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第26行： − + − + − Benoit B. When speaking in French, Mandelbrot pronounced his name .|group=n}} (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"+ − + − 伯努瓦 b。在用法语说话时，曼德布洛特宣布了他的名字。| group = n }(1924年11月20日至2010年10月14日)是波兰出生的法国和美国数学家和博学家，对实用科学有着广泛的兴趣，特别是对他所称的物理现象的”粗糙艺术”和”生活中不受控制的元素”。他说自己是个“分裂主义者”+ − + − + − + − + − + − In 1936, while he was a child, Mandelbrot's family emigrated to France from Warsaw, Poland. After 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. − − 1936年，当曼德布洛特还是个孩子的时候，他的家人从波兰华沙移民到了法国。第二次世界大战结束后，曼德布洛特学习了数学，从巴黎和美国的大学毕业，并在加利福尼亚理工学院获得了航空学硕士学位。他的大部分职业生涯都在美国和法国度过，拥有法国和美国双重国籍。1958年，他在 IBM 开始了35年的职业生涯，在那里他成为了 IBM 研究员，并定期休假到哈佛大学教书。在哈佛大学，他发表了关于美国商品市场与棉花期货的研究成果之后，他开始教授经济学和应用科学。 − − − − − − 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 discovery of the Mandelbrot set in 1980. 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". − − 由于可以访问 IBM 的计算机，曼德尔布洛特是第一个使用计算机图形学图形来创建和显示分形几何图形的人，这导致了他在1980年发现了曼德尔布洛特集合。他展示了如何从简单的规则中创造出视觉的复杂性。他说，通常被认为是“粗糙”、“混乱”或“混乱”的事物，如云或海岸线，实际上有一定程度的秩序。 − − − − − − 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 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 posthumously in 2012. − − 在他职业生涯的最后阶段，他是耶鲁大学数学科学系的斯特林教席，也是耶鲁历史上获得终身教职的最年长的教授。曼德布洛特还在太平洋西北国家实验室、法国里尔北方的大学、高级研究所和法国国家科学研究中心担任过职务。在他的职业生涯中，他获得了超过15个荣誉博士学位，并在许多科学期刊上服务，同时还获得了许多奖项。他的自传《分析家: 科学怪人的回忆录》在他死后于2012年出版。 − − − − − − − − Mandelbrot was born in a Jewish family, in Warsaw during the Second Polish Republic. His father made his living trading clothing; his mother was a dental surgeon. During his first two school years, he was tutored privately by an uncle who despised rote learning: "Most of my time was spent playing chess, reading maps and learning how to open my eyes to everything around me." Later, the family's move to France, the war, and his acquaintance with his father's brother, the mathematician Szolem Mandelbrojt who had moved to Paris around 1920, further prevented a standard education. − − 曼德布洛特出生于第二波兰共和国时期的华沙的一个犹太家庭。他的父亲靠卖衣服为生，母亲是牙科医生。在他的头两个学年里，他的一位叔叔私下辅导他，他鄙视死记硬背: “我的大部分时间都花在下棋、看地图和学习如何睁开眼睛看我周围的一切。”后来，他们全家搬到了法国，战争爆发，他和父亲的兄弟，数学家 Szolem Mandelbrojt 相识，后者在1920年前后移居巴黎，这进一步阻碍了他接受标准教育。 − − + − − − 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. Mandelbrot attended the Lycée Rolin in Paris until the start of World War II, when his family moved to Tulle, France. He was helped by Rabbi David Feuerwerker, the Rabbi of Brive-la-Gaillarde, to continue his studies. Much of France was occupied by the Nazis at the time, and Mandelbrot recalls this period: − − 1936年，当他11岁的时候，他们一家从波兰移民到法国。他写道: “我的父母作为经济和政治难民加入了 Szolem，这一事实挽救了我们的生命。”。曼德布洛特曾在巴黎的 Lycée Rolin 工作，直到二战爆发，他的家人搬到了法国的图勒。他在布里夫拉盖亚尔德的犹太教教士 David Feuerwerker 的帮助下继续学业。当时法国大部分地区都被纳粹占领，曼德布洛特回忆起这段时期: − + + − {{quote|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? 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. − {引用 | 我们一直担心的是，一个足够坚定的敌人可能会向某个权威举报我们，我们将会被送去送死。来自巴黎的密友吉娜 · 莫汉格就遇到了这种情况，她是附近一个县城的内科医生。仅仅是为了消除竞争，另一个医生谴责她... 我们逃过了这个命运。谁知道为什么呢？1958年，这对夫妇移居美国，在那里曼德尔布洛特加入了位于约克敦海茨的 IBM 汤玛士·Ｊ·华生研究中心的研究人员。 − + − − 在他的早期工作中，他发现金融市场的价格变化并不遵循正态分布，而是具有无限方差的 l é vy 稳定分布。