The symmetry of the tetrahedron

The tetrahedron is the least symmetrical of the Platonic solids.   However it does

share some of the cube's symmetry elements.  As shown previously in Fig. 4a it can be

inscribed in the cube to reveal its symmetry in a straightforward manner.  As Fig. 4a demonstrated, the tetrahedron shares the cube's four  rotational axes and (111) planes of 3-fold symmetry.  And, like the cube, it has six (110) mirror planes.  But it has only three  axes and (110) planes of 2-fold rotational symmetry and no (100) planes - thirteen symmetry elements in all.

Likewise, the octahedron can also be inscribed in the tetrahedron to reveal its sym-

metry by association.  In fact a Polymorf model can be built of an octahedron inscribed

in a tetrahedron inscribed in a cube to associate the symmetry of all three polyhedra.  Fig. 8a - Octahedron inscribed Fig. 8b - Octahedron inscribed in in a tetrahedron a tetrahedron inscribed in a cube Required parts 20 triangles 20 large triangles, 12 small squares 24 pinges 24 right triangles, 72 pinges
.

Fig. 8 - Associating the symmetry of the tetrahedron by inscription

. Exercise:   The (110) mirror planes of the cube and tetrahedron have the same orienta- tion but the  axes and (110) planes of rotational symmetry do not. Explain why.

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