Simple octagonal and dodecagonal quasicrystals.

Penrose tilings have become the canonical model for quasicrystal structure, primarily because of their simplicity in comparison with other decagonally symmetric quasiperiodic tilings of the plane. Four remarkable properties of the Penrose tilings have been exploited in the analysis of the physical issues: (1) the class of Penrose tilings is invariant under deflation (a type of self-similarity transformation); (2) the class contains all tilings consistent with a set of matching rules governing the orientations of neighboring tiles; (3) a certain decoration of the tiles produces grids of quasiperiodically spaced Ammann lines; and (4) the tile vertices can be obtained by projection of a subset of hypercubic lattice points. Each of the first three properties can be explicitly displayed by means of a simple decoration of the tiles, a decoration in which all the marked tiles of a given shape are related by the operations in the orientational symmetry group of the tiling.