On Seven Conjectures of Fuchs on Billiard Trajectories in Regular Polygons and Geodesics on the Regular Dodecahedron

Abstract

In a problem contribution (Arnold Math. J. 7 (2021)), D. Fuchs stated seven conjectures on the billiard trajectories in a regular n-gon which join two vertices (short trajectories) and on their n − 2 types A_0, ..., A_{n−3}, on closed trajectories, and on geodesics on the regular dodecahedron. We decide all seven. The tool is the double n-gon: the types are the orbits of saddle connections under an explicit group of affine automorphisms, and they are computed by an angular invariant. Conjecture 2.6 (types of parallel short trajectories) and Conjecture 2.7 (ratios of their lengths) are proved for all n ≥ 4, and Conjecture 2.3 (types of the vertices of a reachable n-gon) for all n ≥ 5. Conjecture 2.5 (reachable points on the two lines through a unitary pair) is true as printed for n = 5 and false for every n ≥ 6; it is proved for all n ≥ 5 with λ² and λ² − 1, λ = 2 cos(π/n), in place of λ + 1 and λ, which are the constants given by the source's own argument for n = 6. Conjecture 2.4 (every reachable point is a vertex of infinitely many reachable n-gons) is false for every n ≥ 5: a reachable point whose type is not A_0 is a vertex of at most one reachable n-gon, and an explicit point is a vertex of none; exactly the points of type A_0 are vertices of infinitely many. Conjecture 3.2 is proved: a short geodesic of type A_0 on the regular dodecahedron never ends at graph distance 2 from its first vertex; its ends are exchanged by a half-turn about the midpoint of an edge, an argument which Athreya, Aulicino and Hooper used for closed saddle connections and Troubetzkoy for the other regular polyhedra. Conjecture 1.7 (after cancellation of common factors, the quotient of the lengths of a closed trajectory and of its preclosed part is never n) is false for every odd n with at least two distinct prime factors; the smallest case is n = 15, for the trajectories parallel to a side. It is true for odd prime powers, and for even n when it is read for the trajectories in general position in their band. These quotients were determined in all periodic directions by Veech (1992), through the monodromy of the covering of the double n-gon by the surface of the billiard; the answer to Conjecture 1.7 follows from his formula by a short computation which is not in his paper. We also give a second derivation of the formula for odd n, and a proof of the counterexample which uses neither surfaces nor homology. For prime n the conjecture was known: this case follows from a theorem of McMullen, for n = 5 from results of Davis, Fuchs and Tabachnikov and of Davis and Lelièvre, and it is also a consequence of Veech's formula. The proofs are elementary. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifiers: AMR-049-0001, AMR-049-0002, AMR-049-0003, AMR-049-0004, AMR-049-0005, AMR-049-0006, AMR-049-0007 (Fuchs, Billiard Trajectories in Regular Polygons and Geodesics on Regular Polyhedra, Arnold Math. J. 7 (2021)).