Cubic Symmetry Types

A polyhedron with cubic symmetry has the same axes of rotational symmetry as a cube (or an octahedron, so the system is labeled O) or an axis of four-fold symmetry is reduced to a two-fold axis and the result is a tetrahedral system of rotational symmetry (labeled T). If the polyhedron is placed so that an axis of four-fold symmetry points vertically and there is a horizontal reflection plane, the system of symmetry has label Oh. If all reflection symmetries are destroyed, the system of symmetry has label O. If a polyhedron is decorated so that four-fold axes are changed to two-fold axes and there is a horizontal reflection plane, the system is labeled Th. If there is no horizontal reflection plane (and as a consequence, there are no vertical reflection planes), but there are reflection planes that contain axes of three-fold symmetry, the system of symmetry is labeled Td. If all reflection symmetries are destroyed, the system is labeled T.

Dynamic illustrations for cubic symmetry types: P. R. Cromwell, Polyhedra, New York: Cambridge Univ. Press, 1997 pp. 309–311.