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

S. T. Hyde; S. Ramsden

doi:10.1140/epjb/e2003-00032-8

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

S. T. Hyde^*
, S. Ramsden

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

18 Citations (SciVal)

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.

Original language	English
Pages (from-to)	273-284
Number of pages	12
Journal	European Physical Journal B
Volume	31
Issue number	2
DOIs	https://doi.org/10.1140/epjb/e2003-00032-8
Publication status	Published - 2003

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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.",

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AB - 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.

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

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