The Info List - Benoit Mandelbrot

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Benoit B. [n 1] Mandelbrot [n 2] (20 November 1924 – 14 October 2010) was a Polish-born, French and 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".[6][7][8] He referred to himself as a "fractalist".[9] He 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.[10] In 1936, while he was a child, Mandelbrot's family emigrated to France from Warsaw, Poland. After World War II
World War II
ended, Mandelbrot studied mathematics, graduating from universities in Paris and 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 discovering the Mandelbrot set
Mandelbrot set
in 1979. He showed how visual complexity can be created from simple rules. He said that things typically considered to be "rough", a "mess" or "chaotic", like clouds or shorelines, actually had a "degree of order".[11] 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.[12] Toward the end of his career, he was Sterling Professor
Sterling Professor
of Mathematical Sciences at Yale
University, where he was the oldest professor in Yale's history to receive tenure.[13] Mandelbrot also held positions at the Pacific Northwest National Laboratory, Université Lille Nord de France, Institute for Advanced Study
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 in 2012.


1 Early years 2 Research career

2.1 Randomness in financial markets 2.2 Developing "fractal geometry" and the Mandelbrot set 2.3 Fractals and the "theory of roughness"

3 Awards and honors 4 Death and legacy 5 Bibliography

5.1 in English 5.2 In French

6 References in popular culture 7 See also 8 Notes 9 References 10 Bibliography 11 Further reading 12 External links

Early years[edit] Mandelbrot was born in Warsaw
during the Second Polish Republic. His family was Jewish. Although his father made his living trading clothing, the family had a strong academic tradition and his mother was a dental surgeon.[14] He was first introduced to mathematics by two of his uncles, one of whom, Szolem Mandelbrojt, was a mathematician who resided in Paris. According to Mandelbrot's autobiography, The Fractalist - Memoir of a Scientific Maverick,[15] "[t]he love of his [Szolem's] mind was mathematics".[9]:16 The family emigrated from Poland to France in 1936, when he was 11. "The fact that my parents, as economic and political refugees, joined Szolem in France saved our lives," he writes.[9]:17[16] Mandelbrot attended the Lycée Rolin in Paris until the start of World War II, when his family then moved to Tulle, France. He was helped by Rabbi David Feuerwerker, the Rabbi
of Brive-la-Gaillarde, to continue his studies.[9]:62–63[17] Much of France was occupied by the Nazis at the time, and Mandelbrot recalls this period:

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 the Lycée du Parc in Lyon, and in 1945 to 1947 attended the École Polytechnique, where he studied under Gaston Julia
Gaston Julia
and Paul 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 his PhD degree in Mathematical Sciences at the University of Paris
University of Paris
in 1952.[14] Research career[edit] 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
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, and later to the Université Lille Nord de France.[18] 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.[18] He remained at IBM
for 35 years, becoming an IBM
Fellow, and later Fellow Emeritus.[14] 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 in financial markets[edit] Mandelbrot saw financial markets as an example of "wild randomness", characterized by concentration and long range dependence. He developed several original approaches for modelling financial fluctuations.[19] In his early work, he found that the price changes in financial markets did not follow a Gaussian distribution, but rather 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.[20] Developing "fractal geometry" and the Mandelbrot set[edit] As a visiting professor at Harvard University, Mandelbrot began to study fractals called Julia sets that were invariant under certain transformations of the complex plane. Building on previous work by Gaston Julia
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
Mandelbrot set
which was introduced by him in 1979. In 1982, Mandelbrot expanded and updated his ideas in The Fractal
of Nature.[21] 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 speaking about the Mandelbrot set, during his acceptance speech for the Légion d'honneur
Légion d'honneur
in 2006

In 1975, Mandelbrot coined the term fractal to describe these structures and first published his ideas, and later translated, Fractals: Form, Chance and Dimension.[22] According to mathematics scientist 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."[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 and Romanesco 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 graphic 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.[23]

A Mandelbrot set

Mandelbrot, however, never felt he was inventing a new idea. He describes 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.[24]

According to Clarke, "the Mandelbrot set
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

the name Mandelbrot, and the word "mandala"—for a religious symbol—which I'm sure is a pure coincidence, but indeed the Mandelbrot set
Mandelbrot set
does seem to contain an enormous number of mandalas.[24]

