文档介绍：plex Number Theory of Minkowskian
Geometry and the Quaternion Algebras
Timothy J. Shelton-Jones
Abstract
The manner in which numbered, signed axes can be assigned to a co-ordinate system,
and rules for rotation established between them, is examined for 1, 2, 3 and 4
dimensions (irrespective of whether such a system can be said to constitute a number
system). It is established that, for co-ordinate systems in 1, 2 or 3 dimensions systems,
choices made as to the parity (right- or left-handedness) of the axes have no effect on
the properties of any rotational functions be defined on the co-ordinate system.
Whereas, in the case of 4-dimensional configurations of co-ordinate axes, such choices
give rise to 2 distinct sets of mathematical properties. One of these is shown to be
equivalent to the mutative number system discovered by Hamilton, known as
the quaternions. The other is shown also to be a form of quaternion algebra which,
although not an integral domain, mutative. Referred to here mutative
Quaternions, they evince the imaginary operator k, where k2 = 1, and the numbers of the
k-axis do constitute a conventional number system, referred to here as Opposable
numbers. The properties of mutative Quaternions and Opposable numbers are
examined, and are shown to conform to those of Minkowskian geometry. The
implications of this are studied.
Contents
Abstract
Part 1 Introduction
Part 2 One-, Two-, and Three-dimensional co-ordinate geometries and their
possible number systems.
Part 3 Four-dimensional co-ordinate geometries and their number systems;.
Hamilton‘s quaternions.
Part mutative quaternions: their axial system and general properties:
their 4-space.
Part 5 Conclusions: mutative Quaternions in nature and mathematics.
Appendix 1 Conventions and Perspectives.
Appendix 2 Other information.
Part 1. Introduction
A geometry is a freer construct mathematically than a number system. We
demand of a nu