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分形學和"有秩序的粗糙"

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一棵楓樹的枝幹，顯示了自然形成的分形分叉。

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

自然的霜結晶，展示了純粹物理系統下的分形分支的生長。

雖然曼德博提出了分形這個新名詞，然而在他的《大自然的分形幾何學》問世之前，就曾有其他數學家對其中引進的一些數學對象做過描述：它們被認為是稀有的、 奇特的，與彼時存在的學術領域幾乎沒什麼聯繫，且具有非自然、非本能的特性。曼德博前無古人地分析了這些現象的共同性質——比如自相似性（線性，非線性，抑或統計學意義上的），尺度不變性，以及（通常的）非整數豪斯多夫維數——將它們聚為同類並抽象為可用的基本工具，從而大大拓寬了科學理論對「不光滑」的真實世界的應用。

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.

曼德博還強調了分形思想可用於構造實際可行的模型，來模擬真實世界中的很多「粗糙的」現象。自然界中的分形集有山脈，海岸線，河流流域的形狀；植物，血管和肺部的結構；星系團；還有布朗運動。分形幾何在人類藝術和娛樂中亦有出現，如音樂，藝術，建築和股票市場的價格走勢。曼德博相信，分形幾何不但不是非自然的，相反在很多方面都比人類創造出的歐氏幾何中各種光滑的研究對象更加直觀和自然：

雲不是球體，山不是圓錐體，海岸線不是圓，樹皮不是光滑的，閃電傳播的路徑也不是直線。
  —曼德博，《大自然的分形幾何學》緒論

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

曼德博曾被視為一個空想家和標新立異者。他不拘形式熱情洋溢的文風以及對視覺和幾何直觀的看重（大量插圖的使用）使得《大自然的分形幾何學》對於非專業讀者也有相當可讀性。這本書廣泛激發了人們對分形學的興趣，同時對混沌理論以及科學和數學的其他領域也有貢獻。

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.