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

摘要

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 researchers 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 one, given a fixed generating pattern, to describe the resultant images produced by various programs. Thus, dimension theory can be used as a tool that enables researchers to predict and classify the behavior of certain fractal generating algorithms. The author ties 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. These examples are the Cantor Set, Space-Filling Curves, Snowflake Curves, and the Sierpinski Gasket. Then, definitions and sample calculations of fractal and Hausdorff-Besicovitch dimension are given. The author discusses various methods for extracting dimension from given fractal sets. Finally, dimension theory is used to classify images.