Closed geodesics on doubled polygons

TL;DR

This paper explores special closed geodesics called 1/k-geodesics on doubled polygons, proving existence for regular n-gons and conjecturing minimal k-values for certain cases, advancing understanding of geodesic behavior on these shapes.

Contribution

It demonstrates the existence of 1/2n-geodesics on doubled regular n-gons and conjectures minimal k-values for doubled regular p-gons with p an odd prime.

Findings

Every doubled regular n-gon admits a 1/2n-geodesic.

Conjecture that for doubled regular p-gons with p odd prime, the minimal k is 2p.

Abstract

In this paper 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 $1/2n$-geodesic. For the doubled regular $p$-gons, with $p$ an odd prime, we conjecture that $k=2p$ is the minimum value for $k$ such that the space admits a $1/k$-geodesic.