The Magic Sierpinski Triangle

This design is called Sierpinski's Triangle (or gasket), after the Polish mathematician Waclaw Sierpinski who described some of its interesting properties in 1916. Among these is its fractal or self-similar character. The large blue triangle consists of three smaller blue triangles, each of which itself consists of three smaller blue triangles, each of which ..., a process of subdivision which could, with adequate screen resolution, be seen to continue indefinitely. Fractals and self-similarity are of considerable interest in their own right, but our interest here is in how to construct Sierpinski's triangle. One way to do so is to inscribe a second triangle inside the original one, by joining the midpoints of the three sides, and then repeat the process for the resulting three outer triangles, for the three outer triangles that result from that, and so forth. But there is a more intriguing way to construct Sierpinski's triangle, sometimes called the Chaos Game. http://serendip.brynmawr.edu/playground/sierpinski.html