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

Simple Closed Quasigeodesics on Tetrahedra