The space groups of axial crystals and quasicrystals

Abstract

We compute all the three-dimensional quasicrystallographic space groups with n-fold axial point groups and standard lattices by a method that treats crystals and quasicrystals on an equal footing. We do not rely on projecting higher-dimensional crystallographic space groups, our results are valid for arbitrary n, and our analysis is elementary. We regard space groups as a scheme for classifying diffraction patterns to be carried out in three-dimensional reciprocal space. The familiar three-dimensional crystallographic space groups with axial point groups emerge simply and directly as special cases of the general n-fold three-dimensional quasicrystallophic treatment with n=3, 4, and 6