Some examples of quasiperiodic tilings obtained with a simple grid method

Abstract

A grid method using tiling by fundamental domain of simple bi-dimensional lattices is presented. It refers to a previous work done by Stampfli in 1986 using two grids by regular hexagons, one rotated by relatively to the other. This leads to a quasiperiodic structure with a twelvefold symmetry made of regular triangles, squares and rhombuses. The tessellation of the plane by the overlap domains of two hexagons, each belonging to one of the two grids is considered. Vertices of the quasiperiodic tiling are the mid-point of the centers of the two overlapping hexagons. Edges of the quasiperiodic tiling are obtained by a Delaunay triangulation of the set of reference points. This method is extended to two other types of quasiperiodic tilings with other fundamental domains. A first example uses two square grids leading to the octagonal Ammann-Beenker quasiperiodic tiling. The second example is also based on the hexagonal lattice, but with grids defined by the lattice fundamental rhombic cell.