Benoit B. Mandelbrot (20 November 1924 â€“ 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature. In 1936, at the age of 11, Mandelbrot emigrated with his family to France from Warsaw, Poland. After World War II ended, Mandelbrot studied mathematics, graduating from universities in Paris and in the United States and receiving a master's degree in aeronautics from the California Institute of Technology. He spent most of his career in both the United States and France, having dual French and American citizenship. In 1958, he began a 35-year career at IBM, where he became an IBM Fellow, and periodically took leaves of absence to teach at Harvard University. At Harvard, following the publication of his study of U.S. commodity markets in relation to cotton futures, he taught economics and applied sciences. Because of his access to IBM's computers, Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, leading to his discovery of the Mandelbrot set in 1980. He showed how visual complexity can be created from simple rules. He said that things typically considered to be "rough", a "mess", or "chaotic", such as clouds or shorelines, actually had a "degree of order". His math and geometry-centered research career included contributions to such fields as statistical physics, meteorology, hydrology, geomorphology, anatomy, taxonomy, neurology, linguistics, information technology, computer graphics, economics, geology, medicine, physical cosmology, engineering, chaos theory, econophysics, metallurgy, and the social sciences. Toward the end of his career, he was 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, UniversitÃ© Lille Nord de France, Institute for Advanced Study and Centre National de la Recherche Scientifique. During his career, he received over 15 honorary doctorates and served on many science journals, along with winning numerous awards. His autobiography, The Fractalist: Memoir of a Scientific Maverick, was published posthumously in 2012.

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The Mandelbrot set (black) within a continuously colored environment is the set of complex numbers for which the function__f_{c}(z)=z^{2}+c}{f_{c}(z)=z^{2}+c}__

does not diverge to infinity when iterated from {z=0}z=0, i.e., for which the sequence__{f_{c}(0)}{f_{c}(0)}, {f_{c}(f_{c}(0))}{f_{c}(f_{c}(0))}, etc.,__

remains bounded in absolute value.This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups.[3] Afterwards, in 1980, Benoit Mandelbrot obtained high quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York. Zooming into the boundary of the Mandelbrot setImages of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications; mathematically, one would say that the boundary of the Mandelbrot set is a fractal curve. The "style" of this recursive detail depends on the region of the set boundary being examined. Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point c, whether the sequence__
{f_{c}(0),f_{c}(f_{c}(0))}{f_{c}(0),f_{c}(f_{c}(0))}__

goes to infinity. Treating the real and imaginary parts of c as image coordinates on the complex plane, pixels may then be coloured according to how soon the sequencez
__{|f_{c}(0)|,|f_{c}(f_{c}(0))|}{|f_{c}(0)|,|f_{c}(f_{c}(0))|}__

crosses an arbitrarily chosen threshold (the threshold has to be at least 2, as -2 is the complex number with the largest magnitude within the set, but otherwise the threshold is arbitrary). If c is held constant and the initial value of z is varied instead, one obtains the corresponding Julia set for the point {\displaystyle c}c.The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of mathematical visualization, mathematical beauty, and motif.

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