例如，他发现，棉花价格遵循一个 l é vy 稳定分布，参数 α 等于1.7，而不是像正态分布一样遵循2。“稳定”分布具有这样的性质: 一个随机变量的多个实例的和服从相同的分布，但具有更大的尺度参数。 − − − − 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 discovery of the [[Mandelbrot set]] in 1980. 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".<ref name="Wolfram" /> His math and geometry-centered research career included contributions to such fields as [[statistical physics]], [[meteorology]], [[hydrology]], [[geomorphology]], [[anatomy]], [[taxonomy (biology)|taxonomy]], [[neurology]], [[linguistics]], [[information technology]], [[computer graphics]], [[economics]], [[geology]], [[medicine]], [[physical cosmology]], [[engineering]], [[chaos theory]], [[econophysics]], [[metallurgy]] and the [[social science]]s.<ref>list includes specific sciences mentioned in [[#Hudson|Hudson & Mandelbrot]], the Prelude, p. xvi, and p. 26</ref> − − − − 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.<ref>{{cite web |author=Steve Olson | url=http://archives.yalealumnimagazine.com/issues/2004_11/mandelbrot.html |title=The Genius of the Unpredictable |publisher=Yale Alumni Magazine |date=November–December 2004 |accessdate=22 July 2014| author-link=Steve Olson (writer) }}</ref> Mandelbrot also held positions at the [[Pacific Northwest National Laboratory]], [[Université Lille Nord de France]], [[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 posthumously in 2012. − − 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 Fractal 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". − − 作为哈佛大学的客座教授，Mandelbrot 开始研究被称为 Julia 集的分形，它们在复平面的某些变换下是不变的。在加斯顿 · 朱莉娅和皮埃尔 · 法图之前工作的基础上，曼德尔布洛特使用计算机绘制了朱莉娅集合的图像。在研究这些 Julia 集的拓扑结构时，他研究了他在1979年提出的 Mandelbrot 集。1982年，曼德布洛特在《自然的分形几何》一书中扩展和更新了他的观点。这一有影响力的工作带来了分形进入主流的专业和流行数学，以及沉默批评家，谁曾驳回分形作为“程序人工制品”。 − − − − == Early years == − − Mandelbrot speaking about the [[Mandelbrot set, during his acceptance speech for the Légion d'honneur in 2006]] − − 曼德布洛特谈到了[2006年他在法国荣誉军团勋章的获奖感言中谈到了曼德布洛特集合] − − Mandelbrot was born in a [[Jews|Jewish]] family, in [[Warsaw]] during the [[Second Polish Republic]]. His father made his living trading clothing; his mother was a dental surgeon. During his first two school years, he was tutored privately by an uncle who despised rote learning: "Most of my time was spent playing chess, reading maps and learning how to open my eyes to everything around me."<ref name="wolf">{{cite journal |last=Mandelbrot |first=Benoît |title=The Wolf Prizes for Physics, ''A Maverick's Apprenticeship''|publisher=Imperial College Press |year=2002 |url=http://users.math.yale.edu/~bbm3/web_pdfs/mavericksApprenticeship.pdf}}</ref> Later, the family's move to France, the [[World War II|war]], and his acquaintance with his father's brother, the mathematician [[Szolem Mandelbrojt]] who had moved to Paris around 1920, further prevented a standard education. − − − − 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 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". 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": − 1975年，曼德布洛特创造了分形这个术语来描述这些结构，并首次发表了他的观点，后来被翻译成分形: 形式、机会和维度。计算机科学家、物理学家斯蒂芬•沃尔夫拉姆(Stephen Wolfram)表示，这本书对曼德布洛特来说是一个“突破” ，在此之前，他通常会“将相当简单的数学应用到以前几乎没有认真研究过数学的领域”。