3. The Mandelbrot Set and Julia Sets

3. A. Complex Iteration

Using complex numbers, the process iterated to generate the Mandelbrot set and the Julia sets takes a very simple form:

z → z2 + c

where z and c are complex numbers.

To iterate the process, pick a complex number c and a complex number z0. Then generate the sequence of complex numbers z1, z2, z3, ... by

z1 = z02 + c

z2 = z12 + c

z3 = z22 + c

and in general

zn+1 = zn2 + c

How this process generates the Mandelbrot set and Julia sets is the subject of Julia Sets and the Mandelbrot Set. Here we review the mechanics of z → z2 + c.

First, we reformulate the process without using complex numbers.

Next we do an example of the iteration.

Return to the Mandelbrot set and Julia sets.