Mandelbrot left IBM
in 1987, after 35 years and 12 days, when IBM decided to end pure research in his division.[25] He joined the Department of Mathematics
at Yale, and obtained his first tenured post in 1999, at the age of 75.[26] At the time of his retirement in 2005, he was Sterling Professor
Sterling Professor
of Mathematical Sciences. Fractals and the "theory of roughness"[edit] Mandelbrot created the first-ever "theory of roughness", and he saw "roughness" in the shapes of mountains, coastlines and river basins; the structures of plants, blood vessels and lungs; the clustering of galaxies. 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:

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?[9]:xii

In his paper titled How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension published in Science in 1967 Mandelbrot discusses 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.[27][28] 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 in The Fractal
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 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:

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 to The Fractal
Geometry of Nature

Section of a Mandelbrot set

Mandelbrot has been called a work of art, and a visionary[29] and a maverick.[30] 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
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
Olbers' paradox
(the "dark night sky" riddle), demonstrating the consequences of fractal theory as a sufficient, but not necessary, resolution of the paradox. He postulated that if the stars in the universe were fractally distributed (for example, like Cantor dust), it would not be necessary to rely on the Big Bang
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
Big Bang
had not occurred.[31] Awards and honors[edit] Mandelbrot's awards include the Wolf Prize for Physics in 1993, the Lewis Fry Richardson
Lewis Fry Richardson
Prize of the European Geophysical Society
European Geophysical Society
in 2000, the Japan Prize
Japan Prize
in 2003,[32] and the Einstein Lectureship of the American Mathematical Society
American Mathematical Society
in 2006. The small asteroid 27500 Mandelbrot was named in his honor. In November 1990, he was made a Chevalier in France's Legion of Honour. In December 2005, Mandelbrot was appointed to the position of Battelle Fellow at the Pacific Northwest National Laboratory.[33] Mandelbrot was promoted to an Officer of the Legion of Honour
Legion of Honour
in January 2006.[34] An honorary degree from Johns Hopkins University
Johns Hopkins University
was bestowed on Mandelbrot in the May 2010 commencement exercises.[35] A partial list of awards received by Mandelbrot:[36]

2004 Best Business Book of the Year Award AMS Einstein Lectureship Barnard Medal Caltech Service Casimir Funk
Casimir Funk
Natural Sciences Award Charles Proteus Steinmetz
Charles Proteus Steinmetz
Medal Fellow, American Geophysical Union Fellow of the American Statistical Association[37] Franklin Medal Harvey Prize (1989) Honda Prize Humboldt Preis IBM
Fellowship Japan Prize
Japan Prize
(2003) John Scott Award Légion d'honneur
Légion d'honneur
(Legion of Honour) Lewis Fry Richardson
Lewis Fry Richardson
Medal Medaglia della Presidenza della Repubblica Italiana Médaille de Vermeil de la Ville de Paris Nevada Prize Member of the Norwegian Academy of Science and Letters.[38] Science for Art Sven Berggren-Priset Władysław Orlicz Prize Wolf Foundation Prize for Physics (1993)

Death and legacy[edit]

Wikinews has related news: Mathematician
Benoît Mandelbrot dies aged 85

Mandelbrot died from pancreatic cancer at the age of 85 in a hospice in Cambridge, Massachusetts
Cambridge, Massachusetts
on 14 October 2010.[1][39] Reacting to news of his death, mathematician Heinz-Otto Peitgen
Heinz-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".[40] 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."[41] Mandelbrot's obituary in The Economist points out his fame as "celebrity beyond the academy" and lauds him as the "father of fractal geometry".[42] Best-selling essayist-author Nassim Nicholas Taleb, has remarked that Mandelbrot's book The (Mis)Behavior of Markets is in his opinion "The deepest and most realistic finance book ever published".[10] Bibliography[edit] in English[edit]

Fractals: Form, Chance and Dimension, 1977 The Fractal
of Nature, 1982 Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E, 1997 by Benoit B. Mandelbrot and R.E. Gomory Fractales, hasard et finance, 1959-1997, Nov 1, 1998 Multifractals and 1/ƒ Noise: Wild Self-Affinity in Physics (1963–1976) (Selecta; V.N) Jan 18, 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) Dec 14, 2001 by Benoit Mandelbrot
Benoit Mandelbrot
and F.J. Damerau Fractals and Chaos: The Mandelbrot Set and Beyond, Jan 9, 2004 The Misbehavior of Markets: A Fractal
View of Financial Turbulence, 2006 by Benoit Mandelbrot
Benoit Mandelbrot
and Richard L. Hudson The Fractalist: Memoir of a Scientific Maverick, 2014

