The Chaos Game: Play It

http://tomcat.gdickinson.com/fractals/

This particular design is named after a Polish mathematician, Waclaw Sierpinski, who described various properties of this pattern in 1916.In this demonstration you are able to choose where to place the three vertices, labeled X, Y, and Z. You also choose where to place the starting point, or seed, anywhere inside or outside of the triangle that you wish. For a speedy demonstration the computer then randomly selects which of the three vertices to work with and draws an imaginary line from the seed point you started with to the randomly selected vertices and places a dot at the midpoint of this line. The dot becomes the next starting point or seed. The computer then randomly selects the next vertices, draws a line, and places a dot at the midpoint. The computer completes this process for many thousands of iterations to achieve the triangle but if you follow this same procedure by hand you will get a similar result. As you are beginning to realize, the elegant order of Sierpinski's Triangle is, in this construction, fundamentally dependent on randomness; the randomness that plays a central role in our experience of the universe.