Named after the Polish mathematician Waclaw Sierpinski (1882-1969), the Sierpinski Triangle can be created in a deterministic manner, or by using a random algorithm (chaos). To make one, begin with any triangle. Use the midpoints of each side as the vertices of a new triangle, which we then remove from the original. This leaves us with three triangles, each of which has dimensions exactly one-half the dimensions of the original triangle, and area exactly one-fourth of the original area. Also, each remaining triangle is similar to the original.

Now we continue (or iterate) this process. From each remaining triangle we remove the "middle" leaving behind three smaller triangles each of which has dimensions one-half of those of the parent triangle (and one-fourth of the original triangle). Clearly, 9 triangles remain at this stage. At the next iteration, 27 small triangles, then 81, and, at the Nth stage, 3^N small triangles remain. It is easy to check that the dimensions of the triangles that remain after the Nth iteration are exactly 1/2^N of the original dimensions. (Thanks to Robert L. Devaney of the Department of Mathematics and Statistics at Boston University.)