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, 6 of which were previously unknown and some of which can be described as packings of Platonic deltahedra. We describe also related cubic crystalline packings of equal hyperbolic discs in 3 space that are frustrated analogues of optimally dense hyperbolic disc packings. The 10-coordinated packings are the least “loosened” Euclidean embeddings, although frustration swells all of the hyperbolic disc packings to give less dense arrays than the flat penny-packing even though their unfrustrated analogues in H² are denser.