Benoit Mandelbrot

Benoit B. Mandelbrot(20 November 1924 – 14 October 2010) was a Polish-born French-Americanmathematicianandpolymathwith broad interests in the practical sciences, especially regarding what he labeled as "the art ofroughness" 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 offractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness andself-similarity" in nature.
In 1936, at the age of 11, Mandelbrot and his family emigrated fromWarsaw, Poland, to France. AfterWorld War IIended, Mandelbrot studied mathematics, graduating from universities in Paris and in the United States and receiving a master's degree inaeronauticsfrom theCalifornia Institute of Technology. He spent most of his career in both the United States and France, havingdualFrenchandAmericancitizenship. In 1958, he began a 35-year career atIBM, where he became anIBM Fellow, and periodically took leaves of absence to teach atHarvard 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 theMandelbrot setin 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 included contributions to such fields asstatistical physics,meteorology,hydrology,geomorphology,anatomy,taxonomy,neurology,linguistics,information technology,computer graphics,economics,geology,medicine,physical cosmology,engineering,chaos theory,econophysics,metallurgy, and thesocial sciences.
Toward the end of his career, he wasSterling Professorof Mathematical Sciences atYale University, where he was the oldest professor in Yale's history to receive tenure.Mandelbrot also held positions at thePacific Northwest National Laboratory,Université Lille Nord de France,Institute for Advanced StudyandCentre 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.
Benedykt Mandelbrotwas born in aLithuanian Jewishfamily, inWarsawduring theSecond Polish Republic.His father made his living trading clothing; his mother was a dental surgeon. During his first two school years, he was tutored privately by an uncle who despisedrote learning: "Most of my time was spent playing chess, reading maps and learning how to open my eyes to everything around me."
In 1936, when he was 11, the family emigrated from Poland to France. The move,World War II, and the influence of his father's brother, the mathematicianSzolem Mandelbrojt(who had moved to Paris around 1920), further prevented a standard education. "The fact that my parents, as economic and political refugees, joined Szolem in France saved our lives," he writes.
Mandelbrot attended the Lycée Rollin (now theCollège-lycée Jacques-Decour) in Paris until the start ofWorld War II, when his family moved toTulle, France. He was helped byRabbiDavid Feuerwerker, the Rabbi ofBrive-la-Gaillarde, to continue his studies.Much of France was occupied by the Nazis at the time, and Mandelbrot recalls this period:
In 1944, Mandelbrot returned to Paris, studied at theLycée du ParcinLyon, and in 1945 to 1947 attended theÉcole Polytechnique, where he studied underGaston JuliaandPaul Lévy. From 1947 to 1949 he studied at California Institute of Technology, where he earned a master's degree in aeronautics.Returning to France, he obtained hisPhD degreein Mathematical Sciences at theUniversity of Parisin 1952.
From 1949 to 1958, Mandelbrot was a staff member at theCentre National de la Recherche Scientifique. During this time he spent a year at theInstitute for Advanced StudyinPrinceton, New Jersey, where he was sponsored byJohn von Neumann. In 1955 he married Aliette Kagan and moved toGeneva, Switzerland(to collaborate withJean Piagetat the International Centre for Genetic Epistemology) and later to theUniversité Lille Nord de France.In 1958 the couple moved to the United States where Mandelbrot joined the research staff at theIBMThomas J. Watson Research CenterinYorktown Heights, New York.He remained at IBM for 35 years, becoming an IBM Fellow, and later FellowEmeritus.
From 1951 onward, Mandelbrot worked on problems and published papers not only in mathematics but in applied fields such asinformation theory, economics, andfluid dynamics.
Mandelbrot sawfinancial marketsas an example of "wild randomness", characterized by concentration and long-range dependence. He developed several original approaches for modelling financial fluctuations.In his early work, he found that the price changes infinancial marketsdid not follow aGaussian distribution, but ratherLévystable distributionshaving infinitevariance. He found, for example, that cotton prices followed a Lévy stable distribution with parameterαequal 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 largerscale parameter.The latter work from the early 60s was done with daily data of cotton prices from 1900, long before he introduced the word 'fractal'. In later years, after the concept of fractals had matured, the study of financial markets in the context of fractals became possible only after the availability of high frequency data in finance.
In the late 1980s, Mandelbrot used intra-daily tick data supplied by Olsen & Associates in Zurichto apply fractal theory to market microstructure. This cooperation led to the publication of the first comprehensive papers on scaling law in finance.This law shows similar properties at different time scales, confirming Mandelbrot's insight of the fractal nature of market microstructure. Mandelbrot's own research in this area is presented in his booksFractals and Scaling in FinanceandThe (Mis)behavior of Markets.
As a visiting professor atHarvard University, Mandelbrot began to study mathematical objects calledJulia setsthat wereinvariantunder certain transformations of thecomplex plane. Building on previous work byGaston JuliaandPierre Fatou, Mandelbrot used a computer to plot images of the Julia sets. While investigating the topology of these Julia sets, he studied theMandelbrot setwhich was introduced by him in 1979.
