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Periodic billiard trajectories in regular polygons and closed geodesics on regular polyhedra. (English) Zbl 1296.53085

Move a point along a geodesic with unit velocity inside a face of a regular polyhedron in Euclidean 3-space. As the point hits an edge, we can either continue along another face, as a geodesic, or we can reflect into the same face, as a billiard. The billiard and the geodesic are periodic under exactly the same circumstances. The author raises the problem of relating the periods.
The periods have small integer ratio, but the precise value of that integer is dependent on the particular geodesic. The author has determined previously the relations between the periods in regular tetrahedra and octahedra.
In this paper, he determines these relations for cubes and regular icosahedra, and discusses the problem for regular dodecahedra. He finds a very surprising (for the reviewer) relation to modular groups, with proofs given by explicitly drawing pictures.

MSC:

53C22 Geodesics in global differential geometry
37C27 Periodic orbits of vector fields and flows
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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References:

[1] Davis, D., Fuchs, D., Tabachnikov, S.: Periodic trajectories in the regular pentagon. Mosc. Math. J. 11, 439-461 (2011) · Zbl 1276.37033
[2] Fuchs, D.: Geodesics on a regular dodecahedron. Preprint, MPIM (2009) · Zbl 1276.37033
[3] Fuchs, D., Fuchs, E.: Closed geodesics on regular polyhedra. Mosc. Math. J. 7, 265-279 (2007) · Zbl 1129.53022
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