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10 - Geometry

Published online by Cambridge University Press:  05 January 2013

Chris Doran
Affiliation:
University of Cambridge
Anthony Lasenby
Affiliation:
University of Cambridge
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Summary

In the preceding chapters of this book we have dealt entirely with a single geometric interpretation of the elements of a geometric algebra. But the relationship between algebra and geometry is seldom unique. Geometric problems can be studied using a variety of algebraic techniques, and the same algebraic result can typically be pictured in a variety of different ways. In this chapter, we explore a range of alternative geometric systems, and discover how geometric algebra can be applied to each of them. We will find that there is no unique interpretation forced on the multivectors of a given grade. For example, to date we have viewed bivectors solely as directed plane segments. But in projective geometry a bivector represents a line, and in conformal geometry a bivector can represent a pair of points.

Ideas from geometry have always been a prime motivating factor in the development of mathematics. By the nineteenth century mathematicians were familiar with affine, Euclidean, spherical, hyperbolic, projective and inversive geometries. The unifying framework for studying these geometries was provided by the Kleinian viewpoint. Under this view a geometry consists of a space of points, together with a group of transformations mapping the points onto themselves. Any property of a particular geometry must be invariant under the action of the associated symmetry group.

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Publisher: Cambridge University Press
Print publication year: 2003

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  • Geometry
  • Chris Doran, University of Cambridge, Anthony Lasenby, University of Cambridge
  • Book: Geometric Algebra for Physicists
  • Online publication: 05 January 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9780511807497.012
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  • Geometry
  • Chris Doran, University of Cambridge, Anthony Lasenby, University of Cambridge
  • Book: Geometric Algebra for Physicists
  • Online publication: 05 January 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9780511807497.012
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Geometry
  • Chris Doran, University of Cambridge, Anthony Lasenby, University of Cambridge
  • Book: Geometric Algebra for Physicists
  • Online publication: 05 January 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9780511807497.012
Available formats
×