Triangular billiards

Definitions and mathematical statement

Let $\Delta\subset\mathbb R^2$ be a nondegenerate closed Euclidean triangle. A billiard trajectory moves at constant positive speed on straight segments in its interior. At a collision with the relative interior of a side it reflects specularly: the tangential component of velocity is preserved and the normal component changes sign. Trajectories that meet a vertex are excluded, because the reflection rule is then undefined.

A periodic trajectory is a finite sequence of nonzero segments that returns to its starting position with its starting velocity and thereafter repeats. Equivalently, there are a starting state $(x,v)$ and $T>0$ for which the nonsingular billiard flow satisfies \[ \Phi_T(x,v)=(x,v). \] The question is whether every nondegenerate triangle admits at least one such periodic trajectory. The initial position and direction may depend on the triangle; the assertion is not that every trajectory is periodic.

No rationality condition is imposed on the angles, and the number of reflections is allowed to be arbitrarily large. Self-intersections of the geometric path are allowed, while corner collisions and zero-length segments are not. The classical orthic construction handles acute triangles, and other classes have known positive results. The target includes arbitrary obtuse triangles. A near-return found numerically is not enough: one must verify an exact closed orbit that remains away from all three vertices throughout its reflections.

Short English statement

Does every triangle admit a closed billiard path that avoids its vertices and returns with the same direction?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700022 (AMR-046-0022), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gasull - Some open problems in low dimensional dynamical systems (2020), Problem environment 22: Triangular billiards. Original source indexed by AMR-046-0022. https://arxiv.org/abs/2012.02524.