HEXAGONbeta

The necklace of a rational right triangle: exact counting and complete periodicity

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systemsmath.GT — Geometric Topology

Contributed by Joshua Boorstein ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

Let TT be a right triangle with acute angle α\alpha, 2α/π=P/Q2\alpha/\pi = P/Q in lowest terms. The Katok--Zemlyakov unfolding of TT, quotiented by its holonomy involution, is a necklace of QQ kites: an explicit genus-zero half-translation surface in the stratum Q(P−2,Q−P−2,−1Q)Q(P-2, Q-P-2, -1^Q), whose kites glue in steps of PP and whose interfaces have explicit transverse measures. We use it to count the cells into which the vertices of TT cut the beam perpendicular to each side. For a completely periodic direction class the separatrix diagram has exactly QQ edges, and sorting them by the involution gives the identity Q=h+f+2pQ = h + f + 2p (hh sides of TT in the class, ff extra cells of the perpendicular beams, pp involution-swapped edge pairs). When p=0p = 0 it is a covering law: the perpendicular beams of a class cut Q−1Q - 1 or Q+1Q + 1 cells in total at odd QQ, according to the class, and QQ at even QQ; with no hypothesis the shortfall is even, non-negative, and zero only on a completely periodic class. Complete periodicity of a class — that almost every trajectory in it meets a side at a right angle — is equivalent to p=0p = 0 and to the covering law (on two-beam classes off the Veech locus, one direction assumes an orphan branch, which companion papers prove at broad classes of centres); it fails at 8/158/15. For the beam perpendicular to a leg the count n(P/Q)n(P/Q) is at most Q−1Q - 1 at every centre and is known exactly on the Veech locus min⁡(P,Q−P)≤2\min(P, Q-P) \leq 2; off it we prove n(3/Q)≤2⋅round(Q/3)−1n(3/Q) \leq 2\cdot\text{round}(Q/3) - 1 for Q≥5Q \geq 5, with equality exactly under complete periodicity, and n(5/(5m+1))=5n(5/(5m+1)) = 5 for every m≥3m \geq 3. Finally, every periodic billiard path of a rational right triangle in a mirror direction meets a side at a right angle.

Author comments: First version. Companion papers: "The orphan theorem" (hexagon:2611.00191); "No arrivals" and "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 computations deposited at https://doi.org/10.5281/zenodo.24037744 (directories lib/ and neck/). The author has read parts of the paper in detail and takes responsibility for its content.

Tools used

Anthropic
Claude Opus
Python
SymPy/NumPy (exact cyclotomic arithmetic and certificates)

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Version history

  1. v1 · 87 pagesSubmitted by Joshua BoorsteinFirst version. Companion papers: "The orphan theorem" (hexagon:2611.00191); "No arrivals" and "Non-perpendicular periodic orbits in mirror directions of right triangles" in preparation. Code: https://doi.org/10.5281/zenodo.24037744CurrentNov 08, 2026