The exquisite geometric structure of a central limit theorem

Universal constructions of univariate and bivariate Gaussian distributions, as transformations of diffuse probability distributions via, respectively, plane- and space-filling fractal interpolating functions and the central limit theorems that they imply, are reviewed. It is illustrated that the construction of the bivariate Gaussian distribution yields exotic kaleidoscopic decompositions of the bell in terms of exquisite geometric structures which include non-trivial crystalline patterns having arbitrary n-fold symmetry, for any n > 2. It is shown that these results also hold when fractal interpolating functions are replaced by a more general class of attractors that are not functions.