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Benoit Mandelbrot
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==Research career== From 1949 to 1958, Mandelbrot was a staff member at the [[Centre National de la Recherche Scientifique]]. During this time he spent a year at the [[Institute for Advanced Study]] in [[Princeton, New Jersey]], where he was sponsored by [[John von Neumann]]. In 1955 he married Aliette Kagan and moved to [[Geneva, Switzerland]] (to collaborate with [[Jean Piaget]] at the International Centre for Genetic Epistemology) and later to the [[Université Lille Nord de France]].<ref name="people">{{Cite web|first= B. B. |last= Mandelbrot| interviewer= Anthony Barcellos |title=Mathematical People, ''Interview of B. B. Mandelbrot'' |publisher= Birkhaüser| year=1984|url=http://users.math.yale.edu/~bbm3/web_pdfs/inHisOwnWords.pdf|access-date=25 June 2013|archive-date=27 April 2015|archive-url= https://web.archive.org/web/20150427164851/http://users.math.yale.edu/~bbm3/web_pdfs/inHisOwnWords.pdf|url-status=live}}</ref> In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the [[IBM]] [[Thomas J. Watson Research Center]] in [[Yorktown Heights, New York]].<ref name="people" /> He remained at IBM for 35 years, becoming an IBM Fellow, and later Fellow [[Emeritus]].<ref name="wolf" /> <!-- Deleted image removed: [[File:Mandelbrot-IBM.jpg|thumb|left|Mandelbrot working at IBM]] --> 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]]. ===Randomness and fractals in financial markets=== Mandelbrot saw [[financial market]]s as an example of "wild randomness", characterized by concentration and long-range dependence. He developed several original approaches for modelling financial fluctuations.<ref>{{cite encyclopedia |last1=Cont |first1=Rama |title=Mandelbrot, Benoit |encyclopedia=Encyclopedia of Quantitative Finance |date=15 May 2010 |pages=eqf01006 |doi=10.1002/9780470061602.eqf01006|isbn = 9780470057568}}</ref> In his early work, he found that the price changes in [[financial market]]s did not follow a [[Gaussian distribution]], but rather [[Paul Lévy (mathematician)|Lévy]] [[stable distributions]] having 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]].<ref>{{cite web |url=https://www.newscientist.com/article/mg15420784.700-flight-over-wall-st.html |title= New Scientist'', 19 April 1997 |publisher=Newscientist.com |date=19 April 1997 |access-date=17 October 2010 |archive-date=21 April 2010 |archive-url= https://web.archive.org/web/20100421101729/http://www.newscientist.com/article/mg15420784.700-flight-over-wall-st.html |url-status=live }}</ref> 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<ref>{{Cite journal |last= Davidson |first= Clive |date= 15 December 1997 |title= Wildly Random Market Moves |url= https://www.joc.com/wildly-random-market-moves_19971215.html |journal= Journal of Commerce |via= JOC.com |access-date= |archive-date= 11 July 2021 |archive-url= https://web.archive.org/web/20210711170709/https://www.joc.com/wildly-random-market-moves_19971215.html |url-status= dead }}</ref><ref>{{Cite web| last= Muldoon| first= Oliver |date= 14 October 2019|title=The Wandering Scientist Turned Father of Fractals| url=https://medium.com/swlh/the-wandering-scientist-turned-father-of-fractals-4dcdc867d4dd|access-date=19 March 2021| website= Medium.com |language=en}}</ref> to apply fractal theory to market microstructure. This cooperation led to the publication of the first comprehensive papers on scaling law in finance.<ref>{{Cite journal| last1= Müller| first1= Ulrich A.|last2=Dacorogna|first2=Michel M.|last3=Olsen|first3=Richard B.|last4=Pictet|first4=Oliver V.|last5=Schwarz|first5=Matthias|last6=Morgenegg|first6=Claude|date=Dec 1990|title=Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis|url=https://doi.org/10.1016/0378-4266(90)90009-Q|journal=Journal of Banking and Finance|volume=14|issue=6|pages=1189–1208| doi=10.1016/0378-4266(90)90009-Q|via=Elsevier Science Direct}}</ref><ref>{{Cite journal|last1=Müller|first1=U. A.|last2=Dacorogna|first2=M. M.|last3=Davé|first3=R. D.|last4=Pictet|first4=O. V.| last5= Olsen| first5= R. B.|last6=Ward|first6=J. R.|date=28 June 1995|title=FRACTALS AND INTRINSIC TIME – A CHALLENGE TO ECONOMETRICIANS| journal= Opening Lecture of the XXXIXth International Conference of the Applied Econometrics Association |citeseerx=10.1.1.197.2969}}</ref> 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 books ''Fractals and Scaling in Finance''<ref>{{Cite book| last= Mandelbrot| first= Benoit|title=Fractals and Scaling in Finance|publisher=Springer|year=1997|isbn=978-1-4757-2763-0}}</ref> and ''The (Mis)behavior of Markets''.