Johannes Kepler
Johannes Kepler
Szolem Mandelbrojt
Szolem Mandelbrojt
INFLUENCED
Nassim Nicholas Taleb
NOTABLE AWARDS
Légion d\'honneur
(Chevalier 1990 · Officier 2006) 2003
Japan Prize
Japan Prize
1993
Wolf Prize
1989
Harvey Prize
1986
Franklin Medal
Franklin Medal
1985 Barnard Medal
SPOUSE
Aliette Kagan
married 1955–2010 (his death)
BENOIT B. MANDELBROT (20 November 1924 – 14 October 2010)
was a Polish-born, French and American mathematician 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 migrated to
France. 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
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." 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 .
Toward the end of his career, he was
Sterling Professor of
Mathematical Sciences at
Yale University
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
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.
CONTENTS
* 1 Early years
* 2 Research career
* 2.1 States of randomness and financial markets
* 2.2 Developing "fractal geometry" and the
Mandelbrot set
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
Mandelbrot was born in
Warsaw
Warsaw during the
Second Polish Republic
Second Polish Republic . His
family was
Jewish
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
Szolem Mandelbrojt , was a mathematician
who resided in Paris. According to Mandelbrot's autobiography, The
Fractalist - Memoir of a Scientific Maverick, "he love of his 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
World War II , when his
family then moved to
Tulle
Tulle , France. He was helped by
Rabbi
Rabbi David
Feuerwerker , the
Rabbi
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
Lyon , and in 1945 to 1947 attended the
École Polytechnique
École Polytechnique ,
where he studied under
Gaston Julia and Paul Lévy . From 1947 to 1949
he studied at
California Institute of Technology
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
1952.
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
Institute for Advanced Study in
Princeton, New Jersey
Princeton, New Jersey ,
where he was sponsored by
John von Neumann . In 1955 he married
Aliette Kagan and moved to
Geneva, Switzerland
Geneva, Switzerland , and later to the
Université Lille Nord de France
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
Thomas J. Watson Research Center in
Yorktown Heights, New York
Yorktown Heights, New York . He
remained at
IBM
IBM for 35 years, becoming an
IBM
IBM Fellow , and later
Fellow
Emeritus .
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 .
STATES OF RANDOMNESS AND FINANCIAL MARKETS
Mandelbrot found that price changes in financial markets did not
follow a
Gaussian distribution , but rather Lévy stable distributions
having theoretically 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 .
DEVELOPING "FRACTAL GEOMETRY" AND THE MANDELBROT SET
As a visiting professor at
Harvard University
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
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 fractal that is now named
after him. In 1982, Mandelbrot expanded and updated his ideas in The
Fractal
Fractal
Geometry
Geometry of Nature . This influential work brought fractals
into the mainstream of professional and popular mathematics, as well
as silencing critics, who had dismissed fractals as "program artifacts
". Mandelbrot speaking about the
Mandelbrot set
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
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 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.
Mandelbrot used the term "fractal" as it derived from the Latin word
"fractus", defined as broken or shattered glass. Using the newly
developed
IBM
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
Johannes Kepler , who calculated and described
the orbits of the planets. A
Mandelbrot set
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.
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."
Mandelbrot left
IBM
IBM in 1987, after 35 years and 12 days, when IBM
decided to end pure research in his division. He joined the
Department of
Mathematics
Mathematics at
Yale
Yale , and obtained his first tenured
post in 1999, at the age of 75. 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, 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 seemed to
promise—truthful measurement, not only of cultivated fields along
the Nile River but also of untamed Earth? :xii
In his paper entitled 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
Fractal
Fractal
Geometry
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. :292
Section of a
Mandelbrot set
Mandelbrot set
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
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
Fractal
Geometry
Geometry of Nature
Mandelbrot has been called 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
Fractal
Fractal
Geometry
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
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.
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 in
2000, the
Japan Prize
Japan Prize in 2003, and the Einstein Lectureship of the
American Mathematical Society in 2006.
