We consider the behavior of trajectories for the billiard on a regular polygon.  In three special cases which give rise to lattice tilings of the plane (the triangle, the square and the hexagon), the behavior of trajectories is very simple to analyze: they are either periodic or quasiperiodic.  Can quasiperiodicity be found in the other cases?  Our discussion will take us to the analysis of the renormalization flow for Veech surfaces which are non-arithmetic in the sense that the trace field is a non-trivial finite extension of $\Q$.  We will see that the typical behavior presents no remains of quasiperiodicity, but exceptional behavior can appear (with positive Hausdorff dimension) if the Veech group contains a Salem element.