<ref>{{Cite book| last= Mandelbrot |first= Benoit| title= The (Mis)behavior of Markets|publisher=Profile Books|year=2004|isbn=9781861977656}}</ref> ===Developing "fractal geometry" and the Mandelbrot set=== As a visiting professor at [[Harvard University]], Mandelbrot began to study mathematical objects called [[Julia set]]s that were [[Invariant (mathematics)|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. While investigating the topology of these Julia sets, he studied the [[Mandelbrot set]] which was introduced by him in 1979. [[File:Mandelbrot p1130876.jpg|thumb|right|Mandelbrot speaking about the [[Mandelbrot set]], during his acceptance speech for the [[Légion d'honneur]] in 2006]] In 1975, Mandelbrot coined the term ''[[fractal]]'' to describe these structures and first published his ideas in the French book ''Les Objets Fractals: Forme, Hasard et Dimension'', later translated in 1977 as ''Fractals: Form, Chance and Dimension''.<ref>''Fractals: Form, Chance and Dimension'', by Benoît Mandelbrot; W H Freeman and Co, 1977; {{isbn|0-7167-0473-0}}</ref> According to computer scientist and physicist [[Stephen 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".<ref name=Wolfram>{{cite news| last= Wolfram| first= Stephen| url= https://www.wsj.com/articles/SB10001424127887324439804578107271772910506 |title= The Father of Fractals| archiveurl= https://web.archive.org/web/20170825102714/https://www.wsj.com/articles/SB10001424127887324439804578107271772910506 |archivedate=25 August 2017 | work= [[The Wall Street Journal]]| date= 22 November 2012| access-date= }}</ref> 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": {{blockquote|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.<ref name=Wolfram />}} 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|Fern leaves]] and [[Romanesco broccoli|Romanesque broccoli]] are two examples from nature."<ref name=Wolfram /> He points out an unexpected conclusion: {{blockquote|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.<ref name=Wolfram />}} 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 1960s [[psychedelic art]] with forms hauntingly reminiscent of nature and the human body". He also saw himself as a "would-be Kepler", after the 17th-century scientist [[Johannes Kepler]], who calculated and described the orbits of the planets.<ref>{{cite web| last= Ivry| first= Benjamin| url= http://forward.com/articles/166094/benoit-mandelbrot-influenced-art-and-mathematics/?p=all |title= Benoit Mandelbrot Influenced Art and Mathematics| archiveurl= https://web.archive.org/web/20130602171300/http://forward.com/articles/166094/benoit-mandelbrot-influenced-art-and-mathematics/?p=all |archivedate= 2 June 2013 | website=[[The Jewish Daily Forward]] | date= 17 November 2012| access-date= }}</ref> [[File:Newton-lplane-Mandelbrot.jpg|thumb|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: {{blockquote|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.<ref name=Clarke>{{citation| url= https://www.youtube.com/watch?v=Lk6QU94xAb8 |title= Arthur C Clarke – Fractals – The Colors Of Infinity|date= 25 December 2010| archiveurl= https://web.archive.org/web/20170531193057/https://www.youtube.com/watch?v=Lk6QU94xAb8 |archivedate= 31 May 2017 | access-date= | via= YouTube}}</ref>}} According to Clarke, "the [[Mandelbrot 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 literally ''infinite'' complexity?" Clarke also notes an "odd coincidence": <blockquote>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.<ref name=Clarke /></blockquote> In 1982, Mandelbrot expanded and updated his ideas in ''[[The Fractal Geometry of Nature]]''.<ref>{{cite book| url= https://books.google.com/books?id=xJ4qiEBNP4gC |title= The Fractal Geometry of Nature |archiveurl= https://web.archive.org/web/20151130231048/https://books.google.com/books?id=xJ4qiEBNP4gC&printsec=frontcover |archivedate=30 November 2015 | first= Benoît| last= Mandelbrot| publisher= W H Freeman & Co| year= 1982 |isbn= 0-7167-1186-9}}</ref> This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "[[Artifact (observational)|program artifacts]]". Mandelbrot left IBM in 1987, after 35 years and 12 days, when IBM decided to end pure research in his division.<ref name="wos44">{{cite web|url=http://www.webofstories.com/play/10483|title=Benoît Mandelbrot • IBM: background and policies |last=Mandelbrot |first=Benoît |author2=Bernard Sapoval |author3=Daniel Zajdenweber|date=May 1998|publisher=[[Web of Stories]]|access-date=17 October 2010|archive-date=8 September 2011|archive-url=https://web.archive.org/web/20110908162215/http://www.webofstories.com/play/10483|url-status=live}}</ref> He joined the Department of Mathematics at [[Yale]], and obtained his first [[tenure]]d post in 1999, at the age of 75.