 Crystallographic restriction theorem

A rotation preserving a rank-two lattice has order $1,2,3,4$ or $6$. In a lattice basis its matrix is integral, while its trace is $2\cos \theta$. The only integral values in $[-2,2]$ are $-2,-1,0,1,2$, which give precisely those rotation orders. Finiteness of a symmetry group alone would not impose this restriction.

The Crystallographic Restriction Theorem is a concept in the field of crystallography and solid state physics that describes certain symmetries in crystalline materials. It states that the symmetry operations of a crystal, such as rotations, translations, and reflections, impose restrictions on the types of point groups that can be realized in three-dimensional space. More specifically, the theorem states that the only symmetry operations allowed for a crystal lattice in three dimensions must be compatible with the periodicity of the lattice.