Introduction

In this diaper we introduce the theory and some applications of superfractals. Superfractals are families of sometimes beautiful fractal objects which can be explored by means of the chaos game (see Figure 5.1) and which span the gap between fully ‘random’ fractal objects and deterministic fractal objects. Our presentation is via elementary examples and theory together with brief descriptions of natural feasible extensions. This chapter depends heavily on the earlier material. You may grasp intuitively the key ideas of superfractals and ‘2-variability’ by studying the experiment described in Section 5.2. But be careful notto miss subtleties such as those that enable the construction of superfractals whose elements are vast collections of homeomorphic pictures, as for example those illustrated in Figures 5.13 and 5.17.

A superfractal (see Figure 5.2) is associated with a single underlying hyperbolic IFS. It has its own underlying logical structure, called the ‘V-variability’ of the superfractal, for some V ∈{1, 2,…}, which enables us to sample the superfractal by means of the chaos game and produce generalized fractal objects such as fractal sets, pictures, measures and so on, one after another. The property of V-variability enables us to ‘dance on the superfractal’, sometimes producing wondrous objects in splendid succession.