A Theorem from Topology Unveils the Mystery of Fractals and Power Laws

Authors: Arturo Tozzi, James F Peters

The (spatial) fractals and (temporal) power laws are ubiquitously displayed by large classes of biological systems. Nevertheless, they are controversial phenomena with still unexplained genesis. From the far-flung branch of topology, a helpful concept comes into play, namely the Borsuk-Ulam theorem, shedding new light on the scale-free origin’s long-standing enigma. The theorem states that a single point, if embedded in just one spatial dimension higher, gives rise to two antipodal points that have matching descriptions and similar features. Here we demonstrate that, when we introduce into a system the proper fractal extra-dimension instead of a spatial one, we are able to achieve two antipodal self-similar shapes, corresponding to the distinctive scale-free’s higher and lower magnifications. By showing that the elusive phenomena of fractals and power laws can be explained and analyzed in a topological framework, we make clear why the Borsuk-Ulam theorem is the most general principle underlying their pervasive occurrence in nature.

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[v1] 2016-10-19 03:29:22

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