Crystallographic Restriction Theorem - Formulation in Terms of Isometries

Formulation in Terms of Isometries

The crystallographic restriction theorem can be formulated in terms of isometries of Euclidean space. A set of isometries can form a group. By a discrete isometry group we will mean an isometry group that maps every point to a discrete subset of RN, i.e. a set of isolated points. With this terminology, the crystallographic restriction theorem in two and three dimensions can be formulated as follows.

For every discrete isometry group in two- and three-dimensional space which includes translations spanning the whole space, all isometries of finite order are of order 1, 2, 3, 4 or 6.

Note that isometries of order n include, but are not restricted to, n-fold rotations. The theorem also excludes S₈, S₁₂, D_4d, and D_6d (see point groups in three dimensions), even though they have 4- and 6-fold rotational symmetry only.

Note also that rotational symmetry of any order about an axis is compatible with translational symmetry along that axis.

The result in the table above implies that for every discrete isometry group in four- and five-dimensional space which includes translations spanning the whole space, all isometries of finite order are of order 1, 2, 3, 4, 5, 6, 8, 10, or 12.

All isometries of finite order in six- and seven-dimensional space are of order 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 or 30 .