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

Hexagonal Inflation Tilings and Planar Monotiles

1
, 1
 and 2,*
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)
Download PDF [452 KB, updated 30 November 2012; original version uploaded 22 October 2012]
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 Euclidean monotiles; aperiodicity; local rules; inflation
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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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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