Hexagonal inflation tilings and planar monotiles
arXiv:1210.3967 · doi:10.3390/sym4040581
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 focussed 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.
19 pages,lots of colour figures
References in corpus (11)
- Pure Point Dynamical and Diffraction Spectra
- Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra
- Characterizations of model sets by dynamical systems
- An aperiodic hexagonal tile
- Spectral and topological properties of a family of generalised Thue-Morse sequences
- Computing modular coincidences
- Examples of substitution systems and their factors
- A note on the dynamical zeta function of general toral endomorphisms
- SCD Patterns Have Singular Diffraction
- On the notions of symmetry and aperiodicity for Delone sets
- Taylor-Socolar hexagonal tilings as model sets