Start with any solid triangle (in the case below, a black equilateral triangle). Connect the midpoints of the sides, and remove the middle. This will leave you with three smaller but similar triangles (in the case below, 3 smaller black equilateral triangles). Repeat the process every time you see solid triangles.
Like the Koch Snowflake exhibits symmetry of scale -- no matter how much we zoom in, we always see the same level of complexity, or of detail. Because we see the same thing when we zoom in, we say it is self-similar.
This serves no mathematical purpose, but I thought it was cool:
Pascal's Triangle
Exploring the Mandelbrot Set