Arithmetic equivalence of point groups for quasiperiodic structures

Abstract

Necessary and sufficient conditions are formulated for an n-dimensional arithmetic point group such that it may be the symmetry group of a d-dimensional quasiperiodic but not periodic, i.e. incommensurate, structure with Fourier modulus of rank n. Only point groups leaving invariant a d-dimensional subspace (the physical space) are considered. For an arithmetic point group describing an incommensurate structure, all equivalent choices for the internal space are related by the normalizer in Gl (n, \bb Z) of the point group. Also, the conditions on arithmetic equivalence of two point groups allowing an incommensurate structure are discussed. These conditions yield a further partition of the arithmetic crystal classes.