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.

摘要

我们研究通过考虑柏拉图固体表面的展开获得的平移表面。我们证明它们都是晶格表面，并且我们计算了相关 Teichmüller 曲线的拓扑结构。使用通常可用于计算原始晶格表面平移覆盖的 Teichmüller 曲线的算法，我们表明展开的十二面体的 Teichmüller 曲线具有 131 个属，具有 19 个锥奇异点和 362 个尖点。我们提供了理论和严格的计算机辅助证明，证明在与四面体、八面体、立方体和二十面体相关的表面上没有封闭的鞍形连接。我们证明了在十二面体上恰好有 31 个封闭鞍连接的等价类，其中等价被定义为平移覆盖的仿射自同构。