The Mandelbrot Set

Definition

The Mandelbrot set is the set of all points c for which the sequence z'←z²+c starting at 0 is bounded.

Such a simple rule, such a complex object.

Topology

It's connected. It has a definite boundary, though of infinite length. It has a finite area. It's a fractal, meaning there is self-similarity and that you can zoom in forever. Literally and truly forever.

Here is an example zoom from the book Chaos and Fractals: New Frontiers of Science by Peitgen, Jürgens and Saupe:

Explore

Get a feel for it with James Henstridge's Renderer or David Eck's Renderer.

Videos

There are a zillion videos with deep zooming out there. I've selected a few for you. The first one is pretty long, but it is interesting because it "visits" 16 different areas of the set:

This one is pretty cool:

This one goes pretty fast:

Here's another fast one:

Here is one that zooms past the "x is six" message:

Spirals all the way down....

If you can't get enough of these, just search for more....

Learn About It

Wikipedia is the place to go next.

There are a couple of good television specials, too. Here is a NOVA documentary:

Here is an oldie presented by Arthur C. Clarke: