Open AccessThree‐dimensional Euclidean nets from two‐dimensional hyperbolic tilings: kaleidoscopic examples
Author(s) -
Ramsden S. J.,
Robins V.,
Hyde S. T.
Publication year - 2009
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
eISSN - 1600-5724
pISSN - 0108-7673
DOI - 10.1107/s0108767308040592
Subject(s) - dual polyhedron , euclidean geometry , hyperbolic geometry , combinatorics , mathematics , hyperbolic triangle , substitution tiling , simple (philosophy) , gyroid , hexagonal tiling , pure mathematics , geometry , physics , algebraic geometry , philosophy , epistemology , nuclear magnetic resonance , copolymer , grid , polymer
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 ).