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分形学和"有秩序的粗糙"

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一棵枫树的枝干，显示了自然形成的分形分叉。

File:Frost Water crystal on Mercury 20Feb2010 cu2.jpg

自然的霜结晶，展示了纯粹物理系统下的分形分支的生长。

虽然曼德博提出了分形这个新名词，然而在他的《大自然的分形几何学》问世之前，就曾有其他数学家对其中引进的一些数学对象做过描述：它们被认为是稀有的、 奇特的，与彼时存在的学术领域几乎没什么联系，且具有非自然、非本能的特性。曼德博前无古人地分析了这些现象的共同性质——比如自相似性（线性，非线性，抑或统计学意义上的），尺度不变性，以及（通常的）非整数豪斯多夫维数——将它们聚为同类并抽象为可用的基本工具，从而大大拓宽了科学理论对“不光滑”的真实世界的应用。

Although Mandelbrot coined the term fractal, some of the mathematical objects he presented in The Fractal Geometry of Nature had been described by other mathematicians. Before Mandelbrot, they had been 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 non-smooth objects in the real world. He highlighted their common properties, such as self-similarity (linear, non-linear, or statistical), scale invariance, and a (usually) non-integer Hausdorff dimension.

曼德博还强调了分形思想可用于构造实际可行的模型，来模拟真实世界中的很多“粗糙的”现象。自然界中的分形集有山脉，海岸线，河流流域的形状；植物，血管和肺部的结构；星系团；还有布朗运动。分形几何在人类艺术和娱乐中亦有出现，如音乐，艺术，建筑和股票市场的价格走势。曼德博相信，分形几何不但不是非自然的，相反在很多方面都比人类创造出的欧氏几何中各种光滑的研究对象更加直观和自然：

云不是球体，山不是圆锥体，海岸线不是圆，树皮不是光滑的，闪电传播的路径也不是直线。
  —曼德博，《大自然的分形几何学》绪论

He also emphasized the use of fractals as realistic and useful models of many "rough" phenomena in the real world. Natural fractals include the shapes of mountains, coastlines and river basins; the structures of plants, blood vessels and lungs; the clustering of galaxies; and Brownian motion. Fractals are 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

曼德博曾被视为一个空想家和标新立异者。他不拘形式热情洋溢的文风以及对视觉和几何直观的看重（大量插图的使用）使得《大自然的分形几何学》对于非专业读者也有相当可读性。这本书广泛激发了人们对分形学的兴趣，同时对混沌理论以及科学和数学的其他领域也有贡献。

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 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.