  The study of polygonal billiards, particularly those in the regular pentagon,
has been the subject of two recent papers. One of these papers approaches the
problem of discovering the periodic trajectories on the pentagon by identifying
slopes of periodic directions with points in the Poincar\'e disk generated by
hyperbolic isometric transformations. The other approach, coming from the other
paper, transforms the double pentagon into a rectilinear translation surface
called the 'golden L', where periodic directions are generated by a set of
matrices associated with this surface in a special way. We connect and unify
these two approaches, and use our unification of these results to generalize
them to arbitrary $2g+1$-sided regular polygons.
