Regular Polyhedral Lattices of Genus 2: 11 Platonic Equivalents?

Dirk Huylebrouck
Proceedings of Bridges 2010: Mathematics, Music, Art, Architecture, Culture (2010)
Pages 199–206


The paper observes Euler's formula for genus 2 regular polyhedral lattices is obeyed by at most 11 cases of the Schläfli symbol {p,q}, where p is the number of edges of each face and q the number of faces meeting at each vertex. At least one example is given for the 'first' 6 cases, but not for their 5 'duals'. The examples are known from various sources, but their present classification suggests they are lookalikes of classical Platonic equivalents. An 'artistic' corollary is the observation that hyperbolic geometry models can be constructed using Zometool.