Three-dimensional Euclidean nets from two-dimensional hyperbolic tilings: kaleidoscopic examples

We present a method for geometric construction of periodic three‐dimensional Euclidean nets by projecting two‐dimensional hyperbolic tilings onto a family of triply periodic minimal surfaces (TPMSs). Our techniques extend the combinatorial tiling theory of Dress, Huson & Delgado‐Friedrichs to en...

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Bibliographic Details
Published in:Acta crystallographica. Section A, Foundations of crystallography Foundations of crystallography, 2009-03, Vol.65 (2), p.81-108
Main Authors: Ramsden, S. J., Robins, V., Hyde, S. T.
Format: Article
Language:English
Subjects:
Crystal structure
Crystallography
Euclidean space
Geometry
kaleidoscopic hyperbolic tilings
three-dimensional Euclidean nets
triply periodic minimal surfaces
two-dimensional hyperbolic tilings
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Summary:We present a method for geometric construction of periodic three‐dimensional Euclidean nets by projecting two‐dimensional hyperbolic tilings onto a family of triply periodic minimal surfaces (TPMSs). Our techniques extend the combinatorial tiling theory of Dress, Huson & Delgado‐Friedrichs to enumerate simple reticulations of these TPMSs. We include a taxonomy of all networks arising from kaleidoscopic hyperbolic tilings with up to two distinct tile types (and their duals, with two distinct vertices), mapped to three related TPMSs, namely Schwarz's primitive (P) and diamond (D) surfaces, and Schoen's gyroid (G).
ISSN:0108-7673
1600-5724
2053-2733
DOI:10.1107/S0108767308040592