Crystallography and symmetry groups

Abstract

Crystals are assemblages of very small basic units of matter repeated periodically in 3 dimensions. The connection with group theory is that each pattern can be characterized by its symmetry group. It turns out that there are only 230 of these so-called crystallographic space groups amongst which are 22, which crystallographers prefer to regard as distinct, but which, from an abstract point of view, form 11 pairs of isomorphic groups. Thus the space groups fall into 219 isomorphism classes. The enumeration of these space groups is built upon the 14 lattices determined by Bravais. Since the enumeration is quite complicated, we here look at some of the corresponding ideas involved in the analogous 2-dimensional problem where 17 groups, no two of which are isomorphic, arise. First recall that an isometry of the plane R2 is a distance- preserving mapping of R onto itself. Amongst such isometries are translations, rotations, reflections (in lines) and glide reflections. The latter being the result of an ordinary reflection in some line 1 followed by a translation parallel to 1. Figure 1 adequately describes these movements.