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 language | English (US) |
|---|---|
| Pages (from-to) | 439-461 |
| Number of pages | 23 |
| Journal | Moscow Mathematical Journal |
| Volume | 11 |
| Issue number | 3 |
| State | Published - Aug 5 2011 |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
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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 journal › Article
TY - JOUR
T1 - Periodic trajectories in the regular pentagon
AU - Davis, Diana
AU - Fuchs, Dmitry
AU - Tabachnikov, Sergei
PY - 2011/8/5
Y1 - 2011/8/5
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=79961012166&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=79961012166&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:79961012166
VL - 11
SP - 439
EP - 461
JO - Moscow Mathematical Journal
JF - Moscow Mathematical Journal
SN - 1609-3321
IS - 3
ER -