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 Mosc. Math. J., 2011, Volume 11, Number 3, Pages 439–461 (Mi mmj426)

Periodic trajectories in the regular pentagon

Diana Davisa, Dmitry Fuchsb, Serge Tabachnikovc

a Department of Mathematics, Brown University, Providence, RI, USA
b Department of Mathematics, University of California, Davis, CA, USA
c Department of Mathematics, Pennsylvania State University, University Park, PA, USA

Abstract: We consider periodic billiard trajectories in a regular pentagon. It is known that the trajectory is periodic if and only if the tangent of the angle formed by the trajectory and the side of the pentagon belongs to $(\sin36^\circ)\mathbb Q[\sqrt5]$. Moreover, for every such direction, the lengths of the trajectories, both geometric and combinatorial, take precisely two values. In this paper, we provide a full computation of these lengths as well as a full description of the corresponding symbolic orbits. We also formulate results and conjectures regarding the billiards in other regular polygons.

Key words and phrases: periodic billiard trajectories, regular pentagon, Veech alternative, closed geodesics, regular dodecahedron.

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MSC: Primary 37E35; Secondary 37E05, 37E15
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Citation: Diana Davis, Dmitry Fuchs, Serge Tabachnikov, “Periodic trajectories in the regular pentagon”, Mosc. Math. J., 11:3 (2011), 439–461

Citation in format AMSBIB
\Bibitem{DavFucTab11} \by Diana~Davis, Dmitry~Fuchs, Serge~Tabachnikov \paper Periodic trajectories in the regular pentagon \jour Mosc. Math.~J. \yr 2011 \vol 11 \issue 3 \pages 439--461 \mathnet{http://mi.mathnet.ru/mmj426} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2894424} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000300365900002} 

• http://mi.mathnet.ru/eng/mmj426
• http://mi.mathnet.ru/eng/mmj/v11/i3/p439

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This publication is cited in the following articles:
1. Davis D., “Cutting Sequences, Regular Polygons, and the Veech Group”, Geod. Dedic., 162:1 (2013), 231–261
2. Dmitry Fuchs, Serge Tabachnikov, “Periodic trajectories in the regular pentagon, II”, Mosc. Math. J., 13:1 (2013), 19–32
3. Fuchs D., “Periodic Billiard Trajectories in Regular Polygons and Closed Geodesics on Regular Polyhedra”, Geod. Dedic., 170:1 (2014), 319–333
4. Davis D., “Cutting Sequences on Translation Surfaces”, N. Y. J. Math., 20 (2014), 399–429
5. Bedaride N., Rao M., “Regular Simplices and Periodic Billiard Orbits”, Proc. Amer. Math. Soc., 142:10 (2014), PII S0002-9939(2014)12076-4, 3511–3519
6. Wu ShengJian, Zhong YuMin, “on Cutting Sequences of the l-Shaped Translation Surface Tiled By Three Squares”, Sci. China-Math., 58:6 (2015), 1311–1326