Eur. Phys. J. B 31, 273-284 (2003)
https://doi.org/10.1140/epjb/e2003-00032-8

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

Applied Mathematics Department, Research School of Physical Sciences, Australian National University, Canberra, 0200 A.C.T., Australia

Corresponding author: ^a stephen.hyde@anu.edu.au

Received: 14 January 2002
Revised: 12 August 2002
Published online: 4 February 2003

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.