In French[edit]

La forme d'une vie. Mémoires (1924-2010) by Benoît Mandelbrot (Author), Johan-Frédérik Hel Guedj (Translator)

References in popular culture[edit]

In 2004, the American singer-songwriter Jonathan Coulton
Jonathan Coulton
wrote "Mandelbrot Set". In 2007, the author Laura Ruby published "The Chaos King," which includes a character named Mandelbrot and discussion of chaos theory. In 2017, Zach Weinersmith' webcomic, Saturday Morning Breakfast Cereal, portrayed Mandelbrot.[43]

See also[edit]

External video

Family background and early education, (4:11) Benoit Mandelbrot interview, Part 1 of 144, Web of Stories[44]

"How Long is the Coast of Britain?" Louis Bachelier Zipf-Mandelbrot law Seven states of randomness Skewness risk Kurtosis risk Fractal
dimension Lacunarity

Self-similarity Self-affinity Hurst exponent Fractional Brownian motion Multifractal system 1/f noise Mandelbrot Competition


^ a b In his autobiography, Mandelbrot did not add a circumflex to the "i" (i.e. "î") in his first name. He included "B" as a middle initial. His New 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 in The Guardian.[2] However, other sources conjecture that he intended his middle initial B to recursively stand for Benoit B. Mandelbrot, thereby including a fractal (his mathematical discovery) in his own name, as a mathematical joke.[3] ^ Pronounced /ˈmændəlbrɒt/ MAN-dəl-brot in English.[4] When speaking in French, Mandelbrot pronounced his name [bənwa mɑ̃dɛlbʁot].[5]