沃尔夫勒姆补充说，由于这项新研究的结果，他不再是一个“流浪的科学家” ，后来称他为“分形之父” :+ − 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.<ref name=Mandelbrot />{{rp|17}}<ref name="bbc_obit">{{cite news|url=https://www.bbc.co.uk/news/world-europe-11560101|title=BBC News – 'Fractal' mathematician Benoît Mandelbrot dies aged 85|date=17 October 2010|work=[[BBC Online]] | accessdate=17 October 2010}}</ref> Mandelbrot attended the Lycée Rolin in Paris until the start of [[World War II]], when his family moved to [[Tulle]], France. He was helped by [[Rabbi]] [[David Feuerwerker]], the Rabbi of [[Brive-la-Gaillarde]], to continue his studies.<ref name=Mandelbrot/>{{rp|62–63}}<ref>Hemenway P. (2005) ''Divine proportion: Phi in art, nature and science''. Psychology Press. {{isbn|0-415-34495-6}}</ref> Much of France was occupied by the Nazis at the time, and Mandelbrot recalls this period: + − {{quote|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.+ − {引用 | Mandelbrot 最终做了一项伟大的科学工作，确定了一个更强大、更基本的观点ーー简而言之，有一些被他称为“分形”的 Unicode几何图形列表，在所有尺度上都是同样“粗糙”的。无论你看起来多么近，它们都不会变得更简单，就像你脚下看到的岩石海岸线的部分看起来就像你从太空中看到的那样参差不齐。+ − {{quote|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?<ref name=Mandelbrot />{{rp|49}}}} + − A Mandelbrot set+ − 一个曼德布洛特集合+ − 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 (mathematician)|Paul Lévy]]. From 1947 to 1949 he studied at California Institute of Technology, where he earned a master's degree in aeronautics.<ref name="guardian_obit" /> Returning to France, he obtained his [[Doctor of Philosophy|PhD degree]] in Mathematical Sciences at the [[University of Paris]] in 1952.<ref name="wolf" /> − Mandelbrot, however, never felt he was inventing a new idea. He describes his feelings in a documentary with science writer Arthur C. Clarke: − 然而，曼德布洛特从未觉得自己正在发明一种新思想。他在与科学作家亚瑟·查理斯·克拉克的一部纪录片中描述了自己的感受:+ +
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'''Benoit B.'''<ref group="n" name="Mandelbrot's_name" /> '''Mandelbrot'''{{#tag:ref|Pronounced {{IPAc-en|ˈ|m|æ|n|d|əl|b|r|ɒ|t}} {{respell|MAN|dəl-brot}} in English.<ref>{{OED|Mandelbrot}}</ref> When speaking in French, Mandelbrot pronounced his name {{IPA-fr|bənwa mɑ̃dɛlbʁot|}}.<ref>Recording of the ceremony on 11 September 2006 at which Mandelbrot received the insignia for an Officer of the ''[[Legion of Honour|Légion d'honneur]]''.</ref>|group=n}} (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 [[Benoit Mandelbrot#Fractals and the .22theory of roughness.22|roughness]]" of physical phenomena and "the uncontrolled element in life".<ref>https://www.insidescience.org/news/remembering-father-fractals</ref><ref>[https://www.ted.com/talks/benoit_mandelbrot_fractals_the_art_of_roughness?language=en Benoit Mandelbrot: Fractals and the art of roughness]. ted.com (February 2010)</ref><ref>[[#Hudson|Hudson & Mandelbrot]], Prelude, page xviii</ref> He referred to himself as a "fractalist"<ref name=Mandelbrot/> and 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.<ref name=nature>{{Cite journal
