The tetrahedron is a "self-dual".  The vertices and faces of one can be juxtaposed

with another to form two completely interpenetrating tetrahedra.  This is called a

stellated tetrahedron or stella octangula.  The stella octangula itself can be inscribed in

the cube to demonstrate its symmetry.  Stellating the tetrahedron restores all of the

Fig. 9 - Alternative models of a stella octangula inscribed in a cube

cubic symmetry elements that were lacking in the tetrahedron alone.  Therefore the

stella octangula possesses full cubic symmetry.

Exercise:  Explain why the tetrahedron is missing some cubic

symmetry elements and why they are restored when it is stellated.

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