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Regular simplices and periodic billiard orbits

Authors: Nicolas Bédaride and Michael Rao
Journal: Proc. Amer. Math. Soc. 142 (2014), 3511-3519
MSC (2010): Primary 37E15
Published electronically: June 19, 2014
MathSciNet review: 3238426
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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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Nicolas Bédaride
Affiliation: Laboratoire d’Analyse Topologie et Probabilités UMR 7353, Centre de Mathématiques 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

Michael Rao
Affiliation: Laboratoire de l’Informatique du Parallélisme, équipe MC2, École Normale Supérieure, 46 Avenue d’Italie 69364 Lyon Cedex 7, France

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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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