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 E³. 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.