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

Jam308309111x4stcubetrans134x120res72gif255.gif (8808 bytes)

demonstrated, the tetrahedron shares the cube's four [111]

rotational axes and (111) planes of 3-fold symmetry.  And,

like the cube, it has six (110) mirror planes.  But it has only

three [110] 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.

tetrahedron.gif (10649 bytes)

cube.gif (11144 bytes)

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


page 6 of lesson

Exercise:   The (110) mirror planes of the cube and tetrahedron have the same orienta-

page 8 of lesson

tion but the [110] axes and (110) planes of rotational symmetry do not. Explain why.


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