Rules for computing symmetry, density, and stoichiometry in a quasi-unit-cell model of quasicrystals

The quasi-unit-cell picture describes the atomic structure of quasicrystals in terms of a single, repeating cluster that overlaps neighbors according to specific overlap rules. In this paper, we discuss the precise relationship between a general atomic decoration in the quasi-unit-cell picture of atomic decorations in the Penrose tiling and in related tiling pictures. Using these relations, we obtain a simple, practical method for determining the density, stoichiometry, and symmetry of a quasicrystal based on the atomic decoration of the quasi-unit-cell taking proper account of the sharing of atoms between clusters.