Some novel three-dimensional Euclidean crystalline networks derived from two-dimensional hyperbolic tilings

Abstract

We demonstrate the usefulness of two-dimensional hyperbolic geometry as a tool to generate three-dimensional Euclidean (E3) networks. The technique involves projection of edges of tilings of the hyperbolic plane (H2) onto three-periodic, minimal surfaces, embedded in E 3. Given the extraordinary wealth of symmetries commensurate with H2, we can generate networks in E3 that are difficult to construct otherwise. In particular, we form four-, five- and seven-connected (E3) nets containing three- and five-rings, viz. (3, 7), (5, 4) and (5, 5) tilings in H2.