^ a b c Hoffman, Jascha (16 October 2010). "Benoît Mandelbrot, Mathematician, Dies at 85". The New York Times. Retrieved 16 October 2010.  ^ a b Lesmoir-Gordon, Nigel (17 October 2010). "Benoît Mandelbrot obituary". The Guardian. London. Retrieved 17 October 2010.  ^ Selinker, Mike (18 October 2010). "Never Trend Away: Jonathan Coulton on Benoit Mandelbrot". Wired.  ^ "Mandelbrot". Oxford English Dictionary
Oxford English Dictionary
(3rd ed.). Oxford University Press. September 2005.  (Subscription or UK public library membership required.) ^ Recording of the ceremony on 11 September 2006 at which Mandelbrot received the insignia for an Officer of the Légion d'honneur. ^ https://www.insidescience.org/news/remembering-father-fractals ^ Benoit Mandelbrot: Fractals and the art of roughness. ted.com (February 2010) ^ Hudson & Mandelbrot, Prelude, page xviii ^ a b c d e f g h i Mandelbrot, Benoit (2012). The Fractalist: Memoir of a Scientific Maverick, Pantheon Books. ISBN 978-0-307-38991-6. ^ a b Gomory, R. (2010). "Benoît Mandelbrot (1924–2010)". Nature. 468 (7322): 378. Bibcode:2010Natur.468..378G. doi:10.1038/468378a. PMID 21085164.  ^ a b c d e Wolfram, Stephen. "The Father of Fractals", Wall Street Journal, 22 November 2012 ^ list includes specific sciences mentioned in Hudson & Mandelbrot, the Prelude, p. xvi, and p. 26 ^ Steve Olson (November–December 2004). "The Genius of the Unpredictable". Yale
Alumni Magazine. Retrieved 22 July 2014.  ^ a b c Mandelbrot, Benoît (2002). "The Wolf Prizes for Physics, A Maverick's Apprenticeship" (PDF). Imperial College Press.  ^ Mandelbrot, Benoit (2014-01-14). The Fractalist: Memoir of a Scientific Maverick (Reprint ed.). Vintage. ISBN 9780307389916.  ^ "BBC News – 'Fractal' mathematician Benoît Mandelbrot dies aged 85". BBC Online. 17 October 2010. Retrieved 17 October 2010.  ^ Hemenway P. (2005) Divine proportion: Phi in art, nature and science. Psychology Press. ISBN 0-415-34495-6 ^ a b Barcellos, Anthony (1984). "Mathematical People, Interview of B. B. Mandelbrot" (PDF). Birkhaüser.  ^ Rama Cont (19 April 2010). "'Mandelbrot, Benoit'". Encyclopedia of Quantitative finance. Wiley. doi:10.1002/9780470061602.eqf01006. Retrieved 17 October 2010.  ^ "''New Scientist'', 19 April 1997". Newscientist.com. 19 April 1997. Retrieved 17 October 2010.  ^ The Fractal
of Nature, by Benoît Mandelbrot; W H Freeman & Co, 1982; ISBN 0-7167-1186-9 ^ Fractals: Form, Chance and Dimension, by Benoît Mandelbrot; W H Freeman and Co, 1977; ISBN 0-7167-0473-0 ^ Ivry, Benjamin. " Benoit Mandelbrot
Benoit Mandelbrot
Influenced Art and Mathematics", Forward, 17 November 2012 ^ a b "Arthur C Clarke – Fractals – The Colors Of Infinity", video interviews, 54 min. ^ Mandelbrot, Benoît; Bernard Sapoval; Daniel Zajdenweber (May 1998). " Web of Stories
Web of Stories
• Benoît Mandelbrot • IBM: background and policies". Web of Stories. Retrieved 17 October 2010.  ^ Tenner, Edward (16 October 2010). "Benoît Mandelbrot the Maverick, 1924–2010". The Atlantic. Retrieved 16 October 2010.  ^ "Dr. Mandelbrot traced his work on fractals to a question he first encountered as a young researcher: how long is the coast of Britain?": Benoit Mandelbrot
Benoit Mandelbrot
(1967). "Benoît Mandelbrot, Novel Mathematician, Dies at 85", The New York Times. ^ Mandelbrot, Benoit B. (5 May 1967). "How long is the coast of Britain? Statistical self-similarity and fractional dimension" (PDF). Science. 156 (3775): 636–638. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. PMID 17837158.  ^ Devaney, Robert L. (2004). ""Mandelbrot's Vision for Mathematics" in Proceedings of Symposia in Pure Mathematics. Volume 72.1" (PDF). American Mathematical Society. Archived from the original (PDF) on 9 December 2006. Retrieved 5 January 2007.  ^ Jersey, Bill (24 April 2005). "A Radical Mind". Hunting the Hidden Dimension. NOVA/ PBS. Retrieved 20 August 2009.  ^ Galaxy
Map Hints at Fractal
Universe, by Amanda Gefter; New Scientist; 25 June 2008 ^ Laureates of the Japan Prize. japanprize.jp ^ "PNNL press release: Mandelbrot joins Pacific Northwest National Laboratory". Pnl.gov. 16 February 2006. Retrieved 17 October 2010.  ^ "''Légion d'honneur'' announcement of promotion of Mandelbrot to ''officier''" (in French). Legifrance.gouv.fr. Retrieved 17 October 2010.  ^ "Six granted honorary degrees, Society of Scholars inductees recognized". Gazette.jhu.edu. 7 June 2010. Retrieved 17 October 2010.  ^ Mandelbrot, Benoit B. (2 February 2006). "Vita and Awards (Word document)". Retrieved 6 January 2007.  Retrieved from Internet Archive 15 December 2013. ^ View/Search Fellows of the ASA, accessed 2016-08-20. ^ "Gruppe 1: Matematiske fag" (in Norwegian). Norwegian Academy of Science and Letters. Retrieved 7 October 2010.  ^ "Benoît Mandelbrot, fractals pioneer, dies". United Press International. 16 October 2010. Retrieved 17 October 2010.  ^ "Mandelbrot, father of fractal geometry, dies". The Gazette. Archived from the original on 19 October 2010. Retrieved 16 October 2010.  ^ "Sarkozy rend hommage à Mandelbrot" [Sarkozy pays homage to Mandelbrot]. Le Figaro
Le Figaro
(in French). Retrieved 17 October 2010.  ^ Benoît Mandelbrot's obituary. The Economist (21 October 2010) ^ http://www.smbc-comics.com/comic/mandelbrot ^ Mandelbrot, Benoît; Bernard Sapoval; Daniel Zajdenweber (May 1998). " Web of Stories
Web of Stories
– Benoît Mandelbrot – Family background and early education". Web of Stories. Retrieved 19 October 2010. 


Hudson, Richard L.; Mandelbrot, Benoît B. (2004). The (Mis)Behavior of Markets: A Fractal
View of Risk, Ruin, and Reward. New York: Basic Books. ISBN 0-465-04355-0. 

