Wednesday, May 9 at 2:30 PM
We consider the classic open problem of whether every triangle has a periodic billiard path. While this has been shown for rational-angled triangles [Masur86], the irrational case remains open. Finding periodic billiard paths is an easy exercise for acute triangles, a long computer-aided case analysis when the maximum angle is at most 100 degrees [Schwartz08], and beyond that very little is known.
We examine the inequalities characterizing the set of triangles over which a given billiard path is periodic. We prove polynomial upper bounds on their rate of change, and use these bounds to derive positive radii within which a periodic billiard path must remain valid. We perform a computer search for periodic billiard paths on randomly selected triangles over a fixed rational grid, and use the radius bounds to show (under a Bayesian model with constant prior, assuming the uniformity of the selected grid points but otherwise unconditionally) that the likelihood that ≥ 98% of all obtuse triangles admit a periodic billiard path is > 0.99999. We also discuss recent progress in deterministic approaches to the problem.