  In considering the nature of the basic mathematical structures appropriate
for describing the fundamental elements of particle physics a significant role
for the octonions, as an extension from the complex numbers and uniquely the
largest division algebra, has occasionally been proposed. Rather than being
based initially upon the more abstract grounds of mathematical aesthetics, here
we describe a unified theory motivated conceptually through an elementary
generalisation of the expression for a local proper time interval, beyond that
of 4-dimensional spacetime and also beyond that of the local structure of
models with extra spatial dimensions, for which the explicit mathematical
development naturally incorporates the octonion algebra as an essential
feature. Properties of matter are identified directly through the symmetry
breaking structure entailed in the necessary extraction of the external
4-dimensional spacetime background and are shown to reproduce a series of
characteristic structures of the Standard Model of particle physics. While
already employing octonion-based constructions of the exceptional Lie groups E6
and E7 the uncovering of the full Standard Model, as well as new physics
beyond, is predicted to involve an E8-related structure for which the octonion
algebra is again anticipated to be of fundamental importance.
