Chapter 5: Two Dimensions and Beyond

Section 4: Substitution Systems and Fractals

The Mandelbrot set

The pictures below show Julia sets produced by the procedure of taking the transformation z -> {Sqrt[z - c], -Sqrt[z - c]} discussed above and iterating it starting at z=0 for an array of values of c in the complex plane.

The Mandelbrot set introduced by Benoit Mandelbrot in 1979 is defined as the set of values of c for which such Julia sets are connected. This turns out to be equivalent to the set of values of c for which starting at z=0 the inverse mapping z->z^2+c leads only to bounded values of z. The Mandelbrot set turns out to have many intricate features which have been widely reproduced for their aesthetic value, as well as studied by mathematicians. The first picture below shows the overall form of the set; subsequent pictures show successive magnifications of the regions indicated. All parts of the Mandelbrot set are known to be connected. The whole set is not self-similar. However, as seen in the third and fourth pictures, within the set are isolated small copies of the whole set. In addition, as seen in the last picture, near most values of c the boundary of the Mandelbrot set looks very much like the Julia set for that value of c.

On pages 407 and 1006 I discuss parameter space sets that are somewhat analogous to the Mandelbrot set, but whose properties are in many respects much clearer. And from this discussion there emerges the following interpretation of the Mandelbrot set that appears not to be well known but which I find most illuminating. Look at the array of Julia sets and ask for each c whether the Julia set includes the point z=0. The set of values of c for which it does corresponds exactly to the boundary of the Mandelbrot set. The pictures below show a generalization of this idea, in which gray level indicates the minimum distance Abs[z-z0] of any point z in the Julia set from a fixed point z0. The first picture shows the case z0=0, corresponding to the usual Mandelbrot set.

From Stephen Wolfram: A New Kind of Science [citation]