Crystallographic restriction theorem
The crystallographic restriction theorem is a group theoretic and geometric result which places an important restriction on which point groups can form crystallographic groups.
Let
be a crystallographic group on a lattice of or with point group . Then the order of elements in are in group
Trigonometric proof
Consider two lattice points
with separation vector , and suppose that rotation by an angle is a symmetry operation. Then and are also lattice points. It follows that for some . The vectors form a trapezium, therefore the length of is given by thus letting
whence follows
, thus for .
Generalized theorem
See @bambergCrystallographicRestrictionPermutations2003