## Symmetry Groups in Three Dimensions

### Brian Hall We consider symmetries of two types, ordinary rotations and "improper" rotations. An improper rotation is either a reflection (mirror image transformation) or a combination of a reflection and a rotation. An ordinary or improper rotation is a symmetry of an object if it maps that object onto itself, for example, a 90-degree rotation of cube.

If an object has any symmetries, the collection of symmetries forms a "group." The possible symmetry groups come in two types, called prismatic and polyhedral. If there is some plane (which we might as well take as the (x,y) plane) that is mapped into itself by every element of the group, the group is called prismatic. The prismatic groups are the symmetry groups of various sorts of prisms and antiprisms. Prismatic groups are essentially two-dimensional, since they consist just of rotations and reflections in the (x,y) plane, possibly combined with mapping z to -z.

If there is no single plane mapped into itself by all elements of the group, the group is called polyhedral. As we shall see, the polyhedral groups are all closely related to the symmetry groups of familiar polyhedra. It can be shown that there are only seven polyhedral groups, which I will now describe. The first three groups are the full symmetry groups of the icosahedron, the cube, and the tetrahedron. Here, "full" symmetry group means that we include both pure rotational symmetries and improper (reflection-type) symmetries. The symmetry group of the icosahedron is the same as the symmetry group of the dodecahedron and has 120 elements. The symmetry group of the cube is the same as the symmetry group of the octahedron and has 48 elements. The symmetry group of the tetrahedron is contained in the symmetry group of the cube and has 24 elements.

The next three polyhedral groups are the groups of pure rotational symmetries of the icosahedron, cube, and tetrahedron, that is, excluding the reflection-type symmetries. These groups have 60, 24, and 12 elements respectively.

The seventh and final polyhedral group is the "pyritohedral" group. A "pyritohedron" is a certain shape assumed by some crystals of the mineral pyrite (fool's gold). [a model of the pyritohedron is coming] The pyritohedral group is then (logically enough) the symmetry group of the pyritohedron. The pyritohedral group can also be viewed as the group of symmetries of a cube that also preserve the pattern of "stripes" on faces illustrated in the model below. Click on the image to launch vZome with this model.

The pyritohedral group contains both rotations and reflections and has 24 elements. The pure rotational part of the pyritohedron group is the same as the pure rotational part of the tetrahedron group. The pyritohedral group can be thought of as the intersection of the full cube group with the full icosahedral group. This amounts to saying that the pyritohedral group consists of those symmetries of a cube that are also symmetries of the Zome system.

It can be shown that these seven groups are the only (finite) polyhedral groups. (More precisely, every finite polyhedral group is the symmetry group of a *suitably oriented* icosahedron, cube, tetrahedron, or pyritohedron, or the pure-rotational part of such a group. Technically, you get different groups by taking different orientations of the icosahedron or whatever, but clearly the resulting groups are "equivalent.")