All Kinds of Fractal Curves

We have already encountered fractal geometric forms, such as Julia sets and the Gingerbread Man. We will develop other aspects of their interesting and aesthetically appealing structure in this chapter. We gradually leave the world of dynamical systems, which until now has played a leading role in the formation of Feigenbaum diagrams, Julia sets, and the Gingerbread Man. There exist other mathematical functions with fractal properties. In particular we can imagine quite different functions, which have absolutely nothing to do with the previous background of dynamical systems. In this chapter we look at purely geometric forms, which have only one purpose – to produce interesting computer graphics. Whether they are more beautiful than the Gingerbread Man is a matter of personal taste.

Perhaps you already know about the two most common fractal curves. The typical structure of the Hilbert and Sierpifiski curves is shown in Figures 8.1-1 and 8.1-2. The curves are here superimposed several times.

As with all computer graphics that we have so far drawn, the pictures are ‘static’ representations of a single situation at some moment. This is conveyed by the word ‘genesis’. Depending on the parameter n, the number of wiggles, the two ‘space-filling’ curves become ever more dense. These figures are so well known that in many computer science books they are used as examples of recursive functions. The formulas for computing them, or even the programs for drawing them, are written down there: see for example Wirth (1983). We therefore do not include a detailed description of these two curves.