Self-Affine Fractals and Fractal Dimension

TLDR

The compass exponent, defined by walking a compass along a fractal curve, coincides with other fractal dimensions for self‑similar curves such as coastlines, and each dimension has local and global values separated by a crossover. The study seeks to walk a compass along self‑affine curves, like scalar Brownian records, to demonstrate that a complete fractal‑dimension description is complex. The authors evaluate the compass exponent on self‑affine curves and compare it with box, mass, and similarity dimensions. For self‑affine curves the global fractal dimension is 1, box and mass dimensions are 1.5, and the compass dimension equals 2; for fractional Brownian records BH(t) the global dimension is 1, local dimensions are 2–H, and the compass dimension is 1/H.

Abstract

Evaluating a fractal curve's "approximate length" by walking a compass defines a "compass exponent." Long ago, I showed that for a self-similar curve (e.g., a model of coastline), the compass exponent coincides with all the other forms of the fractal dimension, e.g., the similarity, box or mass dimensions. Now walk a compass along a self-affine curve, such as a scalar Brownian record B(t). It will be shown that a full description in terms of fractal dimension is complex. Each version of dimension has a local and a global value, separated by a crossover. First finding: the basic methods of evaluating the global fractal dimension yield 1: globally, a self-affine fractal behaves as it if were not fractal. Second finding: the box and mass dimensions are 1.5, but the compass dimension is D = 2. More generally, for a fractional Bownian record BH(t), (e.g., a model of vertical cuts of relief), the global fractal dimensions are 1, several local fractal dimensions are 2-H, and the compass dimension is 1/H. This 1/H is the fractal dimension of a self-similar fractal trail, whose definition was already implicit in the definition of the record of BH(t).