The Mandelbrot Set and Julia Sets

Two Critical Points

The function z2/2 + z3/3 + c has two critical points, z = 0 and z = -1.

Consequently, we have three plausible definitions for the Mandelbrot set: all c for which

the iterates of z0 = 0 remain bounded,

the iterates of z0 = -1 remain bounded, and

the iterates of both z0 = 0 and z0 = -1 remain bounded.

Here are pictures of these three Mandelbrot sets. Click each picture to enlarge.

z = 0 z = -1 z = 0 and z = -1

Return to the Mandelbrot set definition.