In 1975, Mandelbrot coined the termfractalto describe these structures and first published his ideas in the French bookLes Objets Fractals: Forme, Hasard et Dimension, later translated in 1977 asFractals: Form, Chance and Dimension.According to computer scientist and physicistStephen Wolfram, the book was a "breakthrough" for Mandelbrot, who until then would typically "apply fairly straightforward mathematics ... to areas that had barely seen the light of serious mathematics before".Wolfram adds that as a result of this new research, he was no longer a "wandering scientist", and later called him "the father of fractals":
Wolfram briefly describes fractals as a form of geometric repetition, "in which smaller and smaller copies of a pattern are successively nested inside each other, so that the same intricate shapes appear no matter how much you zoom in to the whole.Fern leavesandRomanesque broccoliare two examples from nature."He points out an unexpected conclusion:
Mandelbrot used the term "fractal" as it derived from the Latin word "fractus", defined as broken or shattered glass. Using the newly developed IBM computers at his disposal, Mandelbrot was able to create fractal images using graphics computer code, images that an interviewer described as looking like "the delirious exuberance of the 1960spsychedelic artwith forms hauntingly reminiscent of nature and the human body". He also saw himself as a "would-be Kepler", after the 17th-century scientistJohannes Kepler, who calculated and described the orbits of the planets.
Mandelbrot, however, never felt he was inventing a new idea. He described his feelings in a documentary with science writer Arthur C. Clarke:
According to Clarke, "theMandelbrot setis indeed one of the most astonishing discoveries in the entire history of mathematics. Who could have dreamed that such an incredibly simple equation could have generated images of literallyinfinitecomplexity?" Clarke also notes an "odd coincidence":
In 1982, Mandelbrot expanded and updated his ideas inThe 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 atYale, and obtained his firsttenuredpost in 1999, at the age of 75.At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences.
Mandelbrot created the first-ever "theory of roughness", and he saw "roughness" in the shapes of mountains,coastlinesandriver basins; the structures of plants,blood vesselsandlungs; the clustering ofgalaxies. His personal quest was to create some mathematical formula to measure the overall "roughness" of such objects in nature.He began by asking himself various kinds of questions related to nature:
In his paper "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension", published inSciencein 1967, Mandelbrot discussesself-similarcurves that haveHausdorff dimensionthat are examples offractals, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals.
Mandelbrot emphasized the use of fractals as realistic and useful models for describing many "rough" phenomena in the real world. He concluded that "real roughness is often fractal and can be measured."Although Mandelbrot coined the term "fractal", some of the mathematical objects he presented inThe Fractal Geometry of Naturehad been previously described by other mathematicians. Before Mandelbrot, however, they were 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 explaining non-smooth, "rough" objects in the real world. His methods of research were both old and new:
Fractals are also found in human pursuits, such as music, painting, architecture, and in the financial field. Mandelbrot believed that fractals, far from being unnatural, were in many ways more intuitive and natural than the artificially smooth objects of traditionalEuclidean geometry:
Mandelbrot has been called an artist, and a visionaryand a maverick.His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) madeThe Fractal Geometry of Natureaccessible to non-specialists. The book sparked widespread popular interest in fractals and contributed to chaos theory and other fields of science and mathematics.
Mandelbrot also put his ideas to work in cosmology. He offered in 1974 a new explanation ofOlbers' paradox(the "dark night sky" riddle), demonstrating the consequences of fractal theory as asufficient, but not necessary, resolution of the paradox. He postulated that if thestarsin the universe were fractally distributed (for example, likeCantor dust), it would not be necessary to rely on theBig Bangtheory 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.
Mandelbrot's awards include theWolf Prize in Physicsin 1993, theLewis Fry RichardsonPrize of theEuropean Geophysical Societyin 2000, theJapan Prizein 2003,and the Einstein Lectureship of theAmerican Mathematical Societyin 2006.
The small asteroid27500 Mandelbrotwas named in his honor. In November 1990, he was made a Chevalier in France'sLegion of Honour. In December 2005, Mandelbrot was appointed to the position of Battelle Fellow at thePacific Northwest National Laboratory.Mandelbrot was promoted to an Officer of the Legion of Honour in January 2006.An honorary degree fromJohns Hopkins Universitywas bestowed on Mandelbrot in the May 2010 commencement exercises.
A partial list of awards received by Mandelbrot:
Mandelbrot died frompancreatic cancerat the age of 85 in ahospiceinCambridge, Massachusetts, on 14 October 2010.Reacting to news of his death, mathematicianHeinz-Otto Peitgensaid: "[I]f we talk about impact inside mathematics, and applications in the sciences, he is one of the most important figures of the last fifty years."
Chris Anderson,TED conferencecurator, described Mandelbrot as "an icon who changed how we see the world".Nicolas Sarkozy,President of Franceat the time of Mandelbrot's death, said Mandelbrot had "a powerful, original mind that never shied away from innovating and shattering preconceived notions [... h]is work, developed entirely outside mainstream research, led to modern information theory."Mandelbrot's obituary inThe Economistpoints out his fame as "celebrity beyond the academy" and lauds him as the "father of fractal geometry".
Best-selling essayist-authorNassim Nicholas Talebhas remarked that Mandelbrot's bookThe (Mis)Behavior of Marketsis in his opinion "The deepest and most realistic finance book ever published".