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伯努瓦·曼德布洛特 Benoit Mandelbrot (查看源代码)

2021年5月27日 (四) 17:11的版本

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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?

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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?

我们一直担心的是，一旦敌人下定决心将我们报告给当局，等待我们的就是死刑。类似的事情就发生在巴黎的一位密友Zina Morhange深上，她曾是附近某个县城的医生。当时只是为了解决竞争对手，另一位医生就举报了她...而我们逃过了一劫。谁知道是为什么？

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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" />

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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.[2] Returning to France, he obtained his PhD degree in Mathematical Sciences at the University of Paris in 1952.

1944年，曼德布洛特回到巴黎，在里昂的帕克中学学习，并于1945年至1947年考上了巴黎综合理工学院，在加斯顿·朱莉亚Gaston Julia和保罗·列维Paul Lévy的指导下学习。之后的1947年到1949年，他就读于加利福尼亚理工学院，在那里获得了航空硕士学位。返回法国后，他于1952年在巴黎大学获得数学科学博士学位。

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From 1949 to 1958, Mandelbrot was a staff member at the [[Centre National de la Recherche Scientifique]]. During this time he spent a year at the [[Institute for Advanced Study]] in [[Princeton, New Jersey]], where he was sponsored by [[John von Neumann]]. In 1955 he married Aliette Kagan and moved to [[Geneva, Switzerland]] (to collaborate with [[Jean Piaget]] at the International Centre for Genetic Epistemology) and later to the [[Université Lille Nord de France]].<ref name="people">{{Cite web|last=Barcellos|first=Anthony|title=Mathematical People, ''Interview of B. B. Mandelbrot''|publisher=Birkhaüser|year=1984|url=http://users.math.yale.edu/~bbm3/web_pdfs/inHisOwnWords.pdf|access-date=25 June 2013|archive-date=27 April 2015|archive-url=https://web.archive.org/web/20150427164851/http://users.math.yale.edu/~bbm3/web_pdfs/inHisOwnWords.pdf|url-status=live}}</ref> In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the [[International Business Machines|IBM]] [[Thomas J. Watson Research Center]] in [[Yorktown Heights, New York]].<ref name="people" /> He remained at IBM for 35 years, becoming an IBM Fellow, and later Fellow [[Emeritus]].<ref name="wolf" />

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From 1949 to 1958, Mandelbrot was a staff member at the Centre National de la Recherche Scientifique. During this time he spent a year at the Institute for Advanced Study in Princeton, New Jersey, where he was sponsored by John von Neumann. In 1955 he married Aliette Kagan and moved to Geneva, Switzerland (to collaborate with Jean Piaget at the International Centre for Genetic Epistemology) and later to the Université Lille Nord de France.[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.

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自1949年到1958年，曼德布洛特任职于法国国家科学研究中心。在此期间，他得到了约翰·冯·诺伊曼John von Neumann的赞助，在新泽西州普林斯顿的高级研究学院度过了一年。1955年，他与阿利耶特·卡甘结婚，并搬到瑞士日内瓦（与国际遗传认识论中心的让·皮亚杰Jean Piaget合作），后来又迁往法国里尔大学。到了1958年，这对夫妇搬到了美国，在那里曼德布洛特加入了位于纽约约克敦高地的IBM托马斯·沃森研究中心。他在IBM待了35年，成为IBM院士，后来成为了荣誉退休院士。

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自1949年到1958年，曼德布洛特任职于法国国家科学研究中心。在此期间，他得到了约翰·冯·诺伊曼John von Neumann的赞助，在新泽西州普林斯顿的高级研究学院度过了一年。1955年，他与阿利耶特·卡甘结婚，并搬到瑞士日内瓦（与国际遗传认识论中心的让·皮亚杰Jean Piaget合作），后来又迁往法国里尔大学。到了1958年，这对夫妇搬到了美国，在那里曼德布洛特加入了位于纽约约克敦高地的IBM托马斯·沃森研究中心。他在IBM待了35年，并且成为IBM院士，后来成为了荣誉退休院士。

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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]].

