Polymorf - Knowhere - Tetrahedron symmetry

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 [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.


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

Page 7 Geometry rules! - Symmetry association by inscription