Hexagonal Inflation Tilings and Planar Monotiles
Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is...
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Published in | Symmetry (Basel) Vol. 4; no. 4; pp. 581 - 602 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Basel
MDPI AG
01.12.2012
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Abstract | Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is then called a monotile. One strand of the search for a planar monotile has focused on hexagonal analogues of Wang tiles. This led to two inflation tilings with interesting structural details. Both possess aperiodic local rules that define hulls with a model set structure. We review them in comparison, and clarify their relation with the classic half-hex tiling. In particular, we formulate various known results in a more comparative way, and augment them with some new results on the geometry and the topology of the underlying tiling spaces. |
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AbstractList | Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is then called a monotile. One strand of the search for a planar monotile has focused on hexagonal analogues of Wang tiles. This led to two inflation tilings with interesting structural details. Both possess aperiodic local rules that define hulls with a model set structure. We review them in comparison, and clarify their relation with the classic half-hex tiling. In particular, we formulate various known results in a more comparative way, and augment them with some new results on the geometry and the topology of the underlying tiling spaces. |
Author | Gähler, Franz Grimm, Uwe Baake, Michael |
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Cites_doi | 10.2307/120988 10.1007/978-94-015-8784-6 10.1016/j.cam.2004.05.002 10.1007/s00023-002-8646-1 10.3390/sym4040566 10.1007/s00605-009-0118-y 10.1017/S0143385798100457 10.1016/j.jcta.2011.05.001 10.1017/S0143385706000800 10.1090/ulect/046 10.3390/sym5010001 10.1063/1.1842355 10.1007/s004540010083 10.1088/0305-4470/24/19/025 10.1063/1.3688337 10.1007/978-3-662-04253-3 10.1007/s00454-006-1280-9 10.1007/978-94-007-6431-6_9 10.1017/S0143385704000318 10.1007/978-3-662-05028-6 10.1142/9789814533423 |
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