A Fractal Fundamental Domain with 12-fold Symmetry

Square triangle tilings are relevant models for quasicrystals. We introduce a new self-similar tile-substitution which yields the well-known nonperiodic square triangle tilings of Schlottmann. It is shown that the new tilings are locally derivable from Schlottmann’s, but not vice versa, and that they are mutually locally derivable with the undecorated square triangle tilings. Furthermore, the role of the window (acceptance domain) for these tilings as a fundamental domain of the hexagonal lattice is discussed. 1. Square Triangle Substitutions Nonperiodic tilings, like Penrose tilings, are important models for physical quasicrystals. Besides the three 3-dimensional tilings with icosahedral symmetry (see for instance [1] and references therein) the most relevant models for physical quasicrystals are those planar tilings with 5-fold (resp. 10-fold), 8-fold and 12-fold symmetry. The celebrated Penrose tilings show statistical 10-fold symmetry (compare [4]), and there are two particular Penrose tilings showing global 5-fold dihedral symmetry [6]. The most prominent tilings with 8-fold symmetry are certainly the Ammann-Beenker tilings, see Ammann’s P4 in [6]. These two (families of) tilings use two different building blocks (prototiles) only, and both of them can be generated by a tile-substitution. A tile-substitution is given by a set of prototiles T1, . . . , Tm, a substitution factor λ, and a rule how to replace the enlarged prototiles λTi with congruent copies of the prototiles. For an example, see Figure 1; see also [5] for further details, precise definitions and a wealth of examples. A tile-substitution is