From Plato to Borges

Carlos E. Puente, Department of Land, Air and Water Resources, University of California, Davis

The construction of a multitude of interesting patterns, defined as projections of simple multifractal measures supported by the graphs of fractal interpolating functions, is reviewed. It is illustrated how these ideas bring forth a new Platonic vision to address the complexity of some of nature's tangled patterns over one, two, and three dimensions, including relevant applications to geophysics. Then, a universal connection between arbitrary diffuse measures and the univariate and bivariate Gaussian distributions is explained and the infinite class of two-dimensional symmetric crystalline sets (Borges’ alephs) making up exotic kaleidoscopes inside circular bivariate Gaussian distributions is exhibited. It is shown that such mathematical designs include the structure of natural ice crystals and the rosette structure of relevant biochemical units, including even DNA.