TY - JOUR
T1 - Three-dimensional Euclidean nets from two-dimensional hyperbolic tilings
T2 - Kaleidoscopic examples
AU - Ramsden, S. J.
AU - Robins, V.
AU - Hyde, S. T.
PY - 2009
Y1 - 2009
N2 - 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).
AB - 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).
KW - Kaleidoscopic hyperbolic tilings
KW - Three-dimensional Euclidean nets
KW - Triply periodic minimal surfaces
KW - Two-dimensional hyperbolic tilings
UR - http://www.scopus.com/inward/record.url?scp=60749132004&partnerID=8YFLogxK
U2 - 10.1107/S0108767308040592
DO - 10.1107/S0108767308040592
M3 - Article
SN - 0108-7673
VL - 65
SP - 81
EP - 108
JO - Acta Crystallographica Section A: Foundations of Crystallography
JF - Acta Crystallographica Section A: Foundations of Crystallography
IS - 2
ER -