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Periodic billiard trajectories in regular polygons and closed geodesics on regular polyhedra

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Abstract

We consider the relations between the lengths of periodic billiard trajectories in regular triangles, squares, and regular pentagons and those of closed geodesics on the surfaces of regular polyhedra. The cases of regular tetrahedra and octahedra are fully resolved in [3]. The cases of cubes and regular icosahedra are treated below. In the case of regular dodecahedra we can present only preliminary partial results.

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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)

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  2. Fuchs, D.: Geodesics on a regular dodecahedron. Preprint, MPIM (2009)

  3. Fuchs, D., Fuchs, E.: Closed geodesics on regular polyhedra. Mosc. Math. J. 7, 265–279 (2007)

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Acknowledgments

The author is grateful to the Max Planck Institute for Mathematics (MPIM) in Bonn, Germany, and to Institute des Hautes Etudes Scientifique (IHES) in Bures-sur-Yvette, France, for their hospitality and to Serge Tabachnikov and Anton Zorich for a discussion of results of this article.

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  1. University of California, Davis, CA, USA

    Dmitry Fuchs

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  1. Dmitry Fuchs
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Correspondence to Dmitry Fuchs.

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Fuchs, D. Periodic billiard trajectories in regular polygons and closed geodesics on regular polyhedra. Geom Dedicata 170, 319–333 (2014). https://doi.org/10.1007/s10711-013-9883-9

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  • DOI: https://doi.org/10.1007/s10711-013-9883-9

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