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.
- 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.
- 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)
- Abstract
: Geometric algebra is used in an essential way to provide a
coordinate-free approach to Euclidean geometry and rigid body mechanics
that fully integrates rotational and translational dynamics. Euclidean
points are given a homogeneous representation that avoids designating
one of them as an origin of coordinates and enables direct computation
of geometric relations. Finite displacements
of rigid bodies are associated naturally with screw displacements
generated by
bivectors and represented by twistors that combine multiplicatively.
Classical
screw theory is incorporated in an invariant formulation that is less
ambiguous,
easier to interpret geometrically, and manifestly more efficient in
symbolic computation. The potential energy of an arbitrary elastic
coupling is given an invariant
form that promises significant simplifications in practical
applications.
D. Hestenes & E. Fasse. 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.