曼德尔布罗Benoit Mandelbrot

曼德尔布罗在很大程度上促成了当前对分形几何的兴趣。他展示了分形如何在数学和自然界中许多不同的地方出现。

曼德尔布罗于1924年出生在波兰一个有着浓厚学术传统的家庭。然而，他的父亲以买卖衣服为生，而他的母亲是一名医生。小时候，曼德尔布罗由他的两位叔叔引入了数学。

曼德尔布罗的家人于1936年移居法国，他的叔叔索勒姆·曼德尔布罗伊特——法兰西公学院的数学教授，也是雅克·阿达马在该职位的继任者——负责他的教育。事实上，索勒姆·曼德尔布罗伊特的影响既有积极的一面，也有消极的一面，因为他是戈弗雷·哈罗德·哈代和戈弗雷·哈罗德·哈代数学哲学的极大崇拜者。这引起了曼德尔布罗对纯数学的反感，尽管正如曼德尔布罗自己所说，他现在理解戈弗雷·哈罗德·哈代深切的和平主义如何使他担心应用数学在错误的人手中可能在战争时期被用于邪恶目的。

曼德尔布罗 在巴黎的罗林中学就读，直到第二次世界大战开始，当时他的家人搬到了法国中部的图勒。这对 曼德尔布罗 来说是一段极其艰难的时期，他多次担心自己的生命安全。在 [3] 中，强调了这些岁月对他教育的影响：-

战争、贫困的持续威胁和生存的需要使他远离学校和大学，尽管他认可那些“了不起的”中学老师，但他主要是自学成才。

曼德尔布罗 现在将他的大部分成功归功于这种非传统的教育。这使他能够以可能难以被那些通过传统教育被强烈鼓励以标准方式思考的人所采用的方式思考。这也使他能够发展出一种高度几何化的数学方法，而他非凡的几何直觉和视野开始赋予他对数学问题的独特见解。

在里昂学习后，曼德尔布罗进入了巴黎高等师范学院。这是任何人在那里学习的最短时间之一，因为他只待了一天就离开了。在巴黎综合理工学院的入学考试中取得非常成功的表现后，曼德尔布罗于1944年开始在那里学习。在那里，他在保罗·莱维的指导下学习，保罗·莱维是另一位对曼德尔布罗产生强烈影响的人。

此处一段未译出，以下为英文原文After completing his studies at the École Polytechnique, Mandelbrot went to the United States where he visited the California Institute of Technology. After a Ph.D. granted by the University of Paris, he went to the Institute for Advanced Study in Princeton where he was sponsored by John von Neumann.

曼德尔布罗 于1955年回到法国，在国家科学研究中心工作。在此期间，他在法国和日内瓦与 Aliette 韦尼阿明·卡甘 结婚，但他在那里没有停留太久就返回了美国。克拉克给出了他对当时法国数学风格不满的原因 [3]：-

他仍然深切关注 统计力学 的更奇特形式和数学语言学，并充满非标准的创造性想法，他发现法国 尼古拉·布尔巴基 基础学派的巨大主导地位不符合他的科学品味，于是在1958年他永久离开前往美国，并作为IBM研究员在纽约州约克镇高地的世界知名实验室开始了与IBM长期且最富有成果的合作。

IBM 为 曼德尔布罗 提供了一个环境，使他能够探索各种各样的不同想法。他曾谈到，在 IBM 可以自由选择自己想要的研究方向，这给了他任何大学职位都无法提供的机会。从 IBM 退休后，他在耶鲁大学找到了类似的机会，目前他是该校的斯特林数学科学讲席教授。

