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Singular perpendicular orbits and complete periodicity in rational triangles

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

Contributed by Joshua Boorstein ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

Let Δ\Delta be a triangle whose angles are rational multiples of π\pi. A billiard path leaving a side σ\sigma at a right angle either meets a vertex first — its starting point is then singular — or meets a side at a right angle again: σ\sigma itself, at a new point, or another side, where it retraces itself. The singular points cut σ\sigma into finitely many intervals, and each interval retraces or is carried onto another by the return map, reversing orientation (a mirror pair) or preserving it (a translation pair). We count these intervals exactly. The singular points of the sides perpendicular to a direction class are in bijection with the slots — a vertex with a direction of the class pointing inward — whose path meets a side at a right angle before any vertex, so the number of intervals is v+S−2p−s′v + S - 2p - s': SS slots, vv perpendicular sides, pp pairs of mutually reverse vertex-to-vertex paths meeting no side at a right angle, and s′s' slots whose path never reaches a vertex. This bounds the retracing bands and translation pairs, sharply on isosceles triangles, and makes s′=0s' = 0 equivalent to complete periodicity of the class. Our main theorem is that p=0p = 0 forces complete periodicity: a minimal component needs a swapped pair. The converse fails. In every isosceles triangle (a,a,b)(a, a, b) with NN even and a≥3a \geq 3 the class of the base direction has p≥1p \geq 1, and it is completely periodic, by a companion paper (computer-assisted in part), whenever b≥4b \geq 4 and NN has at most one odd prime factor. At odd NN, two infinite families of scalene triangles have p≥1p \geq 1 and are completely periodic in the checked range.

Author comments: First version. Companion papers: "The orphan theorem" (hexagon:2611.00191), "The necklace of a rational right triangle" (hexagon:2611.00213) and "No arrivals" (hexagon:2611.00250), whose results this paper imports. Code: https://doi.org/10.5281/zenodo.24093456

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.24093456 (directories lib/ and polygon/). 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 cyclotomic arithmetic, exact interval exchanges, Sturm root counts)

References

  1. M. Boshernitzan, G. Galperin, T. Krüger, S. Troubetzkoy, Periodic billiard orbits are dense in rational polygons , Trans. Amer. Math. Soc. 350 (1998), 3523--3535. arXiv:math/9408217.arXiv
  2. M. D. Boshernitzan, Billiards and rational periodic directions in polygons , Amer. Math. Monthly 99 (1992), 522--529.
  3. Y. Cheung, P. Hubert, H. Masur, Topological dichotomy and strict ergodicity for translation surfaces , Ergodic Theory Dynam. Systems 28 (2008), 1729--1748. arXiv:math/0607179.arXiv
  4. V. Delecroix, flatsurvey issue #22 (2022), https://github.com/flatsurf/flatsurvey/issues/22.link
  5. V. Delecroix, J. Rüth, sage-flatsurf issue #141 (2022), https://github.com/flatsurf/sage-flatsurf/issues/141; and the sage-flatsurf documentation example Boshernitzan's Conjectures (2022), https://flatsurf.github.io/sage-flatsurf/examples/boshernitzan_conjecture.html.link
  6. R. Kenyon, J. Smillie, Billiards on rational-angled triangles , Comment. Math. Helv. 75 (2000), 65--108.
  7. H. Masur, S. Tabachnikov, Rational billiards and flat structures , Handbook of Dynamical Systems, Vol. 1A, North-Holland, 2002, 1015--1089.
  8. J. Boorstein, No arrivals: complete periodicity and degenerate orbits in rational right triangles , Hexagon, hexagon:2611.00250 (2026).hexagon
  9. J. Boorstein, The necklace of a rational right triangle: exact counting and complete periodicity , Hexagon, hexagon:2611.00213 (2026).hexagon
  10. J. Boorstein, The orphan theorem: degenerate perpendicular orbits in rational right triangles , Hexagon, hexagon:2611.00191 (2026).hexagon
Show all 15 referencesShow fewer references
  1. Th. W. Ruijgrok, Periodic orbits in triangular billiards , Acta Phys. Polon. B 22 (1991), 955--981.
  2. G. Tokarsky, J. Garber, B. Marinov, K. Moore, One hundred and twelve point three degree theorem , arXiv:1808.06667 (2018).arXiv
  3. W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards , Invent. Math. 97 (1989), 553--583.
  4. Ya. B. Vorobets, G. A. Gal'perin, A. M. Stepin, Periodic billiard trajectories in polygons: generating mechanisms , Russian Math. Surveys 47 (1992), no. 3, 5--80.
  5. Ya. B. Vorobets, Billiards in rational polygons: periodic trajectories, symmetries, and d-stability , Math. Notes 62 (1997), 56--63.

Version history

  1. v1 · 26 pagesSubmitted by Joshua BoorsteinFirst version. Companion papers: "The orphan theorem" (hexagon:2611.00191), "The necklace of a rational right triangle" (hexagon:2611.00213) and "No arrivals" (hexagon:2611.00250), whose results this paper imports. Code: https://doi.org/10.5281/zenodo.24093456CurrentNov 11, 2026