Why Don't Crystals Have 5 Fold Symmetry?

Crystals do appear to have 5-fold symmetry but these symmetries are not possible. Crystals can only exist in 2, 3, 4 or 6-fold rotational axis. The reason why this is not possible is because the external shape of the crsytal is dependent on the geometric arrangements of atoms. Therefore, crystals cannot have 5, 7, 8, and other higher-fold rotational axes. Crystals do appear to have 5-fold symmetry but these symmetries are not possible. Crystals can only exist in a 2, 3, 4 or 6-fold rotational axis. The reason why this is not possible is that the external shape of the crystal is dependent on the geometric arrangements of atoms. Therefore, crystals cannot have 5, 7, 8, and other higher-fold rotational axes.

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