In chapter 3, we found the group of isometries of the real plane and some of its subgroups. In this chapter we will identify three finite groups of isometries of real 3-dimensional space. These will be the groups of rotational symmetries of the regular tetrahedron, the cube and the icosahedron respectively. As regular solids, these may be inscribed in a sphere, and then any symmetry of any one of the solids will leave the centre of the sphere fixed and will transform the surface of the sphere onto itself. As we will prove in chapter 20, rotations are the only direct isometries of 3-dimensional space which fix a point.

It is important to make the models suggested in qns 1, 2 and 3 and to keep the models to hand as you explore their rotational symmetries.

Concurrent reading: Coxeter (1969), p. 151; Steinhaus, pp. 208f, 216f; Hilbert and Cohn-Vossen, pp. 89–93;Martin, chapter 17; Klein, chapter 1.

1. 1 A regular solid is a 3-dimensional polyhedron in which each face is a regular polygon. Any two faces may be matched by an isometry and any two vertices may also be matched by an isometry.

2. If enough of the edges are cut to allow the faces of the polyhedron to lie, connected, in a plane, then the outline so formed is called the net of the solid. By examining the possible nets, determine whether a regular solid with each face an equilateral triangle may possibly be constructed with 2, 3, 4, 5, 6 or more faces meeting at a vertex.

3. […]