Using Dimension Theory to Analyze and Classify the Generation of Fractal Sets

Abstract

This article discusses the interplay in fractal geometry occurring between computer programs for developing (approximations of) fractal sets and the underlying dimension theory. The computer is ideally suited to implement the recursive algorithms needed to create these sets, thus giving us a laboratory for studying fractals and their corresponding dimensions. Moreover, this interaction between theory and procedure goes both ways. Dimension theory can be used to classify and understand fractal sets. This allows us, given a fixed generating pattern, to describe the resultant images produced by various programs. Thus, dimension theory can be used as a tool which enables to predict and classify behavior of certain fractal generating algorithms.We also tie these two perspectives in with some of the history of the subject. Four examples of fractal sets developed around the turn of the century are introduced and studied from both classical and modern viewpoints. Then, definitions and sample calculations of fractal and Hausdorff-Besicovitch dimension are given. We discuss various methods for extracting dimension from given fractal sets. Finally, dimension theory is used to classify image