Rational Polygonal Billiards

Rational polygonal billiards is the study of billiards in a polygon whose angles are rational multiples of . A trajectory moves along straight lines and obeys equal incidence and reflection angles at the sides. A regular trajectory never meets a vertex. Reflecting the table instead of the trajectory unfolds the motion into straight-line motion on a compact surface assembled from finitely many reflected copies of the polygon (Tabachnikov 1995). Periodic trajectories are dense in the phase space for every rational polygon (Boshernitzan et al. 1998).

A direction is periodic if it contains a periodic orbit, and purely periodic if every regular orbit in that direction is periodic. The latter condition is also called complete periodicity. The existence of a periodic trajectory alone does not imply complete periodicity. A regular trajectory meeting a side perpendicularly retraces its path and is periodic.

For a triangle with angles , , and , where and , put and place a side horizontally. Boshernitzan (1992) conjectured that every direction which is an integer multiple of is purely periodic when is even, and also when the triangle is right or isosceles. These are parts (a) and (e), respectively, of his Conjecture 2.2. The right triangle with angles , , and is a known counterexample. Its direction perpendicular to one leg contains both periodic trajectories and nonperiodic regular trajectories (Delecroix and Rüth 2022).

Write a rational right triangle's angles as , where and . The directions that are integer multiples of fall into two classes under the side reflections, each containing a direction perpendicular to a side. Boorstein (2026) reports complete periodicity of both classes when and either has at most one distinct odd prime factor or . The first family includes denominators which are powers of two. Further reported families include the right triangles with and , , and the following explicit ranges. For odd , set equal to the odd member of and require . For even which is not a power of two, put and and require . Here is the floor function and is the totient function. All these ranges retain .

The reported arguments use signed identities among roots of unity for trajectories joining vertices without a perpendicular collision, and a binary linear code constraint on the poles of a quotient of the unfolded surface. The code gives sufficient geometric criteria for complete periodicity, not a decision for every rational right triangle. When is odd, the corresponding results also establish existence of the retracing band called an orphan in the companion papers. The periodicity and orphan results discharge the applicable hypotheses of their branch-counting formulas. In particular, the beam launched perpendicularly from the leg opposite the acute angle has branches when is even and either of the first two arithmetic hypotheses holds. Doubling a covered right triangle along a leg gives corresponding isosceles cases. These results do not assert that every right or isosceles triangle satisfies the conjecture.

Boorstein (2026) also reports infinitely many counterexamples to part (a). For a rational number in lowest terms with even, take the angles . The reported construction applies when belongs to any of

(1)

(2)

(3)

Each such triangle is reported to have a direction containing a periodic trajectory and a nonperiodic regular trajectory. Exact return-word identities and interval arithmetic inequalities are supplied for these intervals. For the known right-triangle counterexample, the paper reports a decomposition of the billiard phase space in that direction into three periodic cylinders and one nonperiodic component in which regular trajectories are dense, with exact area accounting.

Boorstein attributes the mathematical proofs and certificate code to Claude under his direction. He reports independent AI review passes and states that he read parts of the paper in detail. The proof uses two companion manuscripts, and specified finite computations use exact arithmetic and interval arithmetic. The deposited computations do not formally certify all of the written proofs. Independent specialist verification of the complete arguments has not been reported.