Length ratios of parallel short trajectories

Definitions and mathematical statement

Let $n\geq3$ and unfold billiard paths in a regular Euclidean $n$-gon by reflecting the polygon across every encountered side. A short trajectory is a straight segment in the unfolding that joins two polygon vertices and encounters no vertex in its interior. Its length is the Euclidean length of this segment. A reachable point is an endpoint vertex arising in such an unfolding.

The source assigns an oriented short trajectory source and target endpoint angles $\alpha,\beta$ according to its Definition 2.1. These produce $n-2$ types $A_k$, indexed modulo $n-2$; the supplied extract specifies only that type $A_0$ has $\beta-\alpha=2\pi/n$. It does not include Definition 2.1 or the remaining type-assignment rule.

For parallel short trajectories $\gamma,\eta$ of types $A_k,A_\ell$, respectively, prove \[ \frac{\operatorname{length}(\gamma)}{\operatorname{length}(\eta)} =\frac{\sin((k+1)\pi/n)}{\sin((\ell+1)\pi/n)}. \] The formula is read using fixed representatives $k,\ell\in\{0,\ldots,n-3\}$ of the type indices. Representatives must be fixed: the displayed sine expression is not intrinsically a function of arbitrary integer representatives modulo $n-2$. The conjecture quantifies over every admissible parallel pair in the source's sense, not merely the existence of a pair with the displayed ratio. Exact endpoint-angle and parallelism conventions require the missing source definition.

Short English statement

In unfolded regular-polygon billiards, prove that parallel vertex-to-vertex trajectories of types $A_k$ and $A_\ell$ have lengths in the ratio $\sin((k+1)\pi/n):\sin((\ell+1)\pi/n)$. The extract omits the full type definition.

Sources

• Dmitry Fuchs, Billiard Trajectories in Regular Polygons and Foliations by Closed Geodesics on Surfaces (2020), Conjecture 2.7 and the referenced Definition 2.1, https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf; accessed 2026-07-29, with PDF-text extraction recorded in the supplied provenance.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $5000006$, source code catalogue entry, Length ratios of parallel short trajectories. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.