Existence of Periodic Orbits
In ordinary polygonal billiards, the existence of periodic orbits is a major unsolved problem. For instance, it is unknown if every triangular shaped table has a periodic billiard path. More progress has been made for outer billiards, though the situation is far from well-understood. As mentioned above, all the orbits are periodic when the system is defined relative to a convex rational polygon in the Euclidean plane. Moreover, it is a recent theorem of C. Culter (written up by S. Tabachnikov) that outer billiards relative to any convex polygon has periodic orbits—in fact a periodic orbit outside of any given bounded region.
Read more about this topic: Outer Billiard
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