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Non-perpendicular periodic orbits in mirror directions of right triangles

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systemsmath.MG — Metric Geometry

Contributed by Joshua Boorstein ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

Let TT be a right triangle with an acute angle aa. A direction is a mirror direction of TT if it is the axis of a reflection in the group generated by the reflections in the sides of TT — 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\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\cos 2a for any exception. When cos⁡2a\cos 2a is algebraic and a/πa/\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 u2−2v2=−1u^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\cos 2a algebraic, and holds whenever cos⁡2a\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

Provenance statement

The text, the proofs and the code were drafted by an AI system (Claude, Anthropic) working under the author's direction over an extended research project, and checked by independent AI review passes and by the exact computations deposited at https://doi.org/10.5281/zenodo.24037744 (directories lib/ and mirror_A/). The author has read parts of the paper in detail and takes responsibility for its content.

Tools used

Anthropic
Claude Opus
Python
SymPy/mpmath (exact arithmetic in Q(2 cos a) with certified signs, certificates)

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Version history

  1. v1 · 35 pagesSubmitted by Joshua BoorsteinFirst version. Companion paper: "The necklace of a rational right triangle" (hexagon:2611.00213). Code: https://doi.org/10.5281/zenodo.24037744CurrentNov 11, 2026