Cubic Symmetry Types

This Demonstration illustrates cubic symmetry types.

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A polyhedron with cubic symmetry has the same axes of rotational symmetry as a cube (or an octahedron, so the system is labeled ) 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 ). 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 . If all reflection symmetries are destroyed, the system of symmetry has label . 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 . 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 . If all reflection symmetries are destroyed, the system is labeled .
Reference
[1] P. R. Cromwell, Polyhedra, New York: Cambridge Univ. Press, 1997 pp. 308–311.
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Related Curriculum Standards

US Common Core State Standards, Mathematics