The Chaos Game and the Sierpinski Triangle
In mathematics, the chaos game is a manipulation of a polygon in such a way that a fractal is often, but not always, created. It involves starting with an initial random point within the polygon, and plotting points within the polygon that are a given fraction between vertices of the polygon and those points.
Using an equilateral triangle as the polygon, and a fraction of 1/2 will result in the Sierpinski Triangle. Huh, no wonder it's called the chaos game! That sounds a bit confusing! Let's step it out using an equilateral triangle and 1/2 as our fraction to better our understanding of the construction process and the chaos game itself. The steps we use in the chaos game to create a Sierpinski Triangle are as follows:
 Start with an equilateral triangle.
 Choose one of the triangles vertices and a random point inside the triangle.
 Plot the point that is midway between the vertex and random point.
 Use the point you just plotted and one of the triangle's vertices and start again at step three.
 Continue this process indefinitely.
This process must be done for quite a while, but believe it or not, after a long while of doing this, the triangle will look more and more like the Sierpinski Triangle.
Example
Because of the fact that using the chaos game to construct the Sierpinski Triangle involves repeating the procedure an extremely large number of times for the triangle to really resemble the Sierpinski Triangle, computers are normally used for this type of construction. However, let's take a look at doing this for a few steps, so we can get a look at the chaos game and understand how the steps are performed better.
The Chaos Game Eventually Gives the Sierpinski Triangle

You have to admit, that is pretty neat! The vertices and initial point are chosen completely at random, yet somehow this chaotic procedure results in the Sierpinski Triangle! That is, the construction resulting in a Sierpinski triangle is completely reliant on chaos and randomness rather than on a specified choice of vertices. This is why the chaos game is named the 'chaos' game.
Lesson Summary
The Sierpinski Triangle is a selfsimilar fractal that can be constructed in a couple of ways. The first is quite simple, and is as follows:
 Start with an equilateral triangle.
 Connect the midpoints of each of the sides of the triangle. Remove the center triangle that you just created.
 Do the same thing for the remaining three triangles that were created within the original triangle.
 Continue this process for each new triangle created indefinitely.
The second involves the chaos game, or a manipulation of polygons used to create fractals, and is completely reliant on randomness. The steps of this construction method are as follows:
 Start with an equilateral triangle.
 Choose one of the triangles vertices and a random point inside the triangle.
 Plot the point that is midway between the vertex and random point.
 Use the point you just plotted and one of the triangle's vertices and start again at step three.
 Continue this process indefinitely.
This is all so neat! Not only are we now more familiar with the Sierpinski Triangle and its construction using the chaos game, but we also have a great new idea of a doodle to create anytime we have some time to kill! Heck, if it's during math class, maybe your instructor will count it as extra credit!