Variance

Benoît B. Mandelbrot is a mathematician, best known as the father of fractal geometry. He was born in Warsaw, 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.

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 stable distributions having theoretically infinite variance. He found, for example, that cotton prices followed a Levy stable distribution with parameter equal to 1.7 rather than 2 as in a Gaussian distribution.

He also put his ideas to work in cosmology. In 1974 he offered a new explanation of Olbers’ Paradox (the “dark night sky” riddle), which states that in an infinite universe, the night sky should blaze with the light of the stars that lie in all directions, even those far away. Mandelbrot postulated that if the stars in the universe were fractally distributed, 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.

Although Mandelbrot coined the term fractal, some of the mathematical objects he presented 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 to extend the scope of science to non smooth objects in the real world. He highlighted their common properties, such as self similarity and scale variance.

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. 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.

Mandelbrot has been called a visionary. His informal and passionate style of writing and his emphasis on visual and geometric intuition made his publication 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.

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