'''Benoit B.'''<ref group="n" name="Mandelbrot's_name" /> '''Mandelbrot'''{{#tag:ref|Pronounced {{IPAc-en|ˈ|m|æ|n|d|əl|b|r|ɒ|t}} {{respell|MAN|dəl-brot}} in English.<ref>{{OED|Mandelbrot}}</ref> When speaking in French, Mandelbrot pronounced his name {{IPA-fr|bənwa mɑ̃dɛlbʁot|}}.<ref>Recording of the ceremony on 11 September 2006 at which Mandelbrot received the insignia for an Officer of the ''[[Legion of Honour|Légion d'honneur]]''.</ref>|group=n}} (20 November 1924 – 14 October 2010) was a Polish-born French-American [[mathematician]] and [[polymath]] with broad interests in the practical sciences, especially regarding what he labeled as "the art of [[Benoit Mandelbrot#Fractals and the "theory of roughness"|roughness]]" of physical phenomena and "the uncontrolled element in life".<ref>{{Cite web |url=https://www.insidescience.org/news/remembering-father-fractals |title=Archived copy |access-date=8 January 2018 |archive-date=8 January 2018 |archive-url=https://web.archive.org/web/20180108174954/https://www.insidescience.org/news/remembering-father-fractals |url-status=live }}</ref><ref>[https://www.ted.com/talks/benoit_mandelbrot_fractals_the_art_of_roughness?language=en Benoit Mandelbrot: Fractals and the art of roughness] {{Webarchive|url=https://web.archive.org/web/20160414183649/https://www.ted.com/talks/benoit_mandelbrot_fractals_the_art_of_roughness?language=en |date=14 April 2016 }}. ted.com (February 2010)</ref><ref>[[#Hudson|Hudson & Mandelbrot]], Prelude, page xviii</ref> He referred to himself as a "fractalist"<ref name=Mandelbrot/> and 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.<ref name=nature>{{Cite journal
| last1 = Gomory | first1 = R.
| author-link1 = Ralph E. Gomory
| title = Benoît Mandelbrot (1924–2010)
| doi = 10.1038/468378a
| journal = Nature
| last1 = Gomory | first1 = R.
| volume = 468
| issue = 7322
| authorlink1 = Ralph E. Gomory
| pages = 378
| year = 2010
| title = Benoît Mandelbrot (1924–2010)
| doi = 10.1038/468378a
| journal = Nature
| volume = 468
| issue = 7322
| pages = 378
| year = 2010
| pmid = 21085164
| pmid = 21085164
|bibcode = 2010Natur.468..378G | s2cid = 4393964
|bibcode = 2010Natur.468..378G | s2cid = 4393964
}}</ref>
}}</ref>
Benoit B.[n 1] Mandelbrot[n 2] (20 November 1924 – 14 October 2010) was a Polish-born French-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".[5][6][7] He referred to himself as a "fractalist"[8] and 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.
伯努瓦 曼德布洛特（1924年11月20日至2010年10月14日）是波兰裔法国裔美国数学家和博学家，对实用科学有着广泛的兴趣。他将其称为物理现象的“粗糙艺术”和“生活中不受控制的元素”。他称自己为“分形主义者”，并因其对分形几何学领域的贡献而受到认可，其中包括创造了“分形Fractal”一词，并发展了自然界中的“粗糙度Roughness和自相似性Self-similarity”理论。
In 1936, while he was a child, Mandelbrot's family emigrated to France from [[Warsaw]], Poland. After [[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 [[Multiple citizenship|dual]] [[French nationality law|French]] and [[United States nationality law#Dual citizenship|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.
In 1936, while he was a child, Mandelbrot's family emigrated to France from [[Warsaw]], Poland. After [[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 [[Multiple citizenship|dual]] [[French nationality law|French]] and [[United States nationality law#Dual citizenship|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.
