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 Teichm\"uller curves. Using an algorithm that can be used generally to compute Teichm\"uller curves of translation covers of primitive lattice surfaces, we show that the Teichm\"uller 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.