Chapter 5 Fractal Dimension
Fractal geometry is concerned with the study of geometrically complicated objects. The concept of “fractal dimension”’ is a quantitative measure of this complexity. The Cantor set, for example, is a set between dimensions. On one hand it seems small enough to be of dimension zero, but on the other hand it is a much richer set than what one might think of as a zero dimensional set. Fractal dimension quantifies its place in this spectrum.
There are many notions of fractal dimension - Hausdorff, similarity, box-counting, and packing dimensions are just a few. In the serious study of fractal geometry it is important to understand the relationships between these ideas. In particular, we would like to know conditions guaranteeing equality of two or more definitions. Perhaps one definition is of theoretical importance, while the other is easier to calculate.
We will focus on the similarity dimension and the box-counting dimension. The box-counting dimension is broadly applicable and widely used, but can be difficult to calculate. The similarity dimension is of much more restricted applicability, but easy to calculate when appropriate. Fortunately, it turns out that these concepts are equivalent on suitably chosen sets.