Platonic Solids and High Genus Covers of Lattice Surfaces

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.

我们研究由柏拉图立体（Platonic solids）表面展开得到的平移曲面（translation surfaces）。我们证明此类曲面均为格曲面（lattice surfaces），并计算了对应泰希米勒曲线（Teichmüller curves）的拓扑结构。借助可通用计算本原格曲面（primitive lattice surfaces）平移覆盖所对应泰希米勒曲线的算法，我们证实展开后的十二面体对应的泰希米勒曲线亏格为131，带有19个锥奇点（cone singularities）与362个尖点（cusps）。我们分别通过理论推导与严格的计算机辅助证明，表明正四面体、正八面体、立方体与正二十面体对应的平移曲面上不存在闭合鞍连接（saddle connections）。我们证明，在十二面体的平移曲面上恰好存在31个闭合鞍连接等价类，其中等价性由平移覆盖的仿射自同构（affine automorphisms）定义。本文建立的技术方法可推广至柏拉图曲面（Platonic surfaces），乃至更一般的本原格曲面平移覆盖及其欧几里得锥面（Euclidean cone surface）与台球桌商空间（billiard table quotients）场景。