Abstract:Notes on vertex atlas of planar Danzer tiling

In 1982 a quasi-crystal with 5-fold rotational symmetry was discovered by Shechtman et al. The most famous 2-dimensional mathematical model for the quasi-crystal may be the Penrose tiling with 5-fold rotational symmetry. In addition, there are the Ammann-Beenker tiling with 8-fold rotational symmetry and the Danzer tiling with 7-fold rotational symmetry(cf.[3]) in typical tilings. Such tilings are called nonperiodic tilings. We prepare several basic definitions. A planar tiling T is a countable family of polygons Ti called tiles: T = {Ti | i = 1, 2, · · · } such that i=1 Ti = R 2 and Int Ti ∩ Int Tj = ∅ if i = j, where R2 denotes the 2-dimensional Euclidean space. A configuration (without gap and overlapping) of tiles around a vertex in a tiling is called vertex atlas. Let S = {S1, S2, · · · , Sl} be a finite set of polygons. When each tile T in a tiling T is congruent to some Si ∈ S, S is called a prototile set of T. A set of matching rules for a prototile set S is a finite set of patches that may appear in the tilings admitted by S. Fix λ(> 1). For a prototile set S, any prototile is decomposed into λ−1 scale-down copies of S. This decomposition is calle