The small asteroid
27500 Mandelbrot was named in his honor. In
November 1990, he was made a Knight in the French
Legion of Honour
Legion of Honour .
In December 2005, Mandelbrot was appointed to the position of Battelle
Fellow at the
Pacific Northwest National Laboratory
Pacific Northwest National Laboratory . Mandelbrot was
promoted to 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
Book of the Year Award
* AMS Einstein Lectureship
* Barnard Medal
* Caltech Service
*
Casimir Funk Natural Sciences Award
*
Charles Proteus Steinmetz
Charles Proteus Steinmetz Medal
* Fellow, American Geophysical Union
*
Fellow of the American Statistical Association
*
Franklin Medal
Franklin Medal
*
Harvey Prize (1989)
* Honda Prize
* Humboldt Preis
*
IBM
IBM Fellowship
*
Japan Prize
Japan Prize (2003)
*
John Scott Award
John Scott Award
*
Légion d'honneur
Légion d'honneur (
Legion of Honour
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 .
* Science for Art
* Sven Berggren-Priset
* Władysław Orlicz Prize
*
Wolf Foundation Prize for Physics (1993)
DEATH AND LEGACY
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. Reacting to news of
his death, mathematician
Heinz-Otto Peitgen said: "f we talk about
impact inside mathematics, and applications in the sciences, he is one
of the most important figures of the last fifty years."
Chris Anderson , TED conference curator, described Mandelbrot as "an
icon who changed how we see the world".
Nicolas Sarkozy
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 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 fractal geometry".
Best-selling essayist-author
Nassim Nicholas Taleb , a Mandelbrot
protégé and a scientific adviser at Universa Investments , has
remarked that Mandelbrot's book The (Mis)Behavior of Markets is in his
opinion "The deepest and most realistic finance book ever published".
BIBLIOGRAPHY
IN ENGLISH
* Fractals: Form, Chance and Dimension , 1977
* The
Fractal
Fractal
Geometry
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.
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
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
* 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 "The
Mandelbrot Set", a song dedicated to Mandelbrot and his famous
fractal. In 2007, the author
Laura Ruby published a sequel to "The
Wall and the Wing" series named "The
Chaos King". One of the main
characters is named Mandelbrot and his work is referenced in the
novel, specifically, the chaos theory .
SEE ALSO
External video
Family background and early education, (4:11) Benoit Mandelbrot
interview, Part 1 of 144,
Web of Stories
Web of Stories
* "How Long is the Coast of Britain? "
*
Louis Bachelier
*
Zipf-Mandelbrot law
*
Seven states of randomness
*
Skewness risk
*
Kurtosis risk
*
Fractal
Fractal dimension
*
Lacunarity
*
Self-similarity
Self-similarity
*
Self-affinity
*
Hurst exponent
*
Fractional Brownian motion
*
Multifractal system
*
1/f noise
1/f noise
*
Mandelbrot Competition
NOTES
* ^ 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 . The New York Times obituary stated that "he added the middle
initial himself, though it does not stand for a middle name". But
other sources suggest that he intended his middle initial B. to
recursively mean Benoit B. Mandelbrot, thereby including a fractal
(his mathematical discovery) in his own name.
* ^ Pronounced /ˈmændəlbrɒt/ MAN-dəl-brot in English. When
speaking in French, Mandelbrot pronounced his name .
REFERENCES
* ^ 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
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 .
* ^ 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. PMID 21085164 .
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
Unpredictable".
Yale
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
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.
* ^ "\'\'New Scientist\'\', 19 April 1997". Newscientist.com. 19
April 1997. Retrieved 17 October 2010.
* ^ The
Fractal
Fractal
Geometry
Geometry of Nature, by Benoît Mandelbrot; W H
Freeman 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
Web of Stories . Retrieved 17 October 2010.
* ^ Tenner, Edward (16 October 2010). "Benoît Mandelbrot the
Maverick, 1924–2010".
The Atlantic
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. PMID 17837158 . doi
:10.1126/science.156.3775.636 .
* ^ 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
Galaxy Map Hints at
Fractal
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 Octob