Simple Closed Quasigeodesics on Tetrahedra

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.