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~~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.~~

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从1951年起，曼德布洛特开始致力于研究相关问题并发表论文，不仅在数学领域，而且在信息论，经济学和流体动力学等应用领域中也发表许多研究论文。

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~~从1951年起，曼德布洛特致力于研究相关问题并发表论文，不仅在数学领域，而且在诸如信息论，经济学和流体动力学等应用领域中也发表了论文。~~

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Mandelbrot saw [[financial market]]s as an example of "wild randomness", characterized by concentration and long range dependence. He developed several original approaches for modelling financial fluctuations.<ref>{{cite book |author= Rama Cont |chapter= Mandelbrot, Benoit |journal= Encyclopedia of Quantitative Finance |publisher= Wiley |date= 19 April 2010 |doi= 10.1002/9780470061602.eqf01006 |isbn = 9780470057568|author-link= Rama Cont }}</ref> In his early work, he found that the price changes in [[financial market]]s did not follow a [[Gaussian distribution]], but rather [[Paul Lévy (mathematician)|Lévy]] [[stable distributions]] having infinite [[variance]]. He found, for example, that cotton prices followed a Lévy stable distribution with parameter ''α'' equal to 1.7 rather than 2 as in a Gaussian distribution. "Stable" distributions have the property that the sum of many instances of a random variable follows the same distribution but with a larger [[scale parameter]].<ref>{{cite web |url=https://www.newscientist.com/article/mg15420784.700-flight-over-wall-st.html |title=''New Scientist'', 19 April 1997 |publisher=Newscientist.com |date=19 April 1997 |access-date=17 October 2010 |archive-date=21 April 2010 |archive-url=https://web.archive.org/web/20100421101729/http://www.newscientist.com/article/mg15420784.700-flight-over-wall-st.html |url-status=live }}</ref>

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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.

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曼德布洛特将金融市场视为“野生随机性”的很好案例，其特征是集中性和长相关性。为此他开发了几种可以模拟财务波动的创新方法。在他的早期工作中，他发现金融市场的价格变化并非遵循高斯分布，而是遵循具有无限方差的''' 列维稳定分布Lévy stable distributions '''。他发现，例如棉花价格遵循列维稳定分布，其参数α等于1.7，而不是高斯分布中的2。该“稳定”分布具有以下性质：随机变量的许多实例总和遵循相同的分布，只是比例参数较大。

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曼德布洛特将金融市场视为“野生随机性”的一个例子，其特征是集中性和长相关性。为此他开发了几种可以模拟财务波动的创新方法。在他的早期工作中，他发现金融市场的价格变化并非遵循高斯分布，而是遵循具有无限方差的''' 列维稳定分布Lévy stable distributions '''。他发现，例如棉花价格遵循列维稳定分布，其参数α等于1.7，而不是高斯分布中的2。该“稳定”分布具有以下性质：随机变量的许多实例总和遵循相同的分布，只是比例参数较大。

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As a visiting professor at [[Harvard University]], Mandelbrot began to study fractals called [[Julia set]]s 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]]''.<ref>[https://books.google.com/books?id=xJ4qiEBNP4gC&printsec=frontcover ''The Fractal Geometry of Nature''] {{Webarchive|url=https://web.archive.org/web/20151130231048/https://books.google.com/books?id=xJ4qiEBNP4gC&printsec=frontcover |date=30 November 2015 }}, by Benoît Mandelbrot; W H Freeman & Co, 1982; {{isbn|0-7167-1186-9}}</ref> This influential work brought fractals into the mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as "[[Artifact (observational)|program artifacts]]".

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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.[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".

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作为哈佛大学的客座教授，曼德布洛特开始研究名为''' 朱莉娅集合Julia sets'''的分形，这些分形在复杂平面的变换下依旧保持不变。在加斯顿·朱莉娅Gaston Julia和皮埃尔·法图Pierre Fatou先前工作的基础上，曼德尔布洛特使用计算机绘制出了朱莉娅集合的图像。在他研究这些朱莉娅集的拓扑时，于1979年提出的曼德布洛特集。1982年，他在《大自然的分形几何学The Fractal Geometry of Nature》一书中扩展并更新了他的思想。这项颇具影响力的著作将分形技术引入到专业数学和大众数学中，同时也进入到了那些将分形技术仅视为“程序工件”的批评者眼中。

