Tilings Encyclopedia | MLD Class Shield and Socolar

In connection with physical quasicrystals, the most interesting 2dim tilings are based on 5-, 8-, 10- and 12-fold rotational symmetry. This 12-fold tiling was studied by F. Gähler, in particular its cut and project scheme, the local matching rules and diffraction properties [Gah88]. The window of the vertex set of the shield tiling is a regular dodecagon. The shield tiling is mld to the Socolar tiling and the Wheel tiling, thus they share many interesting properties.

With Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Mld Class Shield And Socolar Matching Rules

Socolar

In connection with physical quasicrystals, the most interesting 2dim tilings are based on 5-, 8-, 10- and 12-fold rotational symmetry. This 12-fold tiling was studied thoroughly in [Soc89], where J. Socolar described the generating substitution as well as the local matching rules and the cut and project scheme, As well as the Penrose Rhomb tilings (5- resp. 10-fold) and the Ammann-Beenker tilings (8-fold), it allows a decoration by Ammann bars (see [GS87]).

Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Parallelogram Tiles Canonical Substitution Tiling Mld Class Shield And Socolar Matching Rules

Wheel Tiling

There is a very simple rule to transform the wheel tiling into the shield tiling: Replace each edge in the tiling by an edge orthogonal to it, of equal length, such that the old and new edge intersect in their midpoints. Applying this rule to the wheel tiling yields the shield tiling and vice versa. This is a very simple example of tilings which are mld.

With Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Mld Class Shield And Socolar