My advisor, Shmuel Weinberger, was teaching Math 113, and asked for some pictures of the following procedure:

• Start with a triangle in the plane.
• Reflect that triangle across its three sides.
• And repeat, reflecting the resulting triangles through their sides, and so forth.

I made a couple movies of this, illustrating this procedure as you move through the space of triangles. Observe how, for only four shapes of triangles, the resulting set of triangle vertices is discrete.

Movie with only a few triangles

Download the original as a QuickTime movie.

Movie with more triangles

Download the original as a QuickTime movie.