  Historically, there have been many attempts to produce an appropriate
mathematical formalism for modeling the nature of physical space, such as
Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand
plane, Hamilton's quaternions and Gibbs' vector system using the dot and cross
products. We illustrate however, that Clifford's geometric algebra (GA)
provides the most elegant description of physical space. Supporting this
conclusion, we firstly show how geometric algebra subsumes the key elements of
the competing formalisms and secondly how it provides an intuitive
representation of the basic concepts of points, lines, areas and volumes. We
also provide two examples where GA has been found to provide an improved
description of two key physical phenomena, electromagnetism and quantum theory,
without using tensors or complex vector spaces. This paper also provides
pedagogical tutorial-style coverage of the various basic applications of
geometric algebra in physics.
