HEXAGONbeta

No arrivals: complete periodicity and degenerate orbits in rational right triangles

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systemsmath.GT — Geometric Topologymath.NT — Number Theory

Contributed by Joshua Boorstein ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

Boshernitzan conjectured in 1992 that in a rational right or isosceles triangle every direction of an explicit finite set is purely periodic: every billiard orbit in it that avoids the vertices is periodic. For a right triangle with acute angle Pπ/2QP\pi/2Q in lowest terms the set is two classes of directions, each containing the perpendicular to a side, and the conjecture says that both are completely periodic. It fails at one known right triangle, (4/15,7/30,1/2)π(4/15, 7/30, 1/2)\pi. We prove it, when min⁡(P,Q−P)≥2\min(P, Q - P) \geq 2, whenever P+2P + 2 and P−2P - 2 are both prime to QQ, a set of density about 0.620.62, and whenever QQ has at most one odd prime factor, with a computer-assisted check of finitely many cases; and, through the fold, for the corresponding isosceles triangles. Everything turns on one object, an orbit from a vertex to a vertex that never strikes a side at a right angle: if there is none, both classes are completely periodic, and at odd PP the retracing band of a companion paper exists. Such an orbit satisfies an identity among roots of unity whose coefficients have signs fixed by the geometry, and Galois conjugation — or, at a single odd prime, a slowest-rotation argument — makes it impossible. When neither P+2P + 2 nor P−2P - 2 is a unit modulo QQ, no argument from these inputs can succeed, and the counterexample lies there. A second method reads the poles of a minimal component as a word of a binary code attached to QQ; it reaches further composite QQ and, with interval arithmetic, the triangles (4,Q−4,Q)(4, Q - 4, Q) at every Q≡3(mod4)Q \equiv 3 \pmod{4}. Beyond right triangles we prove, with computer assistance, that Boshernitzan's Conjecture 2.2(a) fails for infinitely many triangles.

Author comments: First version. Companion papers: "The orphan theorem" (hexagon:2611.00191) and "The necklace of a rational right triangle" (hexagon:2611.00213), whose results this paper imports; "Non-perpendicular periodic orbits in mirror directions of right triangles" in preparation. 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 and interval computations deposited at https://doi.org/10.5281/zenodo.24037744 (directories lib/ and method/). 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, interval arithmetic, certificates)

References

  1. P. Apisa, Billiards in right triangles and orbit closures in genus zero strata , arXiv:2110.07540 (2021).arXiv
  2. M. D. Boshernitzan, Billiards and rational periodic directions in polygons , Amer. Math. Monthly 99 (1992), 522--529.
  3. F. Beukers, C. J. Smyth, Cyclotomic points on curves , in Number Theory for the Millennium I (Urbana, IL, 2000), A K Peters, Natick, MA (2002), 67--85.
  4. K. Calta, J. Smillie, Algebraically periodic translation surfaces , J. Mod. Dyn. 2 (2008), 209--248.
  5. V. Delecroix, J. Rüth, Triangle (2, 3, 6) and (7, 8, 15) and others , sage-flatsurf issue #141 (2022), https://github.com/flatsurf/sage-flatsurf/issues/141; and the sage-flatsurf documentation example Boshernitzan's Conjectures (version 0.8.0, November 2025), https://flatsurf.github.io/sage-flatsurf/examples/boshernitzan_conjecture.html.link
  6. N. G. de Bruijn, On the factorization of cyclic groups , Indag. Math. 15 (1953), 370--377.
  7. G. Dresden, On the middle coefficient of a cyclotomic polynomial , Amer. Math. Monthly 111 (2004), 531--533.
  8. C. Freiling, D. Rinne, Tiling a square with similar rectangles , Math. Res. Lett. 1 (1994), 547--558.
  9. W. P. Hooper, Periodic billiard paths in right triangles are unstable , Geom. Dedicata 125 (2007), 39--46.
  10. M. Keane, Interval exchange transformations , Math. Z. 141 (1975), 25--31.
Show all 23 referencesShow fewer references
  1. R. Kenyon, J. Smillie, Billiards on rational-angled triangles , Comment. Math. Helv. 75 (2000), 65--108.
  2. M. Laczkovich, Tilings of polygons with similar triangles , Combinatorica 10 (1990), 281--306.
  3. S. Lang, Fundamentals of Diophantine Geometry , Springer (1983).
  4. T. Y. Lam, K. H. Leung, Vanishing sums of mth roots of unity in finite fields , Finite Fields Appl. 2 (1996), 422--438.
  5. M. Laczkovich, G. Szekeres, Tilings of the square with similar rectangles , Discrete Comput. Geom. 13 (1995), 569--572.
  6. C. T. McMullen, Teichmüller geodesics of infinite complexity , Acta Math. 191 (2003), 191--223.
  7. M. Mirzakhani, A. Wright, Full rank affine invariant submanifolds , Duke Math. J. 167 (2018), 1--40.
  8. J. Boorstein, The necklace of a rational right triangle: exact counting and complete periodicity , companion paper II, Hexagon, hexagon:2611.00213 (2026).hexagon
  9. J. Boorstein, The orphan theorem: degenerate perpendicular orbits in rational right triangles , companion paper I, Hexagon, hexagon:2611.00191 (2026).hexagon
  10. J.-C. Puchta, On triangular billiards , Comment. Math. Helv. 76 (2001), 501--505.
  11. K. Strebel, Quadratic differentials , Springer (1984).
  12. S. Troubetzkoy, Periodic billiard orbits in right triangles , Ann. Inst. Fourier 55 (2005), 29--46.
  13. W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards , Invent. Math. 97 (1989), 553--583. -\/-\/-

Version history

  1. v1 · 94 pagesSubmitted by Joshua BoorsteinFirst version. Companion papers: "The orphan theorem" (hexagon:2611.00191) and "The necklace of a rational right triangle" (hexagon:2611.00213), whose results this paper imports; "Non-perpendicular periodic orbits in mirror directions of right triangles" in preparation. Code: https://doi.org/10.5281/zenodo.24037744CurrentNov 09, 2026