Mathematics Colloquia and Seminars

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A geodesic on a polyhedral surface is a locally shortest curve which does not pass through the vertices of the polyhedron. It can easily be shown using the Gauss-Bonnet Theorem that on a generic polyhedron there are no closed non self-intersecting geodesics (there are no closed self-intersecting geodesics either, but this is more difficult to show). However, on regular polyhedron there are lots of closed geodesics. The case of a regular tetrahedron is very simple: all closed geodesics are non-self-intersecting, and they can be arbitrarily long. On a cube, however, closed geodesics are usually self-intersecting; there are only 3 possible lengths of non-self- intersecting geodesics on a cube with side length 1. If time permits, I will discuss the classification of closed geodesics on a cube and a regular octahedron.