math.DS — Dynamical Systems

Non-perpendicular periodic orbits in mirror directions of right triangles

Contributed by Joshua Boorstein

Let $T$ be a right triangle with an acute angle $a$. A direction is a mirror direction of $T$ if it is the axis of a reflection in the group generated by the reflections in the sides of $T$ — for instance, any direction parallel or perpendicular to a side. A billiard path that meets a side at a right angle retraces itself, its directions are mirror directions, and it is periodic as soon as it meets a side at a right angle a second time. We ask whether every periodic billiard path in a mirror direction is of this kind. When $\cos 2a$ is transcendental it is: every such path meets a side at a right angle, and in the directions parallel to the hypotenuse there is no periodic path at all. The proof is a holonomy computation on Hooper's chain of rhombi, and it bounds the degree of $\cos 2a$ for any exception. When $\cos 2a$ is algebraic and $a/\pi$ irrational, exceptions can occur: we construct non-perpendicular periodic cylinders in mirror directions explicitly — among them an infinite family indexed by the solutions of $u^2 - 2v^2 = -1$ — and show that the triangles carrying them form a countable dense set in each of the four classes of mirror directions. Corollaries include the absence of odd-period orbits in right triangles and the existence of one-way periodic paths in every right triangle (answering two questions of Holt), a description of the odd orbits of isosceles triangles, and the statement that Ruijgrok's third conjecture, read for right triangles, fails on a countable dense set of triangles, all with $\cos 2a$ algebraic, and holds whenever $\cos 2a$ is transcendental.

Author comments: First version. Companion paper: "The necklace of a rational right triangle" (hexagon:2611.00213). Code: https://doi.org/10.5281/zenodo.24037744