Socolar found the basis for this tiling already in 1987, but recently added a substitution tiling. An interesting feature is that there exists a context-independent Ammann bar decoration of the tiles, similar to the one in the original Socolar 12-fold but with different relative phases, and hence a fairly simple set of matching rules.
Self Similar Substitution
A substitution tiling with six trapezoidal prototiles. The substitution rule is given for only five of the six tiles. The sixth tile (yellow) is substituted by nothing. The discoverer gives credits to Veit Elser for suggesting the shape of the tiles.
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
A substitution yielding tilings with statistical 6-fold symmetry, with inflation factor 2. It is not known whether this one is a cut and project tiling or not. If it is, it has necessarily a p-adic internal space.
Finite Rotations Polytopal Tiles
A substitution tiling with three prototiles. The substitution rule is given for only two of the three tiles. The third tile (yellow) is substituted by nothing. The discoverer gives credits to Veit Elser for suggesting the shape of the tiles.