Platonic Solids and High Genus Covers of Lattice Surfaces

Abstract

We study the translation surfaces obtained by considering the unfoldings of the surfaces of Platonic solids. We show that they are all lattice surfaces and we compute the topology of the associated Teichmüller curves. Using an algorithm that can be used generally to compute Teichmüller curves of translation covers of primitive lattice surfaces, we show that the Teichmüller curve of the unfolded dodecahedron has genus 131 with 19 cone singularities and 362 cusps. We provide both theoretical and rigorous computer-assisted proofs that there are no closed saddle connections on the surfaces associated to the tetrahedron, octahedron, cube, and icosahedron. We show that there are exactly 31 equivalence classes of closed saddle connections on the dodecahedron, where equivalence is defined up to affine automorphisms of the translation cover. Techniques established here apply more generally to Platonic surfaces and even more generally to translation covers of primitive lattice surfaces and their Euclidean cone surface and billiard table quotients.

Keywords:

• Translation surfaces
• Platonic solids
• Platonic surfaces
• saddle connection
• Teichmüller curve

Acknowledgments

We would like to thank Joshua Bowman, Diana Davis, Myriam Finster, Dmitri Fuchs, Samuel Lelièvre, Anja Randecker, and Gabriela Weitze-Schmithüsen for helpful discussions. We would like to thank the anonymous referee for their careful reading of the paper and clarifying remarks, and for pointing us to the references [17, 32].

Notes

1 As observed in the introduction, this observation was stated in the final paragraph of [12] using the terminology of tilings of the plane.

2 Since there are infinitely many choices for a fundamental domain of a discrete group, Sage can change its choice with each computation. This will lead to different points in the cusp list from those listed in Column 3. The cusps listed here represent one possible choice.

3 This answer and the list of cosets were independently checked by code produced by Myriam Finster through personal communication.

Additional information

Funding

J.S.A. was partially supported by NSF CAREER grant DMS-1559860. D.A. was partially supported by NSF DMS-1738381, DMS-1600360 and PSC-CUNY grants 60571-00 48 and 61639-00 49. W.P.H. was partially supported by NSF DMS-1500965 and PSC-CUNY grant 60708-00 48.