Why are only onefold, twofold, threefold, fourfold, and sixfold rotations compatible with ordinary crystal lattices?

In ordinary crystals, the restriction to onefold, twofold, threefold, fourfold, and sixfold rotational symmetry is fundamentally dictated by the requirement of translational periodicity.1

A crystal lattice is defined by an orderly, repeating arrangement of atoms that fills space without gaps. For a rotational symmetry to be compatible with this periodic structure, the rotation must allow the lattice points to coincide after displacement. If a structure were to possess other rotational symmetries—such as fivefold symmetry—it would be mathematically impossible to maintain the translational periodicity characteristic of a crystal lattice.21

Specifically, attempting to combine translational symmetry with forbidden rotations leads to a contradiction where the spacing between atoms would have no lower limit, as the combination of symmetry operations would generate new, smaller lattice vectors. Consequently, only the aforementioned rotations allow for the consistent, long-range order and minimum interatomic separation required for a stable crystal lattice. While quasicrystals can exhibit other symmetries, they lack the translational periodicity found in ordinary crystalline solids.1