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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Regular simplices and periodic billiard orbits
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by Nicolas Bédaride and Michael Rao PDF
Proc. Amer. Math. Soc. 142 (2014), 3511-3519 Request permission

Abstract:

A simplex is the convex hull of $n+1$ points in $\mathbb {R}^{n}$ which form an affine basis. A regular simplex $\Delta ^n$ is a simplex with sides of the same length. We consider the billiard flow inside a regular simplex of $\mathbb {R}^n$. We show the existence of two types of periodic trajectories. One has period $n+1$ and hits each face once. The other one has period $2n$ and hits $n$ times one of the faces while hitting any other face once. In both cases we determine the exact coordinates for the points where the trajectory hits the boundary of the simplex.
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Additional Information
  • Nicolas Bédaride
  • Affiliation: Laboratoire d’Analyse Topologie et Probabilités UMR 7353, Centre de Mathéma- tiques et Informatique, Université Aix Marseille, 29 Avenue Joliot Curie, 13453 Marseille Cedex, France
  • Address at time of publication: Aix Marseille Université CNRS, Centrale Marseille, I2M UMR 7373, 13453 Marseille, France
  • Email: nicolas.bedaride@univ-amu.fr
  • Michael Rao
  • Affiliation: Laboratoire de l’Informatique du Parallélisme, équipe MC2, École Normale Supéri- eure, 46 Avenue d’Italie 69364 Lyon Cedex 7, France
  • MR Author ID: 714149
  • Email: michael.rao@ens-lyon.fr
  • Received by editor(s): April 30, 2012
  • Received by editor(s) in revised form: October 13, 2012, and October 21, 2012
  • Published electronically: June 19, 2014
  • Communicated by: Nimish Shah
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 142 (2014), 3511-3519
  • MSC (2010): Primary 37E15
  • DOI: https://doi.org/10.1090/S0002-9939-2014-12076-4
  • MathSciNet review: 3238426