Crystallographic restriction theorem

The crystallographic restriction theorem characterizes the possible orders of rotational symmetry in a lattice. In 2 or 3 dimensions, the rotational symmetries are restricted to 2-fold, 3-fold, 4-fold, and 6-fold.

The theorem's name comes from geological crystals, whose rotational symmetries are generally limited to these same values. However, quasicrystals can occur with other diffraction pattern symmetries, such as 5-fold; these were not discovered until 1982 by Dan Shechtman.

Crystals are modeled as discrete lattices, generated by a list of independent finite translations (Coxeter 1989). Because discreteness requires that the spacings between lattice points have a lower bound, the group of rotational symmetries of the lattice at any point must be a finite group (alternatively, the point is the only system allowing for infinite rotational symmetry). The strength of the theorem is that not all finite groups are compatible with a discrete lattice; in any dimension, we will have only a finite number of compatible groups.