Chaos and Fractals

Algorithm 3.3.1. The escape time algorithm for filled Julia sets.

1. Choose and fix a number \(c\in\mathbb C\text{.}\) We'll generate the filled Julia set of \(f_c\text{,}\)

2. Choose some rectangular region in the complex plane bound on the lower left by, say, \(z_{\text{min}}\) and on the upper right by \(z_{\text{max}}\text{.}\)

3. Partition this region into large number of rows and columns. We'll call the intersection of a row and column a "pixel", which corresponds to a complex value \(z_0\) which will be used an initial seed for the iteration of \(f_c\text{.}\)

4. For each pixel, iterate \(f_c\) from \(z_0\) until one of two things happens:

   1. We exceed the escape radius \(R\) in absolute value, in which case we shade the pixel according to how many iterates it took to escape.

   2. We exceed some pre-specified maximum number of iterations, in which case we color the pixel black.