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 $ℝ 2$ or $ℝ 3$ with point group $𝑆 / 𝐿$ . Then the order of elements in $𝑆 / 𝐿$ are in ${1, 2, 3, 4, 6}$ #m/thm/group

Trigonometric proof

Consider two lattice points $𝐴, 𝐵$ with separation vector $𝐴 𝐵$ , and suppose that rotation by an angle $𝛼$ is a symmetry operation. Then $𝐵' = 𝐴 + 𝑅 𝛼 𝐴 𝐵$ and $𝐴' = 𝐵 + 𝑅 - 𝛼 𝐵 𝐴$ are also lattice points. It follows that $𝐴' 𝐵' = 𝑚 𝐴 𝐵$ for some $𝑚 \in ℤ$ . The vectors $𝐴 𝐵, 𝑅 𝛼 𝐴 𝐵, 𝑅 - 𝛼 𝐵 𝐴, 𝐴' 𝐵'$ form a trapezium, therefore the length of $𝐴' 𝐵'$ is given by

| 𝐴' 𝐵' | = | 𝐴 𝐵 | (2 c o s 𝛼 - 1)

thus letting $𝑀 = 𝑚 + 1$

c o s 𝛼 = 𝑀 2

whence follows $𝑀 \in {- 2, - 1, 0, 1, 2}$ , thus $𝛼 = 2 𝜋 𝑛$ for $𝑛 \in {1, 2, 3, 4, 6}$ .

Generalized theorem

See @bambergCrystallographicRestrictionPermutations2003

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