Geometric Algebra: A natural representation of three-spaceJames M. Chappell, Azhar Iqbal, Derek Abbott

(19 Jan 2011)

Abstract: Historically, there have been many attempts to define the correct algebra for modeling the properties of three dimensional physical space, such as Descartes' system of Cartesian coordinates in 1637, the quaternions of Hamilton representing rotation in three-space that built on the Argand diagram for two-space, and Gibbs' vector calculus employing the dot and cross products. We illustrate however, that Clifford's geometric algebra developed in 1873, but largely overlooked by the science community, provides the simplest and most natural algebra for three-space and hence has general applicability to all fields of science and engineering. To support this thesis, we firstly show how geometric algebra naturally produces all the properties of complex numbers and quaternions and the vector cross product in a single formalism, whilst still maintaining a strictly real field and secondly we show in two specific cases how it simplifies analysis in regards to electromagnetism and Dirac's equation of quantum mechanics. This approach thus has the immediate advantage of removing complex fields from analysis, so that algebraic entities have a geometric expression in real space. As an example, we show how quadratic equations now can be given two real solutions using GA. This viewpoint also has something interesting to say about the concept of number itself, because numbers in GA can encapsulate scalars, lines, areas and volumes, into a single entity, which can be multiplied and divided, just like normal numbers. This simple and elegant formalism would also seem very appropriate for students learning vectors, algebra and geometry, giving a more natural and intuitive understanding of the properties of three-space.

CiteULike: Geometric Algebra: A natural representation of three-space