Some novel three-dimensional Euclidean crystalline networks derived from two-dimensional hyperbolic tilings
Date
2003
Authors
Hyde, Stephen
Ramsden, Stuart
Journal Title
Journal ISSN
Volume Title
Publisher
Springer
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.
Description
Keywords
Keywords: Geometry; Mathematical models; Three dimensional; Two dimensional; Crystalline networks; Hyperbolic geometry; Hyperbolic planes; Tiling; Crystalline materials
Citation
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Source
European Physical Journal B
Type
Journal article