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.
In 1936, while he was a child, Mandelbrot's family emigrated to France from Warsaw, Poland. After 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.
1936年，当曼德布罗特还是个孩子时，一家人从波兰华沙移民到了法国。第二次世界大战结束后，曼德布洛特学习了数学，从巴黎和美国的大学毕业，并获得了加州理工学院的航空硕士学位。他的职业生涯大部分时间都是在美国和法国度过，拥有法国和美国双重国籍。1958年，他在IBM开始了35年的职业生涯，并在那里成为了IBM研究员，定期请假到哈佛大学任教。在哈佛大学发表关于棉花期货的美国商品市场研究之后，他开始教授经济学和应用科学。
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 discovery of the [[Mandelbrot set]] in 1980. He showed how visual complexity can be created from simple rules. He said that things typically considered to be "rough", a "mess", or "chaotic", such as clouds or shorelines, actually had a "degree of order".<ref name="Wolfram" /> His math and geometry-centered research career included contributions to such fields as [[statistical physics]], [[meteorology]], [[hydrology]], [[geomorphology]], [[anatomy]], [[taxonomy (biology)|taxonomy]], [[neurology]], [[linguistics]], [[information technology]], [[computer graphics]], [[economics]], [[geology]], [[medicine]], [[physical cosmology]], [[engineering]], [[chaos theory]], [[econophysics]], [[metallurgy]], and the [[social science]]s.<ref>list includes specific sciences mentioned in [[#Hudson|Hudson & Mandelbrot]], the Prelude, p. xvi, and p. 26</ref>
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 discovery of the Mandelbrot set in 1980. He showed how visual complexity can be created from simple rules. He said that things typically considered to be "rough", a "mess", or "chaotic", such as clouds or shorelines, actually had a "degree of order".[10] 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.
由于曼德布罗特可以使用IBM的计算机，因此他是最早使用计算机图形来创建和显示分形几何图像的人之一，因此他于1980年发现了曼德布洛特集合。他展示了如何从简单的规则图形创建出视觉复杂性。他认为那些通常被认为是“粗糙”，“杂乱”或“混乱”的事物，例如云层或海岸线，实际上都具有“有序度Degree of order”。他以数学和几何学为中心的延申研究领域包括了统计物理学，气象学，水文学，地貌学，解剖学，分类学，神经学，语言学，信息技术，计算机图形学，经济学，地质学，医学，物理宇宙学，工程学，混沌理论等领域的贡献 ，经济物理学，冶金学和社会科学。
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.<ref>{{cite web |author=Steve Olson |url=http://archives.yalealumnimagazine.com/issues/2004_11/mandelbrot.html |title=The Genius of the Unpredictable |publisher=Yale Alumni Magazine |date=November–December 2004 |access-date=22 July 2014 |author-link=Steve Olson (writer) |archive-date=22 October 2014 |archive-url=https://web.archive.org/web/20141022092605/http://archives.yalealumnimagazine.com/issues/2004_11/mandelbrot.html |url-status=live }}</ref> Mandelbrot also held positions at the [[Pacific Northwest National Laboratory]], [[Université Lille Nord de France]], [[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 posthumously in 2012.
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.[12] Mandelbrot also held positions at the Pacific Northwest National Laboratory, Université Lille Nord de France, 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 posthumously in 2012.
在他职业生涯后期，他是耶鲁大学数学科学系的斯特林教授，在那里他被授予了耶鲁历史上最年长的终身教授职位。曼德布罗特还在西北太平洋国家实验室，里尔-北法兰西院校联盟，普林斯顿高等研究院和法国国家科学研究中心担任过职务。他的自传《分形主义者:一个科学特立独行者的回忆录》于2012年死后出版。
== Early years 早年生活 ==
[[文件:Mandelbrot p1130861.jpg|缩略图|右|家庭背景和早期教育，（4:11）伯努瓦 曼德布洛特访谈，《Web of Stories》 144第1部分]]