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第104行： − + − − 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."<ref name=Mandelbrot />{{rp|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: − − 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: − − 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。 − − − − 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. − − − − 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：+ 第136行：
第121行： − Mandelbrot has been called an artist, and a visionary<ref name="RLD">{{cite web|author=Devaney, Robert L.|author-link= Robert L. Devaney |title="Mandelbrot's Vision for Mathematics" in ''Proceedings of Symposia in Pure Mathematics''. Volume 72.1 |publisher=American Mathematical Society |year=2004 |url=http://www.math.yale.edu/mandelbrot/web_pdfs/jubileeletters.pdf |access-date=5 January 2007 |url-status=dead |archive-url=https://web.archive.org/web/20061209093734/http://www.math.yale.edu/mandelbrot/web_pdfs/jubileeletters.pdf |archive-date=9 December 2006 }}</ref> and a maverick.<ref>{{cite web|url=https://www.pbs.org/wgbh/nova/fractals/mandelbrot.html|title=A Radical Mind|last=Jersey|first=Bill|date=24 April 2005|work=Hunting the Hidden Dimension|publisher=NOVA/ PBS|access-date=20 August 2009|archive-date=22 August 2009|archive-url=https://web.archive.org/web/20090822022402/http://www.pbs.org/wgbh/nova/fractals/mandelbrot.html|url-status=live}}</ref> 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 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 has been called an artist, and a visionary[29] and a maverick.[30] His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made The Fractal 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. − − 曼德布洛特被称为艺术家，他是一位有远见和特立独行的人。他非正式和热情的写作风格以及对视觉和几何直觉的重视（并辅以大量插图的支持）使非专业人士可以亲身体会《大自然的分形几何学》。这本书引起了大众对分形的广泛兴趣，并为混沌理论以及科学和数学的其他领域做出了贡献。 − − − − 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> − − +
→Fractals and the "theory of roughness" 分形与“粗糙度理论”
[[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？<ref name=Mandelbrot>不仅能测量尼罗河沿岸的耕地，还能测量未开发的土地？
[[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？<ref name=Mandelbrot>不仅能测量尼罗河沿岸的耕地，还能测量未开发的土地？
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.
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.
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'''<font color="#ff8000"> [[豪斯多夫维数]]Hausdorff dimension</font>'''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。<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>
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'''<font color="#ff8000"> [[豪斯多夫维数]]Hausdorff dimension</font>'''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。<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>
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。” <ref name=Mandelbrot />{{rp|296}}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。
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。
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。
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.<br /> —Mandelbrot, in his introduction to ''[[The Fractal Geometry of Nature]]''</blockquote>
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.<br /> —Mandelbrot, in his introduction to ''[[The Fractal Geometry of Nature]]''</blockquote>
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。
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。
[[文件:Mandel zoom 08 satellite antenna.jpg|缩略图|右|曼德布洛特集合的部分]]
[[文件:Mandel zoom 08 satellite antenna.jpg|缩略图|右|曼德布洛特集合的部分]]
曼德布洛特被称为艺术家，他是一位有远见<ref name="RLD">{{cite web|author=Devaney, Robert L.|author-link= Robert L. Devaney |title="Mandelbrot's Vision for Mathematics" in ''Proceedings of Symposia in Pure Mathematics''. Volume 72.1 |publisher=American Mathematical Society |year=2004 |url=http://www.math.yale.edu/mandelbrot/web_pdfs/jubileeletters.pdf |access-date=5 January 2007 |url-status=dead |archive-url=https://web.archive.org/web/20061209093734/http://www.math.yale.edu/mandelbrot/web_pdfs/jubileeletters.pdf |archive-date=9 December 2006 }}</ref> 和特立独行的人<ref>{{cite web|url=https://www.pbs.org/wgbh/nova/fractals/mandelbrot.html|title=A Radical Mind|last=Jersey|first=Bill|date=24 April 2005|work=Hunting the Hidden Dimension|publisher=NOVA/ PBS|access-date=20 August 2009|archive-date=22 August 2009|archive-url=https://web.archive.org/web/20090822022402/http://www.pbs.org/wgbh/nova/fractals/mandelbrot.html|url-status=live}}</ref> 。他非正式和热情的写作风格以及对视觉和几何直觉的重视（并辅以大量插图的支持）使非专业人士可以亲身体会《大自然的分形几何学》。这本书引起了大众对分形的广泛兴趣，并为混沌理论以及科学和数学的其他领域做出了贡献。
曼德布洛特也将他的想法运用到了宇宙学中。他在1974年对'''<font color="#ff8000"> 奥伯斯佯谬Olbers' paradox </font>'''（“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'''<font color="#ff8000"> 康托尔尘埃Cantor dust</font>'''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。
曼德布洛特也将他的想法运用到了宇宙学中。他在1974年对'''<font color="#ff8000"> [[奥伯斯佯谬]]Olbers' paradox </font>'''（“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'''<font color="#ff8000"> 康托尔尘埃Cantor dust</font>'''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。<ref>''Galaxy Map Hints at Fractal Universe'', by Amanda Gefter; New Scientist; 25 June 2008</ref>
== Awards and honors 奖项与荣誉 ==
== Awards and honors 奖项与荣誉 ==