<ref name="Tenner">{{cite news|url=https://www.theatlantic.com/technology/archive/2010/10/benoit-mandelbrot-the-maverick-1924-2010/64684/|title=Benoît Mandelbrot the Maverick, 1924–2010|last=Tenner|first=Edward|date=16 October 2010|work=[[The Atlantic]]|access-date=16 October 2010|archive-date=18 October 2010|archive-url= https://web.archive.org/web/20101018132145/http://www.theatlantic.com/technology/archive/2010/10/benoit-mandelbrot-the-maverick-1924-2010/64684/|url-status=live}}</ref> At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences. ===Fractals and the "theory of roughness"=== Mandelbrot created the first-ever "theory of roughness", and he saw "roughness" in the shapes of mountains, [[coastline]]s and [[river basin]]s; the structures of plants, [[blood vessel]]s and [[lung]]s; the clustering of [[galaxy|galaxies]]. His personal quest was to create some mathematical formula to measure the overall "roughness" of such objects in nature.<ref name= maverick />{{rp|xi}} He began by asking himself various kinds of questions related to nature: {{blockquote|Can [[geometry]] 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?<ref name= maverick>{{cite book| last= Mandelbrot| first= Benoit |year= 2012| title= The Fractalist: Memoir of a Scientific Maverick| publisher= Pantheon Books |isbn= 978-0-307-38991-6}}</ref>{{rp|xii}}}} In his paper "[[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension]]", published in [[Science (journal)|''Science'']] in 1967, Mandelbrot discusses [[self-similarity|self-similar]] curves that have [[Hausdorff dimension]] that are examples of ''fractals'', 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.<ref>{{cite news| quote= Dr. Mandelbrot traced his work on fractals to a question he first encountered as a young researcher: how long is the coast of Britain?"| url= https://www.nytimes.com/2010/10/17/us/17mandelbrot.html?adxnnl=1&adxnnlx=1332064840-/vD0Sjafcl9t9BNghRf8Qw |title= Benoît Mandelbrot, Novel Mathematician, Dies at 85| archiveurl= https://web.archive.org/web/20181231150228/https://www.nytimes.com/2010/10/17/us/17mandelbrot.html?adxnnl=1&adxnnlx=1332064840-%2FvD0Sjafcl9t9BNghRf8Qw |archivedate=31 December 2018 | work= The New York Times| date= 17 October 2010| access-date= }}</ref><ref name="Mandelbrot_Science_1967">{{cite journal | title=How long is the coast of Britain? Statistical self-similarity and fractional dimension | journal=Science | date=5 May 1967 | last= Mandelbrot| first= Benoit B. | pages=636–638 | volume=156 | issue=3775 | doi=10.1126/science.156.3775.636 | pmid=17837158 | url= http://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf | bibcode=1967Sci...156..636M | s2cid=15662830 | access-date=11 January 2016 | archive-date=13 July 2015 | archive-url=https://web.archive.org/web/20150713023120/http://www.sciencemag.org/content/156/3775/636 | url-status=live }}</ref> 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."<ref name= maverick />{{rp|296}} 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, 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: {{blockquote|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.<ref name= maverick />{{rp|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 traditional [[Euclidean geometry]]: <blockquote>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.<br /> —Mandelbrot, in his introduction to ''The Fractal Geometry of Nature''</blockquote> [[File:Mandel zoom 08 satellite antenna.jpg|thumb|right|Section of a Mandelbrot set]] Mandelbrot has been called an artist, and a visionary<ref name="RLD">{{cite journal| last= Devaney| first= Robert L.|author-link= Robert L. Devaney |title= Mandelbrot's Vision for Mathematics| journal= Proceedings of Symposia in Pure Mathematics| volume= 72| number= 1 |publisher=American Mathematical Society |year=2004 |url=http://www.math.yale.edu/mandelbrot/web_pdfs/jubileeletters.pdf |access-date=5 January 2007 |url-status=dead |archive-url= https://web.archive.org/web/20061209093734/http://www.math.yale.edu/mandelbrot/web_pdfs/jubileeletters.pdf |archive-date=9 December 2006 }}</ref> and a maverick.<ref>{{cite web| url= https://www.pbs.org/wgbh/nova/fractals/mandelbrot.html| title=A Radical Mind| last=Jersey| first=Bill |date=24 April 2005|work=Hunting the Hidden Dimension, NOVA|publisher= PBS|access-date=20 August 2009 |archive-date= 22 August 2009|archive-url=https://web.archive.org/web/20090822022402/http://www.pbs.org/wgbh/nova/fractals/mandelbrot.html|url-status=live}}</ref> His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the 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. 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 [[Necessity and sufficiency|sufficient, but not necessary]], resolution of the paradox. He postulated that if the [[star]]s 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.<ref>{{cite journal| title= Galaxy Map Hints at Fractal Universe| first= Amanda |last= Gefter| journal= New Scientist| date= 25 June 2008| url= | access-date= }}</ref>
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