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作为哈佛大学的客座教授，曼德布洛特开始研究名为''' 朱莉娅集合Julia sets'''的分形，这些分形在复杂平面的变换下依旧保持不变。在加斯顿·朱莉娅Gaston Julia和皮埃尔·法图Pierre Fatou先前工作的基础上，曼德尔布洛特使用计算机绘制出了朱莉娅集合的图像。在他研究这些朱莉娅集的拓扑时，他于1979年提出的曼德布洛特集。1982年，他在《大自然的分形几何学The Fractal Geometry of Nature》一书中扩展并更新了他的思想。但是这项颇具影响力的著作不仅将分形技术引入到专业数学和大众数学中，也同时进入到了那些将分形技术仅视为“程序工件”的批评者眼中。

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In 1975, Mandelbrot coined the term ''[[fractal]]'' to describe these structures and first published his ideas, and later translated, ''Fractals: Form, Chance and Dimension''.<ref>''Fractals: Form, Chance and Dimension'', by Benoît Mandelbrot; W H Freeman and Co, 1977; {{isbn|0-7167-0473-0}}</ref> According to computer scientist and physicist [[Stephen Wolfram]], the book was a "breakthrough" for Mandelbrot, who until then would typically "apply fairly straightforward mathematics ... to areas that had barely seen the light of serious mathematics before".<ref name=Wolfram>Wolfram, Stephen. [https://www.wsj.com/articles/SB10001424127887324439804578107271772910506 "The Father of Fractals"] {{Webarchive|url=https://web.archive.org/web/20170825102714/https://www.wsj.com/articles/SB10001424127887324439804578107271772910506 |date=25 August 2017 }}, ''Wall Street Journal'', 22 November 2012</ref> Wolfram adds that as a result of this new research, he was no longer a "wandering scientist", and later called him "the father of fractals":

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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 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".[10] 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”一词来描述这些结构，并首先发表了他的想法，其翻译为《分形：形式，机会和维度Fractals: Form, Chance and Dimension》。根据计算机科学家和物理学家斯蒂芬·沃尔夫拉姆Stephen Wolfram的说法，这本书对曼德尔布洛特来说是一个“突破”，他在那之前通常会“将相当简单的数学应用于……以前几乎没有看到过的严肃数学领域”。沃尔夫拉姆补充说，由于这项新研究，他不再是“流浪的科学家”，他被称为“分形之父”：

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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.

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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.

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曼德布洛特最终完成了一部伟大的科学著作。他找到了更强大，更根本的概念，简单地说，就是有些几何形状（他称之为“分形”）在各个尺度上都同样“粗糙”。不管您凑多近去看，它们都永远不会变得简单，就像您在脚下看到的多岩石的海岸线与从太空中看到的伸展部分一样参差不齐。

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事实证明曼德布洛特最终完成了一部伟大的科学著作。他建立了一个基本的概念，简单地说，有些几何形状（他称之为“分形”）在各个尺度上都同样“粗糙”。不管您凑多近去看，它们都永远不会变得简单，就像您在脚下看到的多岩石的海岸线与从太空中看到的伸展部分一样参差不齐。

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Wolfram briefly describes fractals as a form of geometric repetition, "in which smaller and smaller copies of a pattern are successively nested inside each other, so that the same intricate shapes appear no matter how much you zoom in to the whole. [[Fern|Fern leaves]] and [[Romanesco broccoli|Romanesque broccoli]] are two examples from nature."<ref name=Wolfram /> He points out an unexpected conclusion:

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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 Romanesque broccoli are two examples from nature."[10] He points out an unexpected conclusion:

沃尔夫拉姆简要地将分形描述为几何重复的一种形式，“在其中，越来越少的相同复制图案相继被嵌套在彼此内部，因此无论您放大多少，同样的复杂形状都会展现出来。蕨叶和罗马式西兰花是自然界的两个例子。“他指出了一个出乎意料的结论：

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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.

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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.

可能有人以为，这种简单而基本的规律已经被研究了数百年甚至数千年，但事实并非如此。实际上，它仅在过去30多年中才受到关注，而且几乎完全是通过一个人的努力，即数学家伯努瓦曼德布洛特。

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Mandelbrot used the term "fractal" as it derived from the Latin word "fractus", defined as broken or shattered glass. Using the newly developed IBM computers at his disposal, Mandelbrot was able to create fractal images using graphics computer code, images that an interviewer described as looking like "the delirious exuberance of the 1960s [[psychedelic art]] with forms hauntingly reminiscent of nature and the human body". He also saw himself as a "would-be Kepler", after the 17th-century scientist [[Johannes Kepler]], who calculated and described the orbits of the planets.<ref>Ivry, Benjamin. [http://forward.com/articles/166094/benoit-mandelbrot-influenced-art-and-mathematics/?p=all "Benoit Mandelbrot Influenced Art and Mathematics"] {{Webarchive|url=https://web.archive.org/web/20130602171300/http://forward.com/articles/166094/benoit-mandelbrot-influenced-art-and-mathematics/?p=all |date=2 June 2013 }}, ''Forward'', 17 November 2012</ref>

