Hexagonal Inflation Tilings and Planar Monotiles

• Main Authors: Michael Baake, Franz Gähler, Uwe Grimm
• Format: Article
• Language: English
• Published: MDPI AG 2012-10-01
• Series: Symmetry
• Subjects: Euclidean monotiles; aperiodicity; local rules; inflation
• Online Access: http://www.mdpi.com/2073-8994/4/4/581

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.