Further reading[edit]

Mandelbrot, Benoit B. (2010). The Fractalist, Memoir of a Scientific Maverick. New York: Vintage Books, Division of Random House. ISBN 978-0-307-38991-6 Mandelbrot, Benoît B. (1983). The Fractal
of Nature. San Francisco: W.H. Freeman. ISBN 0-7167-1186-9.  Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe and Cornelia Zahlten: Fractals: An Animated Discussion (63 min video film, interviews with Benoît Mandelbrot and Edward Lorenz, computer animations), W.H. Freeman and Company, 1990. ISBN 0-7167-2213-5 (re-published by Films for the Humanities & Sciences, ISBN 978-0-7365-0520-8) Mandelbrot, Benoit B. (1997) Fractals and Scaling in Finance: Discontinuity, Concentration, Risk, Springer. Mandelbrot, Benoît (February 1999). "A Multifractal Walk down Wall Street". Scientific American. 280 (2): 70. Bibcode:1999SciAm.280b..70M. doi:10.1038/scientificamerican0299-70.  Mandelbrot, Benoit B., Gaussian Self-Affinity and Fractals, Springer: 2002. Mandelbrot, Benoît; Taleb, Nassim (23 March 2006). "A focus on the exceptions that prove the rule". Financial Times. Archived from the original on 23 October 2010. Retrieved 17 October 2010.  "Hunting the Hidden Dimension: mysteriously beautiful fractals are shaking up the world of mathematics and deepening our understanding of nature", NOVA, WGBH Educational Foundation, Boston for PBS, first aired 28 October 2008. Frame, Michael; Cohen, Nathan (2015). Benoit Mandelbrot: A Life in Many Dimensions. Singapore: World Scientific Publishing Company. ISBN 978-981-4366-06-9.  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.

External links[edit]

Wikimedia Commons has media related to Benoît Mandelbrot.

Wikiquote has quotations related to: Benoit Mandelbrot

Benoit Mandelbrot
Benoit Mandelbrot
at the Mathematics
Genealogy Project Mandelbrot's page at Yale "Benoît Mandelbrot: Fractals and the art of roughness" (TED address). Fractals in Science, Engineering
and Finance (lecture). FT.com interview on the subject of the financial markets which includes his critique of the "efficient market" hypothesis. Taylor, Richard (2011). "Obituaries: Benoit Mandelbrot". Physics Today. 64 (6): 63. Bibcode:2011PhT....64f..63T. doi:10.1063/1.3603925. [permanent dead link] Mandelbrot relates his life story (Web of Stories). Interview (1 January 1981, Ithaca, NY) held by the Eugene Dynkin Collection of Mathematics
Interviews, Cornell University Library. Video animation of Mandelbrot set, zoom factor 10342. Video animation of Mandelbulb
on YouTube, a three-dimensional Mandelbrot-set projection. Video fly-through an animated Mandelbulb
world on YouTube Benoit Mandelbrot
Benoit Mandelbrot
on IMDb Benoit Mandelbrot
Benoit Mandelbrot
at TED

v t e

Laureates of the Wolf Prize in Physics


Chien-Shiung Wu
Chien-Shiung Wu
(1978) George Uhlenbeck / Giuseppe Occhialini
Giuseppe Occhialini


Michael Fisher / Leo Kadanoff
Leo Kadanoff
/ Kenneth G. Wilson (1980) Freeman Dyson
Freeman Dyson
/ Gerardus 't Hooft / Victor Weisskopf (1981) Leon M. Lederman
Leon M. Lederman
/ Martin Lewis Perl (1982) Erwin Hahn / Peter Hirsch / Theodore Maiman
Theodore Maiman
(1983–84) Conyers Herring / Philippe Nozières (1984–85) Mitchell Feigenbaum
Mitchell Feigenbaum
/ Albert J. Libchaber (1986) Herbert Friedman / Bruno Rossi
Bruno Rossi
/ Riccardo Giacconi
Riccardo Giacconi
(1987) Roger Penrose
Roger Penrose
/ Stephen Hawking
Stephen Hawking


Pierre-Gilles de Gennes / David J. Thouless
David J. Thouless
(1990) Maurice Goldhaber
Maurice Goldhaber
/ Valentine Telegdi (1991) Joseph H. Taylor Jr. (1992) Benoît Mandelbrot (1993) Vitaly Ginzburg
Vitaly Ginzburg
/ Yoichiro Nambu
Yoichiro Nambu
(1994–95) John Wheeler (1996–97) Yakir Aharonov
Yakir Aharonov
/ Michael Berry (1998) Dan Shechtman
Dan Shechtman


