The space groups of icosahedral quasicrystals and cubic, orthorhombic, monoclinic, and triclinic crystals

In 1962 Bienenstock and Ewald described a simple and systematic method for computing all the crystallographic space groups in Fourier space. Their approach is reformulated and further simplified, starting from the definition of the point group of a structure as the set of operations that take it into something indistinguishable and not merely identical to within a translation. The reformulation does not require periodicity, making it possible to define and compute on an equal footing the space groups for crystals, quasicrystals, and incommensurately modulated structures, without having to digress into the crystallography of unphysically many dimensions, and using only simple geometry and the most elementary properties of symmetry groups. The general scheme is illustrated by a unified computation of all the icosahedral, cubic, orthorhombic, monoclinic, and triclinic space groups. The remaining (axial) crystallographic and quasicrystallographic space groups have been discussed in a companion paper.