Crystallographic group

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 𝑆 be a crystallographic group on a lattice 𝐿 of ℝ2 or ℝ3 with point group 𝑆/𝐿. Then the order of elements in 𝑆/𝐿 are in {1,2,3,4,6} group

Trigonometric proof

Consider two lattice points 𝐴,𝐵 with separation vector 𝐴𝐵, and suppose that rotation by an angle 𝛼 is a symmetry operation. Then 𝐵′ =𝐴 +𝑅𝛼𝐴𝐵 and 𝐴′ =𝐵 +𝑅−𝛼𝐵𝐴 are also lattice points. It follows that 𝐴′𝐵′ =𝑚𝐴𝐵 for some 𝑚 ∈ℤ. The vectors 𝐴𝐵,𝑅𝛼𝐴𝐵,𝑅−𝛼𝐵𝐴,𝐴′𝐵′ form a trapezium, therefore the length of 𝐴′𝐵′ is given by

|𝐴′𝐵′|=|𝐴𝐵|(2cos⁡𝛼−1)

thus letting 𝑀 =𝑚 +1

cos⁡𝛼=𝑀2

whence follows 𝑀 ∈{ −2, −1,0,1,2}, thus 𝛼 =2𝜋𝑛 for 𝑛 ∈{1,2,3,4,6}.

Generalized theorem

See @bambergCrystallographicRestrictionPermutations2003

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