Self-Affine Fractals and Fractal Dimension

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).