THE SYMMETRY OF QUASI-PERIODIC SYSTEMS

A discussion is given of the symmetry groups of quasiperiodic systems. This is done in a formalism that uses space groups with dimension larger than three. Three main types are distinguished: modulated crystal phases, incommensurate composite structures and quasicrystals. For these the differences and similarities are discussed and the canonical embedding in higher-dimensional space is given, which requires some generalizations of earlier definitions. The equivalence relation between space groups for quasiperiodic systems is different from that for ordinary space groups, because of the presence of a distinct physical space. Apart from higher-dimensional space groups, some quasiperiodic systems have self-similarity properties. Examples are given and the relationship with space-group symmetry is discussed.