Lines on polygons, how hard can it be?

We will focus on 3 examples in Richard Schwartz's Survey Lecture on Billiards. Given a polygon in the plane, consider a point mass moving at unit speed which has perfectly elastic collisions on the boundary. One simple first question is "Is there a periodic trajectory?" On a triangle, this is still not a fully solved problem. Schwartz showed via a computer aided proof that all triangles up to an angle of 100 degrees have a periodic trajectory. More recently we now know that periodic trajectories exist on all triangles up to 112.3 degrees. When the polygon has rational angles (for example the regular pentagon) there is always a periodic trajectory. In fact the periodic trajectories on the regular pentagon are completely classified and give some pretty pictures too. We'll finish by discussing what happens when we glue multiple polygons, for example gluing 12 regular pentagons to make a dodecahedron.

This is a preparatory lecture for the ICM talk of Richard Schwartz.