Point Groups and Space Groups in Geometric Algebra

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. A geometric figure or a rigid body is said to be “symmetrical” if there exist 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 such isometries, known as the Euclidean group E(3). As is well known, every symmetry S can be given the mathematical form