Structure of icosahedral quasicrystals within the multiple-cell approach

Abstract

The multiple-cell approach is discussed as an alternative to the higher-dimensional crystallographic description of icosahedral quasicrystals. Four types of quasi-unit cells fill the space without gaps and overlappings. Every cell in the whole tiling is decorated by specific atoms in a particular way and is associated with a triad: type, position, and orientation. The key features of the proposed approach are the subgroup/coset decomposition of icosahedral symmetry groups in accordance with the orbit-stabilizer theorem, a strict mathematical formalization of the substitution rules for all types of quasi-unit cells in the Socolar-Steinhardt tiling, formalization of the recursive inflation/deflation rules, and the eigenvalue-eigenvector analysis of corresponding substitution matrices. The similar approach can be applied to almost all types of quasicrystals.