Projections off fractal functions: A new vision of nature's complexity

The construction of a vast class of deterministic derived measures, defined as transformations of simple multinomial multifractals via fractal interpolating functions, is reviewed.^1,2 It is illustrated that these objects, which are projections of unique measures defined over the graphs of fractal interpolating functions, provide a new vision to address the complexity of some of nature's tangled patterns over one and two dimensions, as they may be used to model: (a) Rainfall time series, (b) energy dissipation in fully developed turbulence, (c) two-dimensional contaminant plumes within a porous medium, and (d) "chaotic" and "stochastic" signals, as encountered in applications. A recently discovered universal connection between arbitrary measures with continuous cumulative distributions and the univariate and bivariate Gaussian distributions, via plane- and space-filling fractal interpolating functions respectively, is also reviewed.^3,4 It is explained how this relationship yields (via projections) an infinite class of two-dimensional symmetric crystalline sets making up exotic kaleidoscopes of arbitrary symmetries, which decompose the bivariate Gaussian distribution.⁵ It is shown that these ideas enhance the vision that projections may be useful in approaching natural complexity, as some of these sets closely resemble key biochemical structures which even include life's own DNA rosette.