Deterministic Geometric Modeling of Natural Complexity

Modeling of nature’s complexity is among the outstanding challenges we are faced with in the geophysical sciences. Although many ideas have been proposed throughout the years, the most common approaches, being based on notions related to chance, turn out to be insufficient to study, on an individual basis, the vast variety of patterns seen in nature. To this end, a novel deterministic fractal geometric procedure producing a host of interesting patterns over one, two or three dimensions, as transformations of multifractal distributions via fractal functions, is reviewed in this work. Inspired by the success in representing fluid turbulence via multiplicative cascades yielding multifractal measures, the novel approach represents other geophysical phenomena as a form of a “fractional integration” of such an underlying turbulence process performed using a suitable fractal function that transforms, say turbulence into rainfall.