Periodic billiard trajectories in regular polygons and closed geodesics on regular polyhedra

Abstract

We consider the relations between the lengths of periodic billiard trajectories in regular triangles, squares, and regular pentagons and those of closed geodesics on the surfaces of regular polyhedra. The cases of regular tetrahedra and octahedra are fully resolved in [3]. The cases of cubes and regular icosahedra are treated below. In the case of regular dodecahedra we can present only preliminary partial results.