Simple closed geodesics on regular spherical polyhedra

TL;DR

This paper classifies all simple closed geodesics on regular spherical octahedra and cubes, and estimates their number on tetrahedra, advancing understanding of geodesic structures on spherical polyhedra.

Contribution

It provides a complete classification for octahedra and cubes, and estimates geodesic counts for tetrahedra, offering new insights into spherical polyhedral geometry.

Findings

All simple closed geodesics on regular spherical octahedra identified

All simple closed geodesics on regular spherical cubes identified

Estimated number of simple closed geodesics on regular spherical tetrahedra

Abstract

In this article, we found all simple closed geodesics on regular spherical octahedra and spherical cubes. In addition, we estimate the number of simple closed geodesics on regular spherical tetrahedra.