Crystallographic restriction theorem

The crystallographic restriction theorem states that rotations preserving a periodic lattice can have only orders 1, 2, 3, 4, or 6 in two and three dimensions.

Bravais lattice: A periodic array of points generated by integer combinations of basis vectors. The theorem constrains rotations that map this repeating point array onto itself.
Lattice (group): A discrete subgroup of Euclidean space generated by linearly independent vectors. The theorem concerns rotational symmetries of such discrete subgroups.
Crystal system: One of seven categories classifying three-dimensional crystal lattices by their symmetry and unit-cell geometry. Permitted rotation orders help organize the symmetry classes of periodic crystals.
Auguste Bravais: A French physicist who classified the fourteen three-dimensional lattice types now called Bravais lattices. His lattice framework supplied the periodic structures to which crystallographic restrictions apply.
Lattice automorphism: A bijective transformation of a lattice that preserves its additive structure. Allowed rotations must act as automorphisms of the lattice.
Point group: A group of linear symmetries that leaves at least one point fixed. Crystal point groups collect the rotations and other point-fixing symmetries subject to the restriction.
Space group: A group combining a crystal's point symmetries with translations and, sometimes, glide or screw operations. Three-dimensional space-group classification incorporates the rotations allowed by the theorem.
William Barlow: A British crystallographer who studied geometrical models of crystal structure in the nineteenth century. His work on crystal symmetry contributed to early formulations of the restriction.
Integer matrix: A matrix whose entries are integers. In a lattice basis, a symmetry rotation has integer entries.
Rotation matrix: A matrix representing a rotation while preserving Euclidean lengths and orientation. The theorem identifies which rotation matrices can also preserve a periodic lattice.

Linked from 3 pages

Hexagonal crystal systemRelated: It explains why sixfold lattice symmetry is possible while fivefold symmetry is not.