Mathematics Colloquia and Seminars

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We consider convex polyhedra in space. A geodesic is a locally shortest curve not passing through the vertices. On an arbitrary polyhedron there are no closed non-self-intersecting geodesics (This follows from Gauss-Bonnet). We describe closed geodesics on regular polyhedra; for regular tetrahedra and octahedra, we have a complete classification. For cubes, our result is almost complete, and for icosahedra, there are some partial results. Very little is known for the case of dodecahedra.