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.

In this paper, analogues of the Penrose tilings are presented for the cases of octagonal and dodecagonal symmetry, the only other cases in two dimensions for which such analogues exist. The octagonal tiling is composed of two types of decorated tiles: a square and a 45° rhombus. The dodecagonal tiling is composed of three tile types: a regular hexagon, a square, and a 30° rhombus. Deflation procedures, matching rules, and Ammann-line decorations are explicitly displayed, secondary Ammann lines are defined and their significance with regard to the long-range order of the tilings is elucidated. The quasiperiodic sequences specifying the positions of the Ammann lines are derived and the appropriate projections from hypercubic lattices are described. Both the sufficiency of the matching rules and the equivalence of tilings produced by deflation and projection are demonstrated. The tilings are then used as a basis for a treatment of the elasticity of octagonal and dodecagonal quasicrystals. The irreducible phason strains are derived and the signatures of the different types of phason strain in real space and reciprocal space are determined. Standard analysis of a harmonic free energy reveals that there are three phason elastic constants for octagonal and dodecagonal quasicrystals (as opposed to two for decagonal ones) and there is no coupling of phason strain to conventional strain in the dodecagonal case. In each case, one linear combination of the phason elastic constants is irrelevant for phason fluctuations of wavelengths smaller than the sample size. Finally, some remarks are made concerning the applicability of standard elasticity theory to quasicrystals that are well described as simple decorations of tilings.