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Mandelbrot used the term "fractal" as it derived from the Latin word "fractus", defined as broken or shattered glass. Using the newly developed IBM computers at his disposal, Mandelbrot was able to create fractal images using graphics computer code, images that an interviewer described as looking like "the delirious exuberance of the 1960s psychedelic art with forms hauntingly reminiscent of nature and the human body". He also saw himself as a "would-be Kepler", after the 17th-century scientist Johannes Kepler, who calculated and described the orbits of the planets.

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曼德布洛特使用“分形”一词，因为它源自拉丁语 fractus”，意为碎玻璃。通过使用新开发的IBM计算机和图形计算机代码，曼德布洛特创建了分形图像，这些图像被采访者描述为“1960年代迷幻艺术的疯狂，其形式令人不禁联想到自然和人体”。他还将自己视为“未来的开普勒”（在17世纪，科学家约翰尼斯·开普勒Johannes Kepler计算并描述了行星的轨道）。

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~~曼德布洛特使用了“分形”一词，因为它源自拉丁语~~ fractus”，意为碎玻璃。通过使用新开发的IBM计算机和图形计算机代码，曼德布洛特创建了分形图像，这些图像被采访者描述为“1960年代迷幻艺术的疯狂，其形式令人不禁联想到自然和人体”。他还将自己视为“未来的开普勒”（在17世纪，科学家约翰尼斯·开普勒Johannes Kepler计算并描述了行星的轨道）。

[[文件:Newton-lplane-Mandelbrot.jpg|缩略图|右|一个德布洛特集合]]

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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.

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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

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在探索这个集合的时候我并未感觉在发明一个新概念。我也从没有感觉到我的想象力足以发现所有这些非凡的事物。其实他们就一直呈现在那里，只是过去没有人注意过他们。一个如此简单的公式就可以解释所有这些异常复杂的事物，太难以置信了。因此，科学的目标是从一团乱开始，再由一个简单的公式来解释它。我想这是研究科学的梦想。

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According to Clarke, "the [[Mandelbrot set]] is indeed one of the most astonishing discoveries in the entire history of mathematics. Who could have dreamed that such an incredibly simple equation could have generated images of literally ''infinite'' complexity?" Clarke also notes an "odd coincidence<blockquote>the name Mandelbrot, and the word "[[mandala]]"—for a religious symbol—which I'm sure is a pure coincidence, but indeed the Mandelbrot set does seem to contain an enormous number of mandalas.<ref name=Clarke /></blockquote>

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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 mandalas.

克拉克Clarke说：“曼德布洛特集确实是整个数学史上最惊人的发现之一。谁能想到，如此难以置信的简单方程式就可以生成视觉上无限复杂的图像？“克拉克还注意到了一个奇怪的巧合”：我确信曼德布洛特的名称和“曼陀罗”（一个宗教象征）一词纯属巧合，但确实曼德布洛特集似乎包含了大量的曼陀罗。

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Mandelbrot left IBM in 1987, after 35 years and 12 days, when IBM decided to end pure research in his division.<ref name="wos44">{{cite web|url=http://www.webofstories.com/play/10483|title=Web of Stories • Benoît Mandelbrot • IBM: background and policies|last=Mandelbrot|first=Benoît|author2=Bernard Sapoval|author3=Daniel Zajdenweber|date=May 1998|publisher=[[Web of Stories]]|access-date=17 October 2010|archive-date=8 September 2011|archive-url=https://web.archive.org/web/20110908162215/http://www.webofstories.com/play/10483|url-status=live}}</ref> He joined the Department of Mathematics at [[Yale]], and obtained his first [[tenure]]d post in 1999, at the age of 75.<ref name="Tenner">{{cite news|url=https://www.theatlantic.com/technology/archive/2010/10/benoit-mandelbrot-the-maverick-1924-2010/64684/|title=Benoît Mandelbrot the Maverick, 1924–2010|last=Tenner|first=Edward|date=16 October 2010|work=[[The Atlantic]]|access-date=16 October 2010|archive-date=18 October 2010|archive-url=https://web.archive.org/web/20101018132145/http://www.theatlantic.com/technology/archive/2010/10/benoit-mandelbrot-the-maverick-1924-2010/64684/|url-status=live}}</ref> At the time of his retirement in 2005, he was Sterling Professor of Mathematical Sciences.

