TY - JOUR
T1 - Three-periodic nets, tilings and surfaces. A short review and new results
AU - Delgado-Friedrichs, Olaf
AU - O’Keeffe, Michael
AU - Proserpio, Davide M.
AU - Treacy, Michael M.J.
N1 - Funding Information:
The authors thank David Palmer for making changes to the CrystalMaker software (https://crystalmaker.com/), which was used to generate the framework figures. We thank Professor Vladislav A. Blatov at the Samara Center for Theoretical Materials Science for providing the free ToposPro software (https://topospro.com/). 3dt generated the tiling graphics and, with Systre, is freely available at http://gavrog.org/. The free software POV-Ray (http://www.povray.org/) generated Figs. 2(c) and 2(d).
Publisher Copyright:
© 2023 International Union of Crystallography. All rights reserved.
PY - 2023/2/13
Y1 - 2023/2/13
N2 - A brief introductory review is provided of the theory of tilings of 3-periodic nets and related periodic surfaces. Tilings have a transitivity [p q r s] indicating the vertex, edge, face and tile transitivity. Proper, natural and minimal-transitivity tilings of nets are described. Essential rings are used for finding the minimal-transitivity tiling for a given net. Tiling theory is used to find all edge- and face-transitive tilings (q = r = 1) and to find seven, one, one and 12 examples of tilings with transitivity [1 1 1 1], [1 1 1 2], [2 1 1 1] and [2 1 1 2], respectively. These are all minimal-transitivity tilings. This work identifies the 3-periodic surfaces defined by the nets of the tiling and its dual and indicates how 3-periodic nets arise from tilings of those surfaces.
AB - A brief introductory review is provided of the theory of tilings of 3-periodic nets and related periodic surfaces. Tilings have a transitivity [p q r s] indicating the vertex, edge, face and tile transitivity. Proper, natural and minimal-transitivity tilings of nets are described. Essential rings are used for finding the minimal-transitivity tiling for a given net. Tiling theory is used to find all edge- and face-transitive tilings (q = r = 1) and to find seven, one, one and 12 examples of tilings with transitivity [1 1 1 1], [1 1 1 2], [2 1 1 1] and [2 1 1 2], respectively. These are all minimal-transitivity tilings. This work identifies the 3-periodic surfaces defined by the nets of the tiling and its dual and indicates how 3-periodic nets arise from tilings of those surfaces.
KW - 3-periodic nets
KW - 3-periodic tilings
KW - essential rings
KW - nets
KW - tilings
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U2 - 10.1107/S2053273323000414
DO - 10.1107/S2053273323000414
M3 - Article
C2 - 36862044
AN - SCOPUS:85149344299
SN - 0108-7673
VL - 79
SP - 192
EP - 202
JO - Acta Crystallographica Section A: Foundations and Advances
JF - Acta Crystallographica Section A: Foundations and Advances
IS - Pt 2
ER -