Book contents
- Frontmatter
- Contents
- List of figures
- Preface
- 1 Euclidean geometry
- 2 Composing maps
- 3 Spherical and hyperbolic non-Euclidean geometry
- 4 Affine geometry
- 5 Projective geometry
- 6 Geometry and group theory
- 7 Topology
- 8 Quaternions, rotations and the geometry of transformation groups
- 9 Concluding remarks
- Appendix A Metrics
- Appendix B Linear algebra
- References
- Index
6 - Geometry and group theory
Published online by Cambridge University Press: 11 November 2010
- Frontmatter
- Contents
- List of figures
- Preface
- 1 Euclidean geometry
- 2 Composing maps
- 3 Spherical and hyperbolic non-Euclidean geometry
- 4 Affine geometry
- 5 Projective geometry
- 6 Geometry and group theory
- 7 Topology
- 8 Quaternions, rotations and the geometry of transformation groups
- 9 Concluding remarks
- Appendix A Metrics
- Appendix B Linear algebra
- References
- Index
Summary
The substance of this chapter can be expressed as the slogan
Group theory is geometry and geometry is group theory.
In other words, every group is a transformation group: the only purpose of being a group is to act on a space. Conversely, geometry can be discussed in terms of transformation groups. Given a space X and a group G made up of transformations of X, the geometric notions are quantities measured on X which are invariant under the action of G. This chapter formalises the relation between geometry and groups, and discusses some geometric issues for which group theory is a particularly appropriate language.
The action of a transformation group on a space is another way of saying symmetry. To say that an object has symmetry means that it is taken into itself by a group action: rotational symmetry means symmetry under the group of rotations about an axis. As a frivolous example, Coventry market pictured in Figure 6.0 has (approximate) rotational symmetry: if you stand at the centre, all directions outwards are virtually indistinguishable; you can understand a coordinate frame as a signpost to break the symmetry, and to enable people to find their way around.
Each of the geometries studied in previous chapters had transformations associated with it: Euclidean motions of E2, orthogonal transformations as motions of S2, Lorentz transformations as motions of H2, and affine and projective linear transformations of An and ℙn.
- Type
- Chapter
- Information
- Geometry and Topology , pp. 92 - 106Publisher: Cambridge University PressPrint publication year: 2005