A Survey of Mathematics Related Topics: Pg. 482, Chaos Theory

So why is this game interesting? What happens when we spill out a bag of marbles onto the floor? After they stop rolling around, we get a pattern formed by the marbles - perhaps not a very interesting one. If we spill the same marbles out on the same floor again, do we expect to get the same pattern of marbles? No. The pattern is pretty random, and we expect that. The chaos game was proposed by Michael Barnsley in the mid-1980s as a way to see how patterns can result from certain random events.

The chaos game begins with a set of dots on a page called vertices. The classic game starts with three vertices numbered 1, 2, and 3 placed on the corners of an equilateral triangle.

Once the user has selected the number of dots to be placed, and set the probability of where the dots will land, the computer chooses a random point on the screen and rolls a three-sided die. Once the die has been rolled, the computer moves half the distance from the last point to the selected vertex and places a dot. Since real life three-sided dice are hard to find (in fact impossible!) you can fake one by using a regular six-sided die, letting rolls of 1 or 2 move towards vertex 1, rolls of 3 or 4 towards vertex 2, and rolls of 5 or 6 moving towards vertex 3.

For example, suppose we chose point P below and then rolled a 1, 6, 2, 2, and 4 in that order this is what we would get.:

It looks fairly random right?

But what happens if we continue plotting points in this way?

After five hundred points a pattern starts to appear:

Here the arrows have been left off and the dots have been colored the same as the vertex moved toward when placing each dot. Recognize the pattern? It is Sierpinski's Triangle. By using this method Barnsley demonstrated, randomness can sometimes generate a very precise pattern.