1945年，曼德尔布罗的叔叔向他介绍了茹利亚1918年的重要论文，声称这是一部杰作，是有趣问题的潜在来源，但曼德尔布罗不喜欢它。事实上，他对叔叔提出的建议反应相当糟糕，因为他觉得他对数学的整个态度与叔叔的态度如此不同。相反，曼德尔布罗选择了自己非常不同的道路，然而，这条道路在经历了许多不同科学之后，在1970年代将他带回了茹利亚的论文，有些人将这条道路描述为高度个人主义或游牧式的。事实上，曼德尔布罗决定为许多不同的科学分支做出贡献，是在年轻时就做出的非常深思熟虑的决定。令人瞩目的是，他能够在如此多的领域中以如此显著的成功实现这一抱负。

借助计算机图形学，当时在IBM沃森研究中心工作的曼德尔布罗能够展示茹利亚的工作如何成为当今已知一些最美丽的分形的来源。为了做到这一点，他不仅必须发展新的数学思想，还必须开发一些最早的打印图形的计算机程序。

曼德尔布罗 集是复平面上的一个连通点集。在复平面中选取一点 $z_{0}$。

计算：
$z_{1} = z_{0}^{2} + z_{0}$
$z_{2} = z_{1}^{2} + z_{0}$
$z_{3} = z_{2}^{2} + z_{0}$
……

如果序列 $z_{0} , z_{1} , z_{2} , z_{3} , ...$ 始终与原点保持不超过 2 的距离，则称点 $z_{0}$ 属于 曼德尔布罗 集。如果序列远离原点发散，则该点不属于该集合。

你可以在THIS LINK看到曼德尔布罗集。

他的工作最初在他的著作Les objets fractals, forn, hasard et dimensionⓉ（分形、形状、机遇与维数）（1975年）中得到阐述，并在1982年的The fractal geometry of nature中更为完整地展开。
你可以在THIS LINK看到更多关于这些书的内容。

1999 年 6 月 23 日，曼德尔布罗 获得了圣安德鲁斯大学授予的荣誉理学博士学位。在典礼上，Peter Clark 发表了演讲 [3]，其中他恰当地评价了 曼德尔布罗 的成就。我们引用该演讲中的一段话：-

……在一个世纪的尾声，人类在知识、政治和道德上的进步观念也许被视为至多是模糊而含混的，但至少有一个人类活动领域，其中真正进步的观念和成就是明确无误、清晰透彻的。那就是数学。1900年，在Paris David举行的国际数学家大会上，大卫·希尔伯特在一次著名演讲中列出了约25个具有重大意义的未解决问题。其中许多问题已被彻底解决，或被证明不可解，正如我们众所周知，最近在九十年代中期以费马大定理的证明的发现而达到顶峰。大卫·希尔伯特的第一个问题涉及关于连续统或实数线本质的一系列问题，这是19世纪乃至20世纪分析的一个主要关注点。这个问题既是几何学问题，关乎被视为由点构成的线的本质，也是算术问题，关乎实数理论。这两个领域的整合是理查德·戴德金和格奥尔格·康托尔的伟大成就之一，我们[圣安德鲁斯大学]足够明智地在1911年授予后者荣誉。

可以说，在这一成就的灌木丛中潜伏着某些非常非凡的几何对象。对当时所有人来说，它们显得奇怪，确实是相当病态的怪物。它们确实古怪，有曲线——实际上是一维线——填满了二维空间，有曲线表现良好，即漂亮且连续，但在任何点都没有斜率（不是某些点，而是任何点），它们有着奇怪的名字，朱塞佩·皮亚诺空间填充曲线、瓦茨瓦夫·谢尔宾斯基垫圈、海里格·冯·科赫曲线、格奥尔格·康托尔三分集。尽管它们具有病态性质，其非凡复杂性，尤其是在越来越详细地观察时，它们往往描述起来非常简单，因为生成它们的规则简单得荒谬。这些对象如此古怪，以至于数学家开始排斥这些怪物，它们被搁置一旁，被认为太奇怪而无趣。直到我们的荣誉学位获得者从中创造出一门全新的科学——分形几何理论：正是他的洞察力和远见，在这些对象以及他发现的许多新对象中（其中一些现在以他的名字命名），看到的不是数学奇珍，而是通往新数学宇宙的路标，一种与欧几里得的几何同样系统和普遍的新几何，以及一门新的物理科学。

