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Zeitschrift für Kristallographie - Crystalline Materials

Editor-in-Chief: Pöttgen, Rainer

Ed. by Antipov, Evgeny / Bismayer, Ulrich / Boldyreva, Elena V. / Huppertz, Hubert / Petrícek, Václav / Tiekink, E. R. T.

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Volume 223, Issue 11-12 (Dec 2008)


Restricted square-triangle tilings

Johannes Roth
Published Online: 2009-09-25 | DOI: https://doi.org/10.1524/zkri.2008.1049


Restricted square-triangle tilings are tilings of regular squares and triangles where the number of vertex types has been limited to a subset of all possible vertex types. Such structures have recently been observed in several places: for example as strip phases in colloidal patterns on laser fields with pentagonal symmetry, and as square-pair-free tilings in patterns formed by dipolar magnetic binary colloidal mixtures. In general there are four vertex configurations: A (4 squares, 44), Z (6 triangles, 36), H (4233), and S (43433). Strip phases belong to type AHZ, square-pair-free phases to SZ, standard inflation generated quasicrystals to HSZ, and unrestricted dodecagonal random tilings to AHSZ. We study several vertex type combinations and describe the properties of the patterns generated. AHZ tilings can be mapped to binary necklaces and classified completely. A similar mapping is possible for many HS tilings. Defect-free SZ tilings can be transformed into a triangle-square-hexagon (TSH) supertiling with matching rules. Monte-Carlo (MC) simulations are used to examine the properties of several classes of restricted tilings.

Keywords: Quasicrystals; Tilings; Crystallographic symmetry; Monte-Carlo simulations

About the article

* Correspondence address: Stuttgart, Deutschland,

Published Online: 2009-09-25

Published in Print: 2008-12-01

Citation Information: Zeitschrift für Kristallographie International journal for structural, physical, and chemical aspects of crystalline materials, ISSN (Print) 0044-2968, DOI: https://doi.org/10.1524/zkri.2008.1049.

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