Symmetry 2012, 4(4), 581-602; doi:10.3390/sym4040581

Hexagonal Inflation Tilings and Planar Monotiles

1 Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, Bielefeld 33501, Germany 2 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)
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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.
Keywords: Euclidean monotiles; aperiodicity; local rules; inflation

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MDPI and ACS Style

Baake, M.; Gähler, F.; Grimm, U. Hexagonal Inflation Tilings and Planar Monotiles. Symmetry 2012, 4, 581-602.

AMA Style

Baake M, Gähler F, Grimm U. Hexagonal Inflation Tilings and Planar Monotiles. Symmetry. 2012; 4(4):581-602.

Chicago/Turabian Style

Baake, Michael; Gähler, Franz; Grimm, Uwe. 2012. "Hexagonal Inflation Tilings and Planar Monotiles." Symmetry 4, no. 4: 581-602.

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