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Which types of rotational symmetry are forbidden in ordinary crystals?

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quasicrystal Figure 1: Hexagonal lattice of atomic sites.

In ordinary crystals, rotational symmetry is restricted by the requirement of translational periodicity. Because a crystal lattice must allow for a repeating pattern of atoms without leaving gaps or causing overlapping, only specific rotational axes are mathematically compatible with this structure.12

The only rotational symmetries allowed in periodic crystals are twofold, threefold, fourfold, and sixfold (corresponding to rotations of 360° divided by n, where n = 1, 2, 3, 4, or 6).2

Consequently, all other rotational symmetries are forbidden in ordinary crystals, as they cannot coexist with translational periodicity. Specifically, the following axes cannot exist in such materials:
* Fivefold
* Eightfold
* Tenfold
* Twelvefold1

These forbidden symmetries are notable because they have been observed in quasicrystals, which possess long-range order but lack the spatial periodicity found in ordinary crystalline solids.1