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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 of Mathematical Sciences.

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经过35年零12天的努力，曼德布洛特在1987年离开了IBM，当时IBM决定结束其部门的研究。之后他加入了耶鲁大学数学系，并于1999年以75岁的高龄获得了第一任终身职位。在2005年退休之时，他成为了斯特林数学科学教授。

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经过35年零12天的努力，曼德布洛特在1987年离开了IBM，当时IBM决定结束其部门的研究。随后他加入了耶鲁大学数学系，并于1999年以75岁的高龄获得了第一任终身职位。在2005年退休之时，他成为了斯特林数学科学教授。

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Mandelbrot created the first-ever "theory of roughness", and he saw "roughness" in the shapes of mountains, [[coastline]]s and [[river basin]]s; the structures of plants, [[blood vessel]]s and [[lung]]s; the clustering of [[galaxy|galaxies]]. His personal quest was to create some mathematical formula to measure the overall "roughness" of such objects in nature.<ref name=Mandelbrot />{{rp|xi}} He began by asking himself various kinds of questions related to nature:

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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.[8]:xi He began by asking himself various kinds of questions related to nature:

曼德布洛特创造了第一个“粗糙度理论”，他看到了山脉，海岸线和河流盆地形状的“粗糙度”。植物，血管和肺的结构的“粗糙度”；还有星系聚集的“粗糙度”。他个人的追求是创建一些数学公式来测量此类物体在自然界中的整体“粗糙度”。他首先问自己各种与自然有关的问题：

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Can [[geometry]] deliver what the Greek root of its name [geo-] seemed to promise—truthful measurement, not only of cultivated fields along the Nile River but also of untamed Earth?<ref name=Mandelbrot>Mandelbrot, Benoit (2012). ''The Fractalist: Memoir of a Scientific Maverick'', Pantheon Books.

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~~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?~~

几何图形能否传达其希腊词根[geo-]所蕴含的内容，即追求真实的测量数据？不仅能测量尼罗河沿岸的耕地，还能测量未开发的土地？

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In his paper titled [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension]] published in [[Science (journal)|''Science'']] in 1967 Mandelbrot discusses [[self-similarity|self-similar]] curves that have [[Hausdorff dimension]] that are examples of ''fractals'', although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals.<ref>"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). "[https://www.nytimes.com/2010/10/17/us/17mandelbrot.html?adxnnl=1&adxnnlx=1332064840-/vD0Sjafcl9t9BNghRf8Qw Benoît Mandelbrot, Novel Mathematician, Dies at 85] {{Webarchive|url=https://web.archive.org/web/20181231150228/https://www.nytimes.com/2010/10/17/us/17mandelbrot.html?adxnnl=1&adxnnlx=1332064840-%2FvD0Sjafcl9t9BNghRf8Qw |date=31 December 2018 }}", ''The New York Times''.</ref><ref name="Mandelbrot_Science_1967">{{cite journal | title=How long is the coast of Britain? Statistical self-similarity and fractional dimension | journal=Science | date=5 May 1967 | author=Mandelbrot, Benoit B. | pages=636–638 | volume=156 | issue=3775 | doi=10.1126/science.156.3775.636 | pmid=17837158 | url=http://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf | bibcode=1967Sci...156..636M | s2cid=15662830 | access-date=11 January 2016 | archive-date=13 July 2015 | archive-url=https://web.archive.org/web/20150713023120/http://www.sciencemag.org/content/156/3775/636 | url-status=live }}</ref>

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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.