除了担任沃森研究中心的IBM研究员外，曼德尔布罗还是哈佛大学的数学实践教授。他还担任过耶鲁大学的工程学教授、巴黎综合理工学院的数学教授、哈佛大学的经济学教授，以及阿尔伯特·爱因斯坦医学院的生理学教授。正如我们上面提到的，曼德尔布罗涉足如此多不同的科学分支并非偶然，而是他非常深思熟虑的决定。然而，正是分形被如此广泛地发现，在许多情况下提供了进入其他领域的途径[3]：-

我不应该……给人留下这样的印象：我们面前只有一位数学家。让我解释为什么。他的第一个伟大洞见是发现那些长期被忽视的极其复杂、几乎病态的结构，展现出某些普遍特征，需要一种新的维度理论来充分处理它们，他从费利克斯·豪斯多夫和阿布拉姆·萨莫伊洛维奇·贝西科维奇的早期工作中将其推广；但第二个伟大洞见是，如此发现的分形性质，他为其提供了一般理论，几乎普遍存在于自然界中。他看到的是，数学物理试图描述自然所依据的那种压倒性的光滑范式存在根本缺陷且不完整。分形和前分形一旦被注意到，就无处不在。它们出现在物理学中，用于描述一些简单物理系统（如受迫摆）的极其复杂行为，以及湍流和相变的极其复杂行为。它们作为现在所谓的混沌系统的基础出现。它们出现在经济学中，与价格行为有关，并且正如儒勒·昂利·庞加莱曾怀疑但从未证明的那样，出现在交易所或我们伦敦证券交易所的行为中。它们出现在生理学中，与哺乳动物细胞的生长有关。信不信由你……它们出现在花园里。仔细观察，你会看到西兰花和花椰菜的花头之间的差异，这种差异可以在分形理论中精确刻画。

曼德尔布罗因其卓越成就而获得了无数荣誉和奖项。例如，1985年，曼德尔布罗被授予Barnard Medal for Meritorious Service to Science。次年，他获得了Franklin Medal。1987年，他荣获Alexander von Humboldt Prize，1988年获得Steinmetz Medal，以及更多奖项，包括1989年的Légion d'Honneur、1991年的Nevada Medal、1993年的Wolf prize for physics和2003年的Japan Prize for Science and Technology。

他的奖项和荣誉完整列表可在THIS LINK（作为下载）获取。

Benoit Mandelbrot was largely responsible for the present interest in fractal geometry. He showed how fractals can occur in many different places in both mathematics and elsewhere in nature.

Mandelbrot was born in Poland in 1924 into a family with a very academic tradition. His father, however, made his living buying and selling clothes while his mother was a doctor. As a young boy, Mandelbrot was introduced to mathematics by his two uncles.

Mandelbrot's family emigrated to France in 1936 and his uncle Szolem Mandelbrojt, who was Professor of Mathematics at the Collège de France and the successor of Hadamard in this post, took responsibility for his education. In fact the influence of Szolem Mandelbrojt was both positive and negative since he was a great admirer of Hardy and Hardy's philosophy of mathematics. This brought a reaction from Mandelbrot against pure mathematics, although as Mandelbrot himself says, he now understands how Hardy's deep felt pacifism made him fear that applied mathematics, in the wrong hands, might be used for evil in time of war.

Mandelbrot attended the Lycée Rolin in Paris up to the start of World War II, when his family moved to Tulle in central France. This was a time of extraordinary difficulty for Mandelbrot who feared for his life on many occasions. In [3] the effect of these years on his education was emphasised:-

The war, the constant threat of poverty and the need to survive kept him away from school and college and despite what he recognises as "marvellous" secondary school teachers he was largely self taught.

