Symmetry Groups

Symmetry is a fundamental organizational concept in art as well as science. To develop and exploit this concept to its fullest, it must be given a precise mathematical formulation. This has been a primary motivation for developing the branch of mathematics known as “group theory.” There are many kinds of symmetry, but the symmetries of rigid bodies are the most important and useful, because they are the most ubiquitous as well as the most obvious. Moreover, they provide an excellent model for the investigation of other symmetries. We have already developed the mathematical apparatus needed to describe and classify all possible rigid body symmetries. The aim of this section is to show how such a description and classification can be carried out efficiently with geometric algebra. The results have extensive applications in the theory of molecular and crystalline structure. We say that a geometrical figure or a rigid body is “symmetrical” if there exists isometries which permute its parts while leaving the object as a whole unchanged. An isometry of this kind is called a symmetry. The symmetries of a given object form a group called the symmetry group of the object. Obviously, every symmetry group is a subgroup of the group of all isometries, the improper Euclidean group. We know, therefore, from Sec. 5.4 that any symmetry S of a rigid body can be given the mathematical form