https://doi.org/10.1140/epjb/e2004-00202-2
2D hyperbolic groups induce three-periodic Euclidean reticulations
Department of Applied Mathematics, Research School of
Physical Sciences, The Australian National University, Canberra ACT 0200, Australia
Corresponding author: a vanessa.robins@anu.edu.au
Received:
24
December
2003
Revised:
2
April
2004
Published online:
12
July
2004
Many crystalline networks can be viewed as decorations of triply periodic minimal surfaces. Such surfaces are covered by the hyperbolic plane in the same way that the Euclidean plane covers a cylinder. Thus, a symmetric hyperbolic network can be wrapped onto an appropriate minimal surface to obtain a 3d periodic net. This requires symmetries of the hyperbolic net to match the symmetries of the minimal surface. We describe a systematic algorithm to find all the hyperbolic symmetries that are commensurate with a given minimal surface, and the generation of simple 3d nets from these symmetry groups.
PACS: 61.50.Ah – Theory of crystal structure, crystal symmetry; calculations and modeling / 89.75.Hc – Networks and genealogical trees / 02.20.-a – Group theory / 02.40.-k – Geometry, differential geometry, and topology
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2004
