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

<%= 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.