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
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.
PACS: 89.75.Kd – Patterns / 89.75.Hc – Networks and genealogical trees / 82.75.Fq – Synthesis, structure determination, structure modeling
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2003