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曼德布洛特在1967年《科学》杂志上发表的论文《英国的海岸线有多长？统计自相似性和分形维数How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension》中讨论了''' 豪斯多夫维数Hausdorff dimension'''的自相似曲线。这些都是分形的例子，尽管曼德布洛特在论文中并没有使用这个术语，因为他直到1975年才创造这个名词。该论文是曼德布洛特关于分形主题的第一批出版物之一。

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曼德布洛特在1967年《科学》杂志上发表的论文《英国的海岸线有多长？统计自相似性和分形维数How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension》中讨论了''' 豪斯多夫维数Hausdorff dimension'''的自相似曲线。这些都是分形的例子，尽管当时曼德布洛特在论文中并没有使用这个术语，因为他直到1975年才创造这个名词。该论文是曼德布洛特关于分形主题的第一批出版物之一。

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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."[8]:296 Although Mandelbrot coined the term "fractal", some of the mathematical objects he presented in The Fractal Geometry of Nature had been previously described by other mathematicians. Before Mandelbrot, however, they were regarded as isolated curiosities with unnatural and non-intuitive properties. Mandelbrot brought these objects together for the first time and turned them into essential tools for the long-stalled effort to extend the scope of science to explaining non-smooth, "rough" objects in the real world. His methods of research were both old and new:

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曼德布洛特特地强调可以使用分形作为描述现实世界中多数“粗糙”现象的模型，因其能真实且有效地展现出来。他还总结道：“实际粗糙度通常都是分形的，是可以测量出来的。” 不过，尽管他创造了“分形”一词，但他在《大自然的分形几何学》中提出的一些数学现象之前曾被其他数学家描述过。只是在曼德布洛特之前，这些现象被视为不自然和非直觉特性的特例存在。是曼德布洛特首次将这些现象或物体放在一起，并将它们变成了必要的工具，通过长期的努力，以科学的范畴去解释现实世界中这些非光滑的“粗糙”物体。

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曼德布洛特特地强调可以使用分形作为描述现实世界中多数“粗糙”现象的模型，因其能真实且有效地展现出来。他还总结道：“实际粗糙度通常都是分形的，是可以测量出来的。” 不过，尽管他创造了“分形”一词，但他在《大自然的分形几何学》中提出的一些数学现象之前曾被其他数学家描述过。只是在曼德布洛特之前，这些现象被视为不自然并且非直觉特性的特例存在。是曼德布洛特首次将这些现象或物体放在一起，并将它们变成了必要的工具，通过长期的努力，以科学的范畴去解释现实世界中这些非光滑的“粗糙”物体。

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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.

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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.

我越来越喜欢的几何形式是最古老，最具体，最包容的几何形式，特别是由眼睛，由手，甚至由当今计算机提供协助去赋予其力量……为认识和感知世界带来统一的元素……并且，在不经意间，作为创造美的目的，其实这也相当于是额外奖赏。

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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]]: <blockquote>Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.   —Mandelbrot, in his introduction to ''[[The Fractal Geometry of Nature]]''</blockquote>

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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

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分形也存在于人类的追求中，例如音乐，绘画，建筑和股票市场价格。曼德布洛特认为，分形超脱于自然，而且在许多方面比传统欧几里得几何形状的人工光滑物体更直观，更自然：

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分形也同样存在于人类的追求中，例如音乐，绘画，建筑和股票市场价格。曼德布洛特认为，分形超脱于自然，而且在许多方面比传统欧几里得几何形状的人工光滑物体更直观，更自然：

云不是球形，山不是圆锥形，海岸线不是圆形，树皮不是光滑的，闪电也不是直线传播的。

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Mandelbrot also put his ideas to work in cosmology. He offered in 1974 a new explanation of [[Olbers' paradox]] (the "dark night sky" riddle), demonstrating the consequences of fractal theory as a [[Necessity and sufficiency|sufficient, but not necessary]], resolution of the paradox. He postulated that if the [[star]]s in the universe were fractally distributed (for example, like [[Cantor dust]]), it would not be necessary to rely on the [[Big Bang]] theory to explain the paradox. His model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred.<ref>''Galaxy Map Hints at Fractal Universe'', by Amanda Gefter; New Scientist; 25 June 2008</ref>

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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.

曼德布洛特也将他的想法运用到了宇宙学中。他在1974年对''' 奥伯斯佯谬Olbers' paradox '''（“夜空之谜”）提出了新的解释，证明了分形理论的结果是解决悖论的充分非必要条件。他推测，如果宇宙中的恒星是分形分布的（例如像''' 康托尔尘埃Cantor dust'''一样），则不必依靠大爆炸理论来解释这一悖论。他的模型不能排除“大爆炸”的发生，但是即使没有发生“大爆炸”，也可以允许黑暗的天空。

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