Periodic trajectories in the regular pentagon

Diana Davis, Dmitry Fuchs, Sergei Tabachnikov

Research output: Contribution to journalArticle

7 Citations (Scopus)

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°)Q[v√5]- 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.

Original languageEnglish (US)
Pages (from-to)439-461
Number of pages23
JournalMoscow Mathematical Journal
Volume11
Issue number3
StatePublished - Aug 5 2011

Fingerprint

Periodic Trajectories
Pentagon
Trajectory
Billiards
Regular polygon
Tangent line
Orbit
If and only if
Angle

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Davis, Diana ; Fuchs, Dmitry ; Tabachnikov, Sergei. / Periodic trajectories in the regular pentagon. In: Moscow Mathematical Journal. 2011 ; Vol. 11, No. 3. pp. 439-461.
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Periodic trajectories in the regular pentagon. / Davis, Diana; Fuchs, Dmitry; Tabachnikov, Sergei.

In: Moscow Mathematical Journal, Vol. 11, No. 3, 05.08.2011, p. 439-461.

Research output: Contribution to journalArticle

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