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arXiv · 2203.04745

Simple Closed Quasigeodesics on Tetrahedra

Joseph O'Rourke·Costin Vilcu
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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.

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Joseph O'Rourke, Costin Vilcu. 2022-03-09. Simple Closed Quasigeodesics on Tetrahedra. https://arxiv.org/abs/2203.04745

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