ABSTRACT
In Chapter 5 we introduced a number of exotic sets in R, R2, and R3, including the self-similar Cantor set, Sierpin´ski set, and the Koch curve, as well as the spacefilling Peano curve and Hilbert curve. A pertinent question is how to draw, or have the computer draw, approximations to those and other very complicated sets. In Chapter 6 we address this issue. In order to be able to give and understand algorithms for creating those sets, we study the notion of metric space in Section 6.1. Section 6.2 is devoted to the Hausdorff metric that is defined on closed and bounded sets of R and R2. In Section 6.3 we turn to special functions called contractions and affine functions that will be critical in order for the computer to produce fractal images. Section 6.4 presents a discussion of combinations of contractions called iterated function systems that are the major ingredients in the algorithms for creating fractal shapes that occupy Section 6.5. The main goal of the chapter is to understand the technique of creating fractal sets on the computer. We mention that many of the ideas in this chapter appeared in Barnsley’s (1993) book Fractals Everywhere.
