Polyhedra and packings from hyperbolic honeycombs

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.