division. He joined the Department of Mathematics at Yale, and obtained his first tenured post in 1999, at the age of 75. 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 honor. 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 Legion of Honour in January 2006. An honorary degree from Johns Hopkins University was bestowed on Mandelbrot in the May 2010 commencement exercises.

Although Mandelbrot coined the term fractal, some of the mathematical objects he presented in The Fractal Geometry of Nature had been previously described by other mathematicians. Before Mandelbrot, they had often 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, art, 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.

Mandelbrot has been called a visionary and a maverick. His informed & passionate style of writing and his emphasis on visual and geometric intuition (supported bythe 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.When visiting the Museu de la Ciència de Barcelona in 1988, he told its director that the painting The Face of War had given him “the intuition about the transcendence of the fractal geometry when making intelligible the omnipresent similitude in the forms of nature”. He also said that, fractally, Gaudí was superior to Van der Rohe. Sadly Mandelbrot died in a hospice in Cambridge, Massachusetts, on 14th October 2010 from pancreatic cancer, at the age of 85. Reacting to news of his death, mathematician Heinz-Otto Peitgen said “if we talk about impact inside mathematics, and applications in the sciences,he is one of the most important figures of the last 50 years

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