In this talk we raise the following (dynamical) question: "given a billard trajectory on a regular polygon starting from the center of the polygon and eventually ending at a vertex, is it true that the "reverse" trajectory starting from the center of the polygon but with the opposite direction also ends eventually at a vertex ?" One can easily convince itself that the answer is yes when the number of sides of the regular polygon is even, by symmetry. Now, if the number of sides is odd, the question is not trivial and we will see that the answer is still 'yes' for the equilateral triangle and the regular pentagon, but 'no' if the polygon has 7 sides or more! Further, we will see that this question is closely related to  the (difficult) problem of determining the real numbers (in some algebraic field) having a finite 'Hecke continued fraction expansion', or equivalently the parabolic fixed points on a given Hecke group. On the way, we will encounter so-called translation surfaces and their Veech groups, and we will discuss the notion of connection points on such surfaces.