The Mandelbrot Set and Julia Sets

The Mandelbrot Set - Definition

The Mandelbrot set, M, is defined by the same iteration process used to define Julia sets, but applied in a different way.

Instead of using the complex plane to represent the different choices of z0 (the Dynamical Plane), it represents different values of c (the Parameter Plane).

To compute an image of the Mandelbrot set, for each c start with z0 = 0 (why start with z0 = 0?) and generate the sequence z1, z2, z3, ... by the basic iteration rule

zn → zn+1 = zn2 + c

If the sequence does not run away to infinity, then the point c belongs to M; if the sequence does run away to infinity, then c does not belong to M.

As with Julia sets, we paint the pixel black if the sequence produced by the c at its center does not run away to infinity, that is, if this c belongs to M, and otherwise we paint the pixel a color determined by how quickly the sequence gets farther than 2 from the origin (and hence runs away to infinity).

With this applet, the Mandelbrot set iterator, from Bob Devaney's dynamical systems website at BU, we can pick a point on or around the Mandelbrot set and watch the iterates using that c value.

Our applet Mandelbrot-Julia Sets allows similar explorations, and variations using some other functions.

For reference, here is a picture of the Mandelbrot set. The pink circle is the circle of radius 2 centered at the origin.

Return to the Mandelbrot set.