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

z_n+1 = z_n² + 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.