The Chaos Game
Chaos theory deals with the construction of fractals. The Chaos Game is one of the more commonly used algorithms in fractal construction.
The "rules" of the game are simple. A certain number of vertices are drawn, and an arbitrary starting point is selected. By coin flip, die roll, or whatever, a vertex is chosen at random. A new point is then drawn halfway between the previous point and the selected vertex. The process is then repeated multiple times.
Eventually, an attractor will reveal itself. For two vertices, the attractor is a line segment; for three, it is a Sierpinski triangle.
This is a topic which is best learned by doing, so we will explore two Java applets created by Robert devaney, a professor at Boston University.
One applet actually makes a game out of the Chaos Game; it can be found here.
Another applet allows the user to specify starting vertices, and then it plays the Chaos Game with those vertices. Click here to see.
It is also possible to use Mathematica with the Chaos Game. Here, one enters an IFS and the program will play the game with that IFS until it reaches its maximum number of turns. The source code is available here.