The Groups of the Tetrahedron, the Octahedron, and the Icosahedron

After discussing the dihedral group, it is time to post my images of how Klein introduces the symmetry groups of the tetrahedron, the octahedron and the icosahedron.

By duality, this also handles the case of the dodecahedron (it is the dual of the icosahedron) and that of the cube (it is the dual of the octahedron) and thus all the Platonic solids are covered.

Before we start out, we should note that Klein first handles only the rotations in the symmetry groups of the Platonic solids. Reflections are handled in a later chapter. Thus, the following illustrations also deal with rotations only.

The Group of the Tetrahedron

Klein discusses the tetrahedron together with its opposite tetrahedron (see red and blue tetrahedron in the image below). Their vertices form a cube which helps to define all the rotation axes of the symmetry group.

First, we get four axes (yellow, period 3) as the space diagonals of the cube. Then, we get three further axes (green, period 2) through the midpoints of opposite faces of the cube. These midpoints form an octahedron and this is illustrated next.

The Group of the Octahedron

The rotation axes we got for the tetrahedron remain valid for the octahedron (green and yellow). However, rotations around these axes have different periods than in the previous case. In addition to these, we get axes (blue) through the midpoints of opposite edges of the tetrahedron.

The Group of the Icosahedron

Finally, for the icosahedron, we get similar types of axes, however, the picture looks a lot more complicated due to the many vertices, edges, and faces. We get rotations around axes through opposite vertices (yellow, period 5), through the midpoints of opposite faces (green, period 3) and through the midpoints of opposite edges (blue, period 2).

Now, with these illustrations, it should be a lot of fun to go back and read Chapter 1 of Klein’s book.