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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Bibliographic Details
Published inSymmetry (Basel) Vol. 4; no. 4; pp. 581 - 602
Main Authors Baake, Michael, Gähler, Franz, Grimm, Uwe
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.12.2012
Subjects
aperiodicity
Euclidean monotiles
inflation
local rules
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Summary: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.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym4040581