1. Start with any triangle in a plane.
2. Shrink the triangle by ½, make three copies, and translate them so that each triangle touches the two other triangles at a corner.
3. Repeat step 2 ad infinitum.
The gasket can also be made by starting with Pascal's triangle, then coloring the even numbers white and the odd numbers black. Most curiously, it can be generated by a game of chance. Start with three points, labeled 1, 2, and 3, and any starting point, S. Then select randomly 1, 2, or 3, using a die or some other method. Each random number defines a new point halfway between the latest point and the labeled point that the random number indicates. When the game has gone on long enough the pattern produced is the Sierpinski gasket.
The gasket has a Hausdorff dimension of log 3/log 2 = 1.585..., which follows from the fact that it is a union of three copies of itself, each scaled by a factor of ½. Adding rounded corners to the defining curve gives a non-intersecting curve that traverses the gasket from one corner to another and which Benoit Mandelbrot called the Sierpinski arrowhead.