Excerpt

Few foundational concepts underlying complexity science have sparked the imagination and been embraced by both the academic and nonacademic community more than that of fractals. Most people are familiar with the idea. Simply put, fractals are objects that look approximately the same at all scales or at any level of magnification. Each subunit looks like a scaled-down version of the original whole with the same geometrical pattern repeating itself over and over as the resolution changes. For example, any branch of a tree when isolated looks like a scaled-down version of the original tree. This self-similar property embodied in fractals is ubiquitous throughout nature, ranging, to varying degrees, from circulatory and neural systems to transport systems, clouds, financial markets, and river and social networks. Indeed, the fascination with fractals largely derives from the remarkable observation that underlying the crinkliness, roughness, discontinuity, messiness, and apparent arbitrariness of the continuously evolving and adapting complex world around us lies a hidden regularity encapsulated in fractal-like structures.

This insight was pioneered by the mathematician Benoit Mandelbrot, who introduced the term fractal in 1975 when elaborating on ideas he had developed in his seminal 1967 paper “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension.” This work and its rhetorical title were inspired by an equally seminal paper published in 1961 with the marvelously obscure title “The Problem of Contiguity: An Appendix to Statistics of Deadly Quarrels,” written by the little- known polymath Lewis Fry Richardson. A passionate pacifist, Richardson had attempted to develop a quantitative theory for the origins of war and conflict and made the dubious hypothesis that the probability of war between neighboring states was proportional to the length of their common border, so he turned his attention to measuring their lengths . . . and thereby discovered fractals, self-similarity, and power laws.

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