  This paper challenges some of the common assumptions underlying the
mathematics used to describe the physical world. We start by reviewing many of
the assumptions underlying the concepts of real, physical, rigid bodies and the
translational and rotational properties of such rigid bodies. Nearly all
elementary and advanced texts make physical assumptions that are subtly
different from ours, and as a result we develop a mathematical description that
is subtly different from the standard mathematical structure. Using the
homogeneity and isotropy of space, we investigate the translational and
rotational features of rigid bodies in two and three dimensions. We find that
the concept of rigid bodies and the concept of the homogeneity of space are
intrinsically linked. The geometric study of rotations of rigid objects leads
to a geometric product relationship for lines and vectors. By requiring this
product to be both associative and to satisfy Pythagoras' theorem, we obtain a
choice of Clifford algebras. We extend our arguments from space to include
time. By assuming that ct=l and rewriting this in Lorentz invariant form as
c^2t^2-x^2-y^2-z^2=0 we obtain a generalization of Pythagoras to spacetime.
This leads us directly to establishing that the Clifford algebra CL(1,3) is an
appropriate mathematical structure to describe spacetime. Clifford algebras are
not division algebras. We show that the existence of non-invertible elements in
the algebra is not a limitation of the usefulness to physics of the algebra but
rather that it reflects accurately the spacetime properties of physical
systems.
