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". He referred
to himself as a "fractalist". 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.
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
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 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". 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
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,
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
5.1 in English
5.2 In French
6 References in popular culture
7 See also
11 Further reading
12 External links
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. 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,
"[t]he love of his [Szolem's] mind was mathematics".: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.:17 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.:62–63 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?: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 and Paul 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
his PhD degree in Mathematical Sciences at the
University of Paris
University of Paris in
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. 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. He remained
IBM for 35 years, becoming an
IBM Fellow, and later Fellow
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
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.
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
Developing "fractal geometry" and the Mandelbrot set
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 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. In 1982, Mandelbrot expanded and updated his ideas in The
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
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, and later translated,
Fractals: Form, Chance and Dimension. 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." 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.
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." He points out an
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.
Mandelbrot used the term "fractal" as it derived from the Latin word
"fractus", defined as broken or shattered glass. Using the newly
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.
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.
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.
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
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
IBM in 1987, after 35 years and 12 days, when IBM
decided to end pure research in his division. He joined the
Mathematics at Yale, and obtained his first tenured post
in 1999, at the age of 75. At the time of his retirement in 2005,
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, 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.: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?: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.
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.":296 Although Mandelbrot coined the term "fractal", some
of the mathematical objects he presented in The
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.: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
Clouds are not spheres, mountains are not cones, coastlines are not
circles, and bark is not smooth, nor does lightning travel in a
—Mandelbrot, in his introduction to The
Section of a Mandelbrot set
Mandelbrot has been called a work of art, and a visionary and a
maverick. His informal and passionate style of writing and his
emphasis on visual and geometric intuition (supported by the inclusion
of numerous illustrations) made The
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 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 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.
Awards and honors
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
Japan Prize in 2003, 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. Mandelbrot
was promoted to an Officer of the
Legion of Honour
Legion of Honour in January
2006. An honorary degree from
Johns Hopkins University
Johns Hopkins University was
bestowed on Mandelbrot in the May 2010 commencement exercises.
A partial list of awards received by Mandelbrot:
2004 Best Business Book of the Year Award
AMS Einstein Lectureship
Casimir Funk Natural Sciences Award
Charles Proteus Steinmetz
Charles Proteus Steinmetz Medal
Fellow, American Geophysical Union
Fellow of the American Statistical Association
Harvey Prize (1989)
Japan Prize (2003)
John Scott Award
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
Member of the Norwegian Academy of Science and Letters.
Science for Art
Władysław Orlicz Prize
Wolf Foundation Prize for Physics (1993)
Death and legacy
Wikinews has related news:
Mathematician Benoît Mandelbrot dies aged
Mandelbrot died from pancreatic cancer at the age of 85 in a hospice
Cambridge, Massachusetts on 14 October 2010. Reacting to
news of his death, mathematician
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
Chris Anderson, TED conference curator, described Mandelbrot as "an
icon who changed how we see the world". 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." Mandelbrot's obituary in The Economist points out his
fame as "celebrity beyond the academy" and lauds him as the "father of
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".
Fractals: Form, Chance and Dimension, 1977
Geometry 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.
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 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,
Benoit Mandelbrot and Richard L. Hudson
The Fractalist: Memoir of a Scientific Maverick, 2014
La forme d'une vie. Mémoires (1924-2010) by Benoît Mandelbrot
(Author), Johan-Frédérik Hel Guedj (Translator)
References in popular culture
In 2004, the American singer-songwriter
Jonathan Coulton wrote
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.
Family background and early education, (4:11) Benoit Mandelbrot
interview, Part 1 of 144, Web of Stories
"How Long is the Coast of Britain?"
Seven states of randomness
Fractional Brownian motion
^ 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", an
assertion that is supported by his obituary in The Guardian.
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
^ Pronounced /ˈmændəlbrɒt/ MAN-dəl-brot in English. When
speaking in French, Mandelbrot pronounced his name [bənwa
^ a b c Hoffman, Jascha (16 October 2010). "Benoît Mandelbrot,
Mathematician, Dies at 85". The New York Times. Retrieved 16 October
^ 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.
Oxford English Dictionary
Oxford English Dictionary (3rd ed.). Oxford University
Press. September 2005. (Subscription or UK public library
^ Recording of the ceremony on 11 September 2006 at which Mandelbrot
received the insignia for an Officer of the Légion d'honneur.
^ Benoit Mandelbrot: Fractals and the art of roughness. ted.com
^ 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.
