Periodic trajectories in the regular pentagon

Diana Davis*, Dmitry Fuchs, Serge Tabachnikov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


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
Issue number3
StatePublished - 2011
Externally publishedYes


  • Closed geodesies
  • Periodic billiard trajectories
  • Regular dodecahedron
  • Regular pentagon
  • Veech alternative

ASJC Scopus subject areas

  • General Mathematics


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