Mathematics serves as the language with which we try to understand how nature works. Because of limited calculational abilities in the past, the focus of mathematical activity was on smoothly varying and easily calculable quantities and objects. The ability to compute the behavior of many more dynamical systems has shown that ``smoothness'' is a very rare phenomenon in nature. Mandelbrot was the first mathematician to open his eyes to the world and see that the true geometry of nature is very different indeed, and he named this geometry fractal. Nature generally exhibits patterns of extreme intricacy and detail. One aspect of the fractal philosophy is that this intricacy is the result of the evolution of relatively simple dynamical systems over long periods of time. We will see how fractal geometry arises from very simply-stated symbolic dynamical systems called Lindenmayer systems or L-systems, after Aristid Lindenmayer who first used them to model biological phenomena.
A similar idea called Iterated Function Systems leads to even greater freedom in modelling natural objects. In fact, Barnsley and others have developed this idea in the formulation of an Inverse Problem: how to find simple IFS's that reproduce any particular photograph at least reasonably well. It is possible that their work may lead to the most efficient video compression algorithms. We will study the basics of fractal geometry in this chapter.