Mandelbrot now attributed much of his success to this unconventional education. It allowed him to think in ways that might be hard for someone who, through a conventional education, is strongly encouraged to think in standard ways. It also allowed him to develop a highly geometrical approach to mathematics, and his remarkable geometric intuition and vision began to give him unique insights into mathematical problems.

After studying at Lyon, Mandelbrot entered the École Normale in Paris. It was one of the shortest lengths of time that anyone would study there, for he left after just one day. After a very successful performance in the entrance examinations of the École Polytechnique, Mandelbrot began his studies there in 1944. There he studied under the direction of Paul Lévy who was another to strongly influence Mandelbrot.

After completing his studies at the École Polytechnique, Mandelbrot went to the United States where he visited the California Institute of Technology. After a Ph.D. granted by the University of Paris, he went to the Institute for Advanced Study in Princeton where he was sponsored by John von Neumann.

Mandelbrot returned to France in 1955 and worked at the Centre National de la Recherche Scientifique. He married Aliette Kagan during this period back in France and Geneva, but he did not stay there too long before returning to the United States. Clark gave the reasons for his unhappiness with the style of mathematics in France at this time [3]:-

Still deeply concerned with the more exotic forms of statistical mechanics and mathematical linguistics and full of non standard creative ideas he found the huge dominance of the French foundational school of Bourbaki not to his scientific tastes and in 1958 he left for the United States permanently and began his long standing and most fruitful collaboration with IBM as an IBM Fellow at their world renowned laboratories in Yorktown Heights in New York State.

IBM presented Mandelbrot with an environment which allowed him to explore a wide variety of different ideas. He has spoken of how this freedom at IBM to choose the directions that he wanted to take in his research presented him with an opportunity which no university post could have given him. After retiring from IBM, he found similar opportunities at Yale University, where he is presently Sterling Professor of Mathematical Sciences.

In 1945 Mandelbrot's uncle had introduced him to Julia's important 1918 paper claiming that it was a masterpiece and a potential source of interesting problems, but Mandelbrot did not like it. Indeed he reacted rather badly against suggestions posed by his uncle since he felt that his whole attitude to mathematics was so different from that of his uncle. Instead Mandelbrot chose his own very different course which, however, brought him back to Julia's paper in the 1970s after a path through many different sciences which some characterise as highly individualistic or nomadic. In fact the decision by Mandelbrot to make contributions to many different branches of science was a very deliberate one taken at a young age. It is remarkable how he was able to fulfil this ambition with such remarkable success in so many areas.

With the aid of computer graphics, Mandelbrot who then worked at IBM's Watson Research Center, was able to show how Julia's work is a source of some of the most beautiful fractals known today. To do this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics.

The Mandelbrot set is a connected set of points in the complex plane. Pick a point $z_{0}$ in the complex plane.

Calculate:
$z_{1} = z_{0}^{2} + z_{0}$
$z_{2} = z_{1}^{2} + z_{0}$
$z_{3} = z_{2}^{2} + z_{0}$
. . .

If the sequence $z_{0} , z_{1} , z_{2} , z_{3} , ...$ remains within a distance of 2 of the origin forever, then the point $z_{0}$ is said to be in the Mandelbrot set. If the sequence diverges from the origin, then the point is not in the set.

You can see the Mandelbrot Set at THIS LINK.

His work was first put elaborated in his book Les objets fractals, forn, hasard et dimension Ⓣ (1975) and more fully in The fractal geometry of nature in 1982.
You can see more about these books at THIS LINK.

