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 #m/thm/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

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