Empirical Geometry

Introduction: Empirical Geometry

Our physical understanding of 'space' has changed enormously over the last two centuries and indeed is still under continual review. However, the main philosophical changes affect our understanding of geometry on the largest and smallest scales and under extreme conditions of speed and mass, i.e. questions of relativity and quanta. The basic principles of geometry on the intermediate 'human' scale remain essentially unaltered.

Accordingly all the propositions in the empirical geometry developed here are formulated to apply to real configurations, in the physical sense, not the mathematical sense, of 'real'. Our straight lines do not extend without end, they simply go beyond our local domain; and our points do not become vanishingly small, they simply become indistinguishable from one another when they get too close for our instrumentation to distinguish between them.

The idea of this work is thus simply to present geometrical knowledge in a visually appealing manner. It takes the form of a series of specially chosen diagrams, accompanied by explanatory notes. Geometry is treated here first as the art or craft of precise pattern construction, second as a branch of physics, and only thirdly as a basis for abstract mathematical formalisms.

Further, it is recognised that the interconnections of geometrical concepts can be explored in many different directions, not just the one that happens to have been chosen by a systematiser such as Euclid. Any system exists independently of its many possible axiomatisations.

Some attempt is made to develop geometry in a sequential way, starting from simple concepts and developing towards more complex configurations. However this is not done in a rigorous axiomatic fashion. The axiomatic logical development of mathematical systems is better expounded nowadays through the study of algebraic systems (such as groups, rings and fields). Geometry can be relieved of this pedagogic burden that it has borne for two millennia.