An orbit on a polygon is a path that a "billiards ball" (a point) would follow if it obeyed Snell's law of reflection (the angle of incidence is equal to the angle of reflection). A periodic orbit is an orbit that returns to the same point with the same initial angle (we count the smallest such number so as to avoid repeats). In this case, the period of a periodic orbit is the number of edge-hitting events.
My question is this:
Is anyone aware of a (possibly classical) result with proof that tells us what the period of a periodic orbit on a regular hexagon would be given either the initial angle or a vector that defines the angle?