In this chapter we are going to learn how to draw beautiful pictures of fractals, strange sets with some pretty amazing properties. Fractals arise naturally in the study of dynamical systems so we'll begin our discussion there. A dynamical system is simply a system that is changing as opposed to a static system. So everything from the solar system to the flow of blood in our arteries to the pendulum of a clock qualifies as a dynamical system. In order to approach the subject mathematically we usually focus on a single function, or a family of functions, and ask what happens if we repeatedly apply the same function over and over. This simple setup will lead us to Julia Sets as well as the famous Mandelbrot Set. Once we see how to draw these fractals we'll also learn how to make movies with Mathematica.

10.1 Iterating a Function

Suppose we start with a function f(x) and create a "feedback loop" by taking the output of the function and feeding it back in as input. What will happen? For example, suppose f(x) = x ². If we start with an input of 2 we find that the output of f is 4 since f squares the input. If we then square 4 we get 16, squaring 16 gives 256, and so on. Starting from the initial value, or initial input of 2, yields the sequence

We can think of this...