A Trajectory from a Vertex to Itself on the Dodecahedron

Abstract

We prove that there exists a geodesic trajectory on the dodecahedron from a vertex to itself that does not pass through any other vertex.

MSC:

• Primary 51M20

Acknowledgments

We would like to thank Samuel Lelièvre for posing this problem to us. We are very grateful to W. Patrick Hooper for many helpful discussions and for helping us generate Figure 2 using the Sage-FlatSurf [2] package.

J.S.A. gratefully acknowledges support from NSF CAREER grant DMS-1559860 and D.A. from NSF grant DMS-1738381 and PSC-CUNY Award # 60571-00 48.