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

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

<%= movie('sparse.m4v', 400, 400 ) %>

Download the original as a QuickTime movie.

Movie with more triangles

<%= movie('dense.m4v', 400, 400 ) %>

Download the original as a QuickTime movie.