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Which rotational symmetries can occur in ordinary periodic crystals?
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quasicrystal
Figure 1: Hexagonal lattice of atomic sites.
In ordinary periodic crystals, the only rotational symmetries that are compatible with translational periodicity are those that move the crystal through an angle of 360° divided by n, where n is equal to 1, 2, 3, 4, or 6.12
Specifically, the allowed rotational symmetry axes in periodic crystals are:
* Twofold
* Threefold
* Fourfold
* Sixfold12
Other rotational symmetries, such as fivefold, eightfold, tenfold, and twelvefold axes, are forbidden in ordinary crystals because they cannot coexist with the translational symmetry required for a periodic lattice. While these forbidden symmetries have been observed in quasicrystals, those materials lack the spatial periodicity characteristic of ordinary crystalline solids.1