Benoit B. Mandelbrot[a][b] (20 November 1924 – 14 October 2010) was a Polish-born French-Americanmathematician andpolymath with broad interests in the practical sciences, especially regarding what he labeled as "the art ofroughness" of physical phenomena and "the uncontrolled element in life".[6][7][8] He referred to himself as a "fractalist"[9] 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.[10]
In 1936, at the age of 11, Mandelbrot and his family emigrated fromWarsaw, Poland, to France. AfterWorld War II ended, Mandelbrot studied mathematics, graduating from universities in Paris and in the United States and receiving a master's degree inaeronautics from theCalifornia Institute of Technology. He spent most of his career in both the United States and France, havingdualFrench andAmerican citizenship. 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.
Benedykt Mandelbrot[15] was born in aLithuanian Jewish family, inWarsaw during theSecond Polish Republic.[16] 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."[17]
In 1936, when he was 11, the family emigrated from Poland toFrance. The move,World War II, and the influence of his father's brother, the mathematicianSzolem Mandelbrojt (who had moved toParis 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.[9]: 17 [18]
Our constant fear was that a sufficiently determined foe might report us to an authority and we would be sent to our deaths. This happened to a close friend from Paris,Zina Morhange, a physician in a nearby county seat. Simply to eliminate the competition, another physician denounced her ... We escaped this fate. Who knows why?[9]: 49
In 1944, Mandelbrot returned to Paris, studied at theLycée du Parc inLyon, and in 1945 to 1947 attended the prestigiousÉcole Polytechnique, where he studied under French mathematiciansGaston Julia andPaul Lévy. From 1947 to 1949 he studied at California Institute of Technology, where he earned a master's degree in aeronautics.[2] Returning to France, he obtained hisPhD degree in Mathematical Sciences at theUniversity of Paris in 1952.[17]
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 markets as an example of "wild randomness", characterized by concentration and long-range dependence. He developed several original approaches for modelling financial fluctuations.[21] In his early work, he found that the price changes infinancial markets did not follow aGaussian distribution, but ratherLévystable distributions having infinitevariance. He found, for example, that cotton prices followed a Lévy stable distribution with parameterα 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 largerscale parameter.[22] 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 Zurich[23][24] to apply fractal theory to market microstructure. This cooperation led to the publication of the first comprehensive papers on scaling law in finance.[25][26] 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 Finance[27] andThe (Mis)behavior of Markets.[28]
Developing "fractal geometry" and the Mandelbrot set
As a visiting professor atHarvard University, Mandelbrot began to study mathematical objects calledJulia sets that wereinvariant under certain transformations of thecomplex plane. Building on previous work byGaston Julia andPierre Fatou, Mandelbrot used a computer to plot images of the Julia sets. While investigating the topology of these Julia sets, he studied theMandelbrot set which was introduced by him in 1979.
In 1975, Mandelbrot coined the termfractal to 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.[29] 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".[11] 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":
Mandelbrot ended up doing a great piece of science and identifying a much stronger and more fundamental idea—put simply, that there are some geometric shapes, which he called "fractals", that are equally "rough" at all scales. No matter how close you look, they never get simpler, much as the section of a rocky coastline you can see at your feet looks just as jagged as the stretch you can see from space.[11]
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 leaves andRomanesque broccoli are two examples from nature."[11] He points out an unexpected conclusion:
One might have thought that such a simple and fundamental form of regularity would have been studied for hundreds, if not thousands, of years. But it was not. In fact, it rose to prominence only over the past 30 or so years—almost entirely through the efforts of one man, the mathematician Benoit Mandelbrot.[11]
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 art with 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.[30]
A Mandelbrot set
Mandelbrot, however, never felt he was inventing a new idea. He described his feelings in a documentary with science writer Arthur C. Clarke:
Exploring this set I certainly never had the feeling of invention. I never had the feeling that my imagination was rich enough to invent all those extraordinary things on discovering them. They were there, even though nobody had seen them before. It's marvelous, a very simple formula explains all these very complicated things. So the goal of science is starting with a mess, and explaining it with a simple formula, a kind of dream of science.[31]
According to Clarke, "theMandelbrot set is 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 literallyinfinite complexity?" Clarke also notes an "odd coincidence":
the name Mandelbrot, and the word "mandala"—for a religious symbol—which I'm sure is a pure coincidence, but indeed the Mandelbrot set does seem to contain an enormous number of mandalas.[31]
In 1982, Mandelbrot expanded and updated his ideas inThe Fractal Geometry of Nature.[32] 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.[33] He joined the Department of Mathematics atYale, and obtained his firsttenured post in 1999, at the age of 75.[34] He invited his colleagueMichael Frame to work at Yale and co-published various articles with him.[35] At the time of Mandelobrot's 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,coastlines andriver basins; the structures of plants,blood vessels andlungs; the clustering ofgalaxies. His personal quest was to create some mathematical formula to measure the overall "roughness" of such objects in nature.[9]: xi He began by asking himself various kinds of questions related to nature:
Cangeometry deliver what the Greek root of its name [geo-] seemed to promise—truthful measurement, not only of cultivated fields along the Nile River but also of untamed Earth?[9]: xii
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."[9]: 296 Although Mandelbrot coined the term "fractal", some of the mathematical objects he presented inThe Fractal Geometry of Nature had 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:
The form of geometry I increasingly favored is the oldest, most concrete, and most inclusive, specifically empowered by the eye and helped by the hand and, today, also by the computer ... bringing an element of unity to the worlds of knowing and feeling ... and, unwittingly, as a bonus, for the purpose of creating beauty.[9]: 292
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:
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, in his introduction toThe Fractal Geometry of Nature
Section of a Mandelbrot set
Mandelbrot has been called an artist, and a visionary[38] and a maverick.[39] 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 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.
