Discussion and Application of Geometrically and Statistically Self-Similar Fractals

Abstract

This paper will present examples of fractal geometry with their respective techniques of analysis in order to illustrate the foundations for a quantitative study of the natural world. The first section studies dimension, both self-similar measure and Hausdorff-Besicovitch measure, emphasizing the idea of dimension as a measure rather than as a number of degrees of freedom. The second section presents random, statistically self-similar fractals. A topologically one-dimensional construction found in fair games of chance leads to the Levy flight, a topologically two-dimensional example that gives a model for the distribution of galaxies as viewed from Earth.