^ 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
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.
^ "BBC News – 'Fractal' mathematician Benoît Mandelbrot dies
aged 85". BBC Online. 17 October 2010. Retrieved 17 October
^ 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.
Geometry 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 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 (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
^ "''Légion d'honneur'' announcement of promotion of Mandelbrot to
''officier''" (in French). Legifrance.gouv.fr. Retrieved 17 October
^ "Six granted honorary degrees, Society of Scholars inductees
recognized". Gazette.jhu.edu. 7 June 2010. Retrieved 17 October
^ 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
^ "Sarkozy rend hommage à Mandelbrot" [Sarkozy pays homage to
Le Figaro (in French). Retrieved 17 October 2010.
^ Benoît Mandelbrot's obituary. The Economist (21 October 2010)
^ 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.
Mandelbrot, Benoit B. (2010). The Fractalist, Memoir of a Scientific
Maverick. New York: Vintage Books, Division of Random House.
Mandelbrot, Benoît B. (1983). The
Geometry 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,
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.
Mandelbrot, Benoit B., Gaussian Self-Affinity and Fractals, Springer:
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.
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.
Wikimedia Commons has media related to Benoît Mandelbrot.
Wikiquote has quotations related to: 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
Mathematics Interviews, Cornell University Library.
Video animation of Mandelbrot set, zoom factor 10342.
Video animation of
Mandelbulb on YouTube, a three-dimensional
Video fly-through an animated
Mandelbulb world on YouTube
Benoit Mandelbrot on IMDb
Benoit Mandelbrot at TED
Laureates of the
Wolf Prize in Physics
Chien-Shiung Wu (1978)
George Uhlenbeck /
Giuseppe Occhialini (1979)
Michael Fisher /
Leo Kadanoff /
Kenneth G. Wilson (1980)
Freeman Dyson / Gerardus 't Hooft / Victor Weisskopf (1981)
Leon M. Lederman
Leon M. Lederman /
Martin Lewis Perl (1982)
Erwin Hahn /
Peter Hirsch /
Theodore Maiman (1983–84)
Conyers Herring /
Philippe Nozières (1984–85)
Mitchell Feigenbaum /
Albert J. Libchaber (1986)
Herbert Friedman /
Bruno Rossi /
Riccardo Giacconi (1987)
Roger Penrose /
Stephen Hawking (1988)
Pierre-Gilles de Gennes /
David J. Thouless
David J. Thouless (1990)
Maurice Goldhaber /
Valentine Telegdi (1991)
Joseph H. Taylor Jr. (1992)
Benoît Mandelbrot (1993)
Vitaly Ginzburg /
Yoichiro Nambu (1994–95)
John Wheeler (1996–97)
Yakir Aharonov / Michael Berry (1998)
Dan Shechtman (1999)
Raymond Davis Jr.
Raymond Davis Jr. /
Masatoshi Koshiba (2000)
Bertrand Halperin / Anthony Leggett (2002–03)
Robert Brout /
François Englert /
Peter Higgs (2004)
Daniel Kleppner (2005)
Albert Fert /
Peter Grünberg (2006–07)
John F. Clauser /
Alain Aspect /
Anton Zeilinger (2010)
Maximilian Haider /
Harald Rose /
Knut Urban (2011)
Jacob Bekenstein (2012)
Peter Zoller / Juan Ignacio Cirac (2013)
James D. Bjorken / Robert P. Kirshner (2015)
Yoseph Imry (2016)
Michel Mayor /
Didier Queloz (2017)
Charles H. Bennett /
Gilles Brassard (2018)
Chaos theory in organizational development
Control of chaos
Edge of chaos
Santa Fe Institute
Synchronization of chaos
Arnold's cat map
Complex quadratic map
Complex squaring map
Coupled map lattice
Double scroll attractor
Interval exchange map
Swinging Atwood's machine
Van der Pol oscillator
Bouncing ball dynamics
Leon O. Chua
Edward Norton Lorenz
Oleksandr Mykolayovych Sharkovsky
James A. Yorke
Burning Ship fractal
Helge von Koch
Lewis Fry Richardson
"How Long Is the Coast of Britain?"
List of fractals by Hausdorff dimension
The Beauty of Fractals
The Beauty of Fractals (1986 book)
Chaos: Making a New Science (1987 book)
Geometry of Nature (1982 book)
ISNI: 0000 0001 2096 8076
BNF: cb11914221j (data)