Raymond Davis Jr.
Raymond Davis Jr.
/ Masatoshi Koshiba
Masatoshi Koshiba
(2000) Bertrand Halperin
Bertrand Halperin
/ Anthony Leggett (2002–03) Robert Brout
Robert Brout
/ François Englert
François Englert
/ Peter Higgs
Peter Higgs
(2004) Daniel Kleppner (2005) Albert Fert
Albert Fert
/ Peter Grünberg
Peter Grünberg


John F. Clauser / Alain Aspect
Alain Aspect
/ Anton Zeilinger
Anton Zeilinger
(2010) Maximilian Haider / Harald Rose
Harald Rose
/ Knut Urban (2011) Jacob Bekenstein
Jacob Bekenstein
(2012) Peter Zoller
Peter Zoller
/ Juan Ignacio Cirac (2013) James D. Bjorken / Robert P. Kirshner (2015) Yoseph Imry
Yoseph Imry
(2016) Michel Mayor
Michel Mayor
/ Didier Queloz
Didier Queloz
(2017) Charles H. Bennett / Gilles Brassard (2018)

Agriculture Arts Chemistry Mathematics Medicine Physics

v t e

Chaos theory

Chaos theory

Anosov diffeomorphism Bifurcation theory Butterfly effect Chaos theory
Chaos theory
in organizational development Complexity Control of chaos Dynamical system Edge of chaos Fractal Predictability Quantum chaos Santa Fe Institute Synchronization of chaos Unintended consequences

Chaotic maps (list)

Arnold tongue Arnold's cat map Baker's map Complex quadratic map Complex squaring map Coupled map lattice Double pendulum Double scroll attractor Duffing equation Duffing map Dyadic transformation Dynamical billiards


Exponential map Gauss map Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Kaplan–Yorke map Logistic map Lorenz system Multiscroll attractor Rabinovich–Fabrikant equations Rössler attractor Standard map Swinging Atwood's machine Tent map Tinkerbell map Van der Pol oscillator Zaslavskii map

Chaos systems

Bouncing ball dynamics Chua's circuit Economic bubble FPUT problem Tilt-A-Whirl

Chaos theorists

Michael Berry Mary Cartwright Leon O. Chua Mitchell Feigenbaum Celso Grebogi Martin Gutzwiller Brosl Hasslacher Michel Hénon Svetlana Jitomirskaya Sofia Kovalevskaya Bryna Kra Edward Norton Lorenz Aleksandr Lyapunov Benoît Mandelbrot Hee Oh Edward Ott Henri Poincaré Mary Rees Otto Rössler David Ruelle Caroline Series Oleksandr Mykolayovych Sharkovsky Nina Snaith Floris Takens Audrey Terras Mary Tsingou Amie Wilkinson James A. Yorke Lai-Sang Young

v t e




Assouad Box-counting Correlation Hausdorff Packing Topological

Recursion Self-similarity

Iterated function system

Barnsley fern Cantor set Dragon curve Koch snowflake Menger sponge Sierpinski carpet Sierpinski triangle Space-filling curve T-square n-flake

Strange attractor

Multifractal system


canopy Space-filling curve

H tree

Escape-time fractals

Burning Ship fractal Julia set


Lyapunov fractal Mandelbrot set Newton fractal Tricorn

Rendering Techniques

Buddhabrot Orbit trap Pickover stalk

Random fractals

Brownian motion Brownian tree Diffusion-limited aggregation Fractal
landscape Lévy flight Mandelbox Mandelbulb Percolation theory Self-avoiding walk


Georg Cantor Felix Hausdorff Gaston Julia Helge von Koch Paul Lévy Aleksandr Lyapunov Benoit Mandelbrot Lewis Fry Richardson Wacław Sierpiński


"How Long Is the Coast of Britain?"


List of fractals by Hausdorff dimension The Beauty of Fractals
The Beauty of Fractals
(1986 book) Fractal
art Chaos: Making a New Science (1987 book) The Fractal
of Nature (1982 book)

Authority control

WorldCat Identities VIAF: 107586686 LCCN: n81107154 ISNI: 0000 0001 2096 8076 GND: 118974203 SELIBR: 245815 SUDOC: 027005518 BNF: cb11914221j (data) BIBSYS: 90065702 MGP: 60791 NDL: 00448659 NKC: xx0012614 RLS: 000135469 BNE: XX97