Scaling properties for the surfaces of fractal and nonfractal objects: An infinite hierarchy of critical exponents

Abstract

A recent finding of Meakin et al. and Halsey et al. is that the surface of diffusion-limited aggregates (DLA) requires an infinite hierarchy of fractal dimensions for its characterization. In this work, we seek to understand this discovery and to place it into perspective. To this end, we study the distribution of hit probabilities near the surface of a variety of suitably chosen fractal and nonfractal objects---ranging from DLA and screened-growth aggregates on the one hand to simple A-arm stars and S-sided polygons on the other. We show physically how the infinite hierarchy of fractal dimensions arises, even for nonfractal objects. An important difference however, is that the infinite hierarchy is characterized by a constant gap exponent for the nonfractal objects, while for DLA a constant gap exponent is not sufficient.