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 thestars in the universe were fractally distributed (for example, likeCantor dust), it would not be necessary to rely on theBig 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.[40]
The small asteroid27500 Mandelbrot was 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.[42] Mandelbrot was promoted to an Officer of the Legion of Honour in January 2006.[43] An honorary degree fromJohns Hopkins University was bestowed on Mandelbrot in the May 2010 commencement exercises.[44]
A partial list of awards received by Mandelbrot:[45]
Mandelbrot died frompancreatic cancer at the age of 85 in ahospice inCambridge, Massachusetts, on 14 October 2010.[1][51] Reacting to news of his death, mathematicianHeinz-Otto Peitgen said: "[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."[1]
Chris Anderson,TED conference curator, described Mandelbrot as "an icon who changed how we see the world".[52]Nicolas Sarkozy,President of France at 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."[53] Mandelbrot's obituary inThe Economist points out his fame as "celebrity beyond the academy" and lauds him as the "father of fractal geometry".[54]
Best-selling essayist-authorNassim Nicholas Taleb has remarked that Mandelbrot's bookThe (Mis)Behavior of Markets is in his opinion "The deepest and most realistic finance book ever published".[10]
Mandelbrot, B. (1959) Variables et processus stochastiques de Pareto-Levy, et la repartition des revenus. Comptes rendus de l'Académie des Sciences de Paris, 249, 613–615.
Mandelbrot, B. (1960) The Pareto-Levy law and the distribution of income. International Economic Review, 1, 79–106.
Mandelbrot, B. (1961) Stable Paretian random functions and the multiplicative variation of income. Econometrica, 29, 517–543.
Mandelbrot, B. (1964) Random walks, fire damage amount and other Paretian risk phenomena. Operations Research, 12, 582–585.
Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E, 1997 by Benoit B. Mandelbrot and R.E. Gomory
Mandelbrot, Benoit B. (1997)Fractals and Scaling in Finance: Discontinuity, Concentration, Risk, Springer.
Fractales, hasard et finance, 1959–1997, 1 November 1998
Multifractals and 1/ƒ Noise: Wild Self-Affinity in Physics (1963–1976) (Selecta; V.N) 18 January 1999 by J.M. Berger and Benoit B. Mandelbrot
Gaussian Self-Affinity and Fractals: Globality, The Earth, 1/f Noise, and R/S (Selected Works of Benoit B. Mandelbrot) 14 December 2001 by Benoit Mandelbrot and F.J. Damerau
Mandelbrot, Benoit B.,Gaussian Self-Affinity and Fractals, Springer: 2002.
Fractals and Chaos: The Mandelbrot Set and Beyond, 9 January 2004
Theindie music satiristJonathan Coulton's song, "Mandelbrot Set", is dedicated to Mandelbrot, who at the time of the song's release "is still alive and teaching math at Yale." The song incorporates the concepts and images associated with order out of chaos.
The song'spre-chorus describes a mathematical equation that is not the Mandelbrot Set, to which Coulton has said, "if any of you are mathematicians and you’re just waiting anxiously to come up to me afterwards and point out the fact that the equation is not quite right, that I actually described aJulia set, and not a Mandlebrot Set, I already know that shit."[56]
^In his autobiography, Mandelbrot did not add acircumflex to the "i" (i.e. "î") in his first name, as is usual forthe French given name. He included "B" as amiddle initial. HisNew York Times obituary stated that "he added the middle initial himself, though it does not stand for a middle name",[1] an assertion that is supported by his obituary inThe Guardian.[2]
^Müller, U. A.; Dacorogna, M. M.; Davé, R. D.; Pictet, O. V.; Olsen, R. B.; Ward, J. R. (28 June 1995). "FRACTALS AND INTRINSIC TIME – A CHALLENGE TO ECONOMETRICIANS".Opening Lecture of the XXXIXth International Conference of the Applied Econometrics Association.CiteSeerX10.1.1.197.2969.
^Mandelbrot, Benoit (1997).Fractals and Scaling in Finance. Springer.ISBN978-1-4757-2763-0.
^Mandelbrot, Benoit (2004).The (Mis)behavior of Markets. Profile Books.ISBN9781861977656.
^Fractals: Form, Chance and Dimension, by Benoît Mandelbrot; W H Freeman and Co, 1977;ISBN0-7167-0473-0
^Mandelbrot, Benoît B. (2013).The fractalist: memoir of a scientific maverick (First vintage books ed.). New York: Vintage Books.ISBN978-0-307-38991-6.
^"Benoît Mandelbrot, Novel Mathematician, Dies at 85".The New York Times. 17 October 2010. Archived fromthe original on 31 December 2018.Dr. Mandelbrot traced his work on fractals to a question he first encountered as a young researcher: how long is the coast of Britain?"
^Jersey, Bill (24 April 2005)."A Radical Mind".Hunting the Hidden Dimension, NOVA. PBS.Archived from the original on 22 August 2009. Retrieved20 August 2009.
^Gefter, Amanda (25 June 2008). "Galaxy Map Hints at Fractal Universe".New Scientist.
Frame, Michael; Cohen, Nathan (2015).Benoit Mandelbrot: A Life in Many Dimensions. Singapore: World Scientific Publishing Company.ISBN978-981-4366-06-9.
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