Polish mathematician Waclaw Sierpinski (1882-1969) worked in the areas of set theory, topology and number theory, and made important contributions to the axiom of choice and continuum hypothesis. But he is best known for the fractal that bears his name, the Sierpinski triangle, which he introduced in 1916.
The Sierpinski triangle, sometimes referred to as the Sierpinski gasket, is a simple iterated function system that often serves as the first example of a fractal given to elementary school or high school students. There are two main ways to construct the triangle, one of which is obvious, and the other rather incredible.
Construction 1: Begin with a base triangle, and then draw lines connecting the midpoints of each leg, forming three self-similar right-side up subtriangles at each of the base triangle's corners. Then repeat this process for each of the newly formed subtriangles, and so on, ad infinitum.
Construction 2: "The Chaos Game"
Most people expect a fairly random graph of streaks within a triangle. But the resulting graph is not random at all:
A random algorithm produces this! Out of chaos crystallizes order. Mathematicians denote this result, the orbit of the seed, by S. Regardless of the initial value of the seed, the Sierpinski triangle is realized.
Below is an applet made by Jacobo Bulaevsky that lets you see iterations of the Sierpinski triangle using the first style of construction described above. It would behoove you to not venture past 10 iterations, since otherwise your browser will freeze.