Symmetry Group - Three Dimensions

Three Dimensions

Up to conjugacy the set of 3D point groups consists of 7 infinite series, and 7 separate ones. In crystallography they are restricted to be compatible with the discrete translation symmetries of a crystal lattice. This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups (27 from the 7 infinite series, and 5 of the 7 others).

The continuous symmetry groups with a fixed point include those of:

• cylindrical symmetry without a symmetry plane perpendicular to the axis, this applies for example often for a bottle
• cylindrical symmetry with a symmetry plane perpendicular to the axis
• spherical symmetry

For objects and scalar fields the cylindrical symmetry implies vertical planes of reflection. However, for vector fields it does not: in cylindrical coordinates with respect to some axis, has cylindrical symmetry with respect to the axis if and only if and have this symmetry, i.e., they do not depend on φ. Additionally there is reflectional symmetry if and only if .

For spherical symmetry there is no such distinction, it implies planes of reflection.

The continuous symmetry groups without a fixed point include those with a screw axis, such as an infinite helix. See also subgroups of the Euclidean group.