The Mandelbrot Set

The Mandelbrot Set? What the heck is THAT?

Back in 1985, my friend, Mark Lehman, and I began an exploration of something wonderful. Scientific American had published an article about the Mandelbrot Set and the emerging science of Chaos. It was heady stuff! We read about a number set within the complex number plane which contained an infinite variety of images. Images which were amazingly compelling. Seemingly, every image in the natural world could be found replicated within this thing... this Mandelbrot Set.

One of the most poetic descriptions of the Mandelbrot set is still one of the finest:

"The Mandelbrot set broods in silent complexity at the center of a vast two-dimensional sheet of numbers called the complex plane. When a certain operation is applied repeatedly to the numbers, the ones outside the set flee to infinity. The numbers inside remain to drift or dance about. Close to the boundary minutely choreographed wanderings mark the onset of the instability. Here is an infinite regress of detail that astonishes us with its variety, its complexity and its strange beauty"

- A. K. Dewdney, Scientifican American, August 1985

The Mandelbrot set at first appears to be nothing but an ugly bug with warts. But when you zoom in on its edges, incredibly intricate patterns begin to emerge.  Spirals, jellyfish, bow ties, and many other beautiful, bizarre objects abound in infinite numbers on the edge of the Mandelbrot set. And everywhere you look, at every level of magnification, miniature versions of the Mandelbrot set keep popping up.

Zooming in on the edge.

The boundary of the Mandelbrot set is an example of a fractal. A fractal is a mathematical object that is self-similar at every level of magnification. Nowhere will you find a fractal with more intricacy and complexity than the Mandelbrot set. And somehow, this infinite complexity is the product of a ridiculously simple formula:

z = z2 + c     where z and c are complex numbers.

Here's the procedure:

  1. Pick a point c on the complex plane, and set z = 0.
  2. Calculate z = z2 + c.
  3. Repeat step 2 until z > 2 or 1000 iterations

If you reach 1000 iterations and z has remained bounded (i.e. < 2), then the point c is in the Mandelbrot set.

Now the fun starts when colors are applied. The trick is to give the point c a color based on how fast it becomes unbounded. For example, if you bail out after before 50 iterations, you might assign the color blue. Or if you bail out between 50 and 100 iterations, you might assign the color yellow, etc.

Choosing the appropriate color range is how you pick out the detailed patterns on the edge of the Mandelbrot set. In this program, that is what is being accomplished when you adjust the color scroll bars.

What Is It?