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Mandelbrot Set
Mandelbrot defined a new mapping of the complex plan (the Mandelbrot Set) by including all complex points that do not diverge to infinity when iteratively squared and added to the initial point: zi+1 = zi2 + z0. The boundary of the surface thus defined has the peculiar property of having a Hausdorff dimension of 2 while it has (it is a curve) a topological dimension of 1. Mandelbrot Set's complexity actually increases when zooming in!

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