Simple Closed Quasigeodesics on Tetrahedra

TL;DR

This paper proves the existence of three specific simple closed quasigeodesics on any tetrahedron, providing explicit constructions and identifying classes with many such quasigeodesics, advancing understanding of polyhedral surface geometry.

Contribution

It explicitly identifies three simple closed quasigeodesics on any tetrahedron based on vertex inclusion, and finds classes with numerous quasigeodesics, extending Pogorelov's existence results.

Findings

At least one quasigeodesic through 1, 2, and 3 vertices on any tetrahedron.

Isosceles tetrahedra have quasigeodesics but lack 1-vertex quasigeodesics.

An infinite class of tetrahedra with at least 34 quasigeodesics each.

Abstract

Pogorelov proved in 1949 that every every convex polyhedron has at least three simple closed quasigeodesics. Whereas a geodesic has exactly pi surface angle to either side at each point, a quasigeodesic has at most pi surface angle to either side at each point. Pogorelov's existence proof did not suggest a way to identify the three quasigeodesics, and it is only recently that a finite algorithm has been proposed. Here we identify three simple closed quasigeodesics on any tetrahedron: at least one through 1 vertex, at least one through 2 vertices, and at least one through 3 vertices. The only exception is that isosceles tetrahedra have simple closed geodesics but do not have a 1-vertex quasigeodesic. We also identify an infinite class of tetrahedra that each have at least 34 simple closed quasigeodesics.