TY - JOUR
T1 - Polyhedra and packings from hyperbolic honeycombs
JF - Proceedings of the National Academy of Sciences
JO - Proc Natl Acad Sci USA
SP - 6905
LP - 6910
DO - 10.1073/pnas.1720307115
VL - 115
IS - 27
AU - Pedersen, Martin Cramer
AU - Hyde, Stephen T.
Y1 - 2018/07/03
UR - http://www.pnas.org/content/115/27/6905.abstract
N2 - The simplest 2D regular honeycombs are familiar patterns, found in an extraordinary range of natural and designed systems. They include tessellations of the plane by squares, hexagons, and equilateral triangles. Regular triangular honeycombs also form on the sphere; they are the triangular Platonic polyhedra: the tetrahedron, octahedron, and icosahedron. Regular hyperbolic honeycombs adopt an infinite variety of topologies; these must be distorted to be situated in 3D space and are thus frustrated. We construct minimally frustrated realizations of the simplest 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.
ER -