The picture above shows the Mandelbrot set, a set of points in the complex plane, the border of the set is a fractal.
Not only coastlines, fern, and the vegatable romanesco exhibit self-similarity, but also stock price charts. Mandelbrot describes a fascinating, unorthodox view on finacial theory in his book The (Mis)behaviour of Markets: A Fractal View of Risk, Ruin and Reward.
Mandelbrot's TEDTalk from 2010.
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