Closed geodesics on doubled polygons

Adelstein, Ian M.; Fong, Adam Y. W.

doi:10.2140/involve.2019.12.1219

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Closed geodesics on doubled polygons

Ian M. Adelstein and Adam Y. W. Fong

Vol. 12 (2019), No. 7, 1219–1227

DOI: 10.2140/involve.2019.12.1219

Abstract

We study $1 ∕ k$ -geodesics, those closed geodesics that minimize on any subinterval of length $L ∕ k$ , where $L$ is the length of the geodesic. We investigate the existence and behavior of these curves on doubled polygons and show that every doubled regular $n$ -gon admits a $1 ∕ (2 n)$ -geodesic. For the doubled regular $p$ -gons, with $p$ an odd prime, we conjecture that $k = 2 p$ is the minimum value for $k$ such that the space admits a $1 ∕ k$ -geodesic.

Keywords

closed geodesics, regular polygons, billiard paths

Mathematical Subject Classification 2010

Primary: 53C20, 53C22

Milestones

Received: 24 January 2019

Revised: 7 February 2019

Accepted: 18 February 2019

Published: 12 October 2019

Communicated by Frank Morgan

Authors

Ian M. Adelstein
Department of Mathematics Yale University New Haven, CT United States

Adam Y. W. Fong
Department of Mathematics Trinity College Hartford, CT United States

Vol. 12, No. 7, 2019