it stops. This dynamical system is equivalent to the geodesic flow on a flat surface: just double the polygon along the edges and put a flat metric everywhere but at the vertices, which become singular points with cone angle twice the angle of the polygon at the corresponding vertex. This surface is not a translation surface or a half-translation surface, but in some cases it is related to one. Namely, if all angles of the polygon are rational multiples of there is ramified cover of this surface which is a translation surface, which can be constructed from a union of copies of . The dynamics of the billiard flow can then be studied through the geodesic flow on the translation surface. For example, the billiard in a square is related in this way to the billiard on the flat torus constructed from four copies of the square; the billiard in an equilateral triangle gives rise to the flat torus constructed from an hexagon. The billiard in a "L" shape constructed from squares is related to the geodesic flow on a square-tiled surface; the billiard in the triangle with angles is related to the Veech surface constructed from two regular pentagons constructed above. Relation with interval exchange transformations Let be a translation surface and a direction, and let be the geodesic flow on with direction . Let be a geodesic segment in the direction orthogonal to , and defined the first recurrence, or Poincaré map as follows: is equal to where for . Then this map is an interval exchange transformation and it can be used to study the dynamic of the geodesic flow. Notes References Surfaces Dynamical systems