The four dimensional spacetime continuum, as originally conceived by Minkowski, has become the default framework for describing physical laws. Due to its fundamental importance, there have been various attempts to find the origin of this structure fr
om more elementary principles. In this paper, we show how the Minkowski spacetime structure arises naturally from the geometrical properties of three dimensional space when modelled by Clifford geometric algebra of three dimensions $ Cell(Re^3) $. We find that a time-like dimension along with the three spatial dimensions, arise naturally, as well as four additional degrees of freedom that we identify with spin. Within this expanded eight-dimensional arena of spacetime, we find a generalisation of the invariant interval and the Lorentz transformations, with standard results returned as special cases. The value of this geometric approach is shown by the emergence of a fixed speed for light, the laws of special relativity and the form of Maxwells equations, without recourse to any physical arguments.
As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of ele
mentary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions. Complex numbers, quaternions and Cartesian vectors can be described by the various components within a Clifford multivector and from our results we are able to demonstrate new inter-relationships between these algebraic systems. One key relationship that we discover is that a complex number raised to a vector power produces a quaternion thus combining these systems within a single equation. We also find a single formula that produces the square root, amplitude and inverse of a multivector over one, two and three dimensions. Finally, comparing the functions over different dimension we observe that $ Cell left (Re^3 right) $ provides a particularly versatile algebraic framework.
