Point Groups in Three Dimensions - Finite Isometry Groups

Finite Isometry Groups

For point groups, being finite corresponds to being discrete; infinite discrete groups as in the case of translational symmetry and glide reflectional symmetry do not apply.

Symmetries in 3D that leave the origin fixed are fully characterized by symmetries on a sphere centered at the origin. For finite 3D point groups, see also spherical symmetry groups.

Up to conjugacy the set of finite 3D point groups consists of:

• 7 infinite series with at most one more-than-2-fold rotation axis; they are the finite symmetry groups on an infinite cylinder, or equivalently, those on a finite cylinder. They are sometimes called the axial or prismatic point groups.
• 7 point groups with multiple 3-or-more-fold rotation axes; they can also be characterized as point groups with multiple 3-fold rotation axes, because all 7 include these axes; with regard to 3-or-more-fold rotation axes the possible combinations are:
  • 4×3
  • 4×3 and 3×4
  • 10×3 and 6×5

A selection of point groups is compatible with discrete translational symmetry: 27 from the 7 infinite series, and 5 of the 7 others, the 32 so-called crystallographic point groups. See also the crystallographic restriction theorem.