Polyhedra and packings from hyperbolic honeycombs

We derive more than 80 embeddings of 2D hyperbolic honeycombs in Euclidean 3-space, forming 3-periodic infinite polyhedra with cubic symmetry. All embeddings are “minimally frustrated,” formed by removing just enough isometries of the (regular, but unphysical) 2D hyperbolic honeycombs {3, 7}, {3, 8}, {3, 9}, {3, 10}, and {3, 12} to allow embeddings in Euclidean 3-space. Nearly all of these triangulated “simplicial polyhedra” have symmetrically identical vertices, and most are chiral. The most symmetric examples include 10 infinite “deltahedra,” with equilateral triangular faces. We identify the Klein graph among the created structures.