On 23 June 1999 Mandelbrot received the Honorary Degree of Doctor of Science from the University of St Andrews. At the ceremony Peter Clark gave an address [3] in which he put Mandelbrot's achievements into perspective. We quote from that address:-

... at the close of a century where the notion of human progress intellectual, political and moral is seen perhaps to be at best ambiguous and equivocal there is one area of human activity at least where the idea of, and achievement of, real progress is unambiguous and pellucidly clear. That is mathematics. In 1900 in a famous address to the International Congress of mathematicians in Paris David Hilbert listed some 25 open problems of outstanding significance. Many of those problems have been definitively solved, or shown to be insoluble, culminating as we all know most recently in the mid-nineties with the discovery of the proof of Fermat's Last Theorem. The first of Hilbert's problems concerned a thicket of issues about the nature of the continuum or the real line, a major concern of 19th and indeed of 20th century analysis. The problem was both one of geometry concerning the nature of the line thought of as built up of points and of arithmetic thought of as the theory of the real numbers. The integration of those two fields was one of the great achievements of Richard Dedekind and Georg Cantor, the latter of whom we [St Andrews University] were intelligent enough to honour in 1911.

Now lurking about so to speak in the undergrowth of that achievement lay certain very extraordinary geometric objects indeed. To all at the time, they seemed strange, indeed rather pathological monsters. Odd indeed they were, there were curves - one dimensional lines in effect - which filled two dimensional spaces, there were curves which were well behaved, that is nice and continuous but which had no slope at any point (not just some points, ANY points) and they went by strange names, the Peano Space filling curve, the Sierpiński gasket, the Koch curve, the Cantor Ternary set. Despite their pathological qualities, their extraordinary complexity, especially when viewed in greater and greater detail, they were often very simple to describe in the sense that the rules which generated them were absurdly simple to state. So odd were these objects that mathematicians set about barring these monsters and they were set aside as too strange to be of interest. That is until our honorary graduand created out of them an entirely new science, the theory of fractal geometry: it was his insight and vision which saw in those objects and the many new ones he discovered, some of which now bear his name, not mathematical curiosities, but signposts to a new mathematical universe, a new geometry with as much system and generality as that of Euclid and a new physical science.

As well as IBM Fellow at the Watson Research Center Mandelbrot was Professor of the Practice of Mathematics at Harvard University. He also held appointments as Professor of Engineering at Yale, of Professor of Mathematics at the École Polytechnique, of Professor of Economics at Harvard, and of Professor of Physiology at the Einstein College of Medicine. Mandelbrot's excursions into so many different branches of science was, as we mention above, no accident but a very deliberate decision on his part. It was, however, the fact that fractals were so widely found which in many cases provided the route into other areas [3]:-

I should not ... give the impression that we have here before us a mathematician alone. Let me explain why. The first of his great insights was the discovery that the extraordinarily complex almost pathological structures, which had been long ignored, exhibited certain universal characteristics requiring a new theory of dimension to treat them adequately which he had generalised from earlier work of Hausdorff and Besicovitch but the second great insight was that the fractal property so discovered, the general theory of which he had provided, was present almost universally in Nature. What he saw was that the overwhelming smoothness paradigm with which mathematical physics had attempted to describe Nature was radically flawed and incomplete. Fractals and pre-fractals once noticed were everywhere. They occur in physics in the description of the extraordinarily complex behaviour of some simple physical systems like the forced pendulum and in the hugely complex behaviour of turbulence and phase transition. They occur as the foundations of what is now known as chaotic systems. They occur in economics with the behaviour of prices and as Poincaré had suspected but never proved in the behaviour of the Bourse or our own Stock exchange in London. They occur in physiology in the growth of mammalian cells. Believe it or not ... they occur in gardens. Note closely and you will see a difference between the flower heads of broccoli and cauliflower, a difference which can be exactly characterised in fractal theory.

Mandelbrot has received numerous honours and prizes in recognition of his remarkable achievements. For example, in 1985 Mandelbrot was awarded the Barnard Medal for Meritorious Service to Science. The following year he received the Franklin Medal. In 1987 he was honoured with the Alexander von Humboldt Prize, receiving the Steinmetz Medal in 1988 and many more awards including the Légion d'Honneur in 1989, the Nevada Medal in 1991, the Wolf prize for physics in 1993 and the 2003 Japan Prize for Science and Technology.

A full list of his prizes and honours is available (as a download) at THIS LINK.