Realizations of lattice quotients of Petrie-Coxeter polyhedra

Keywords: Regular polyhedron, regular map, Petrie-Coxeter polyhedron, polyhedral embedding, polyhedral 2-manifold, automorphism group


The Petrie-Coxeter polyhedra naturally give rise to several infinite families of finite regular maps on closed surfaces embedded into the 3-torus. For the dual pair of Petrie-Coxeter polyhedra {4, 6 | 4} and {6, 4 | 4}, we describe highly-symmetric embeddings of these maps as geometric, combinatorially regular polyhedra (polyhedral 2-manifolds) with convex faces in Euclidean spaces of dimensions 5 and 6. In each case the geometric symmetry group is a subgroup of index 1 or 2 in the combinatorial automorphism group.

Bled 2019 – Polytopes, Configurations, and Symmetries (Grünbaum Issue)