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Hexagonal Inflation Tilings and Planar Monotiles
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, Bielefeld 33501, Germany
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK
* Author to whom correspondence should be addressed.
Received: 2 September 2012; in revised form: 8 October 2012 / Accepted: 14 October 2012 / Published: 22 October 2012
(This article belongs to the Special Issue Polyhedra
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.
Keywords: Euclidean monotiles; aperiodicity; local rules; inflation
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Baake, M.; Gähler, F.; Grimm, U. Hexagonal Inflation Tilings and Planar Monotiles. Symmetry 2012, 4, 581-602.
Baake M, Gähler F, Grimm U. Hexagonal Inflation Tilings and Planar Monotiles. Symmetry. 2012; 4(4):581-602.
Baake, Michael; Gähler, Franz; Grimm, Uwe. 2012. "Hexagonal Inflation Tilings and Planar Monotiles." Symmetry 4, no. 4: 581-602.