Check out this game, it was invented by a mathematician called John Conway. It's very simple and the patterns that are generated look amazing.
The rules are very simple. You divide up an area into squares and a square can either be dead or alive.
In the next generation, the next click of the clock, the squares are going to change statuses in some way or another. The square which has one or zero neighbors is going to die, a live square that has one or zero neighbors is going to die. A square which has two neighbors is going to survive. A square with three neighbors is going to give birth, is going to come alive, if it isn't already alive. A square with four or more neighbors is going to die of over crowding.