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.

Point Groups and Space Groups in Geometric Algebra

Abstract
: Geometric algebra provides the essential foundation for a new approach to symmetry groups. Each of the 32 lattice point groups and 230 space groups in three dimensions is generated from a set of three symmetry vectors. This greatly facilitates representation, analysis and application of the groups to molecular modeling and crystallography.

D. Hestenes. In: L. Doerst, C. Doran & J. Lasenby (Eds), Applications of Geometric Algebra with Applications in Computer Science and Engineering, © Birkhauser, Boston (2002). p. 3-34.

The Crystallographic Space Groups in Geometric Algebra.

Abstract
: We present a complete formulation of the 2D and 3D crystallographic space groups in the conformal geometric algebra of Euclidean space. This enables a simple new representation of translational and orthogonal symmetries in a multiplicative group of versors. The generators of each group are constructed directly from a basis of lattice vectors that define its crystal class. A new system of space group symbols enables one to unambiguously write down all generators of a given space group directly from its symbol.

D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages)