# Empirical Geometry

## Contents

 Part 1: Geometry in General Points Angles and Rotations Figures Congruence Pythagoras Golden Ratio Part 2: Geometry of Lines Straight Lines Part 3: Geometry of Triangles Triangles General properties Triangle Centres Midcentre or Median Centre or Centroid and Medians Circumcentre and Circumcircle Incentre and Incircle and Angle Bisectors Orthocentre and Altitudes Fermat Centre and Napoleon Triangle Lemoine Centre Ninepoints Centre Centre of Similitude Relating Incircle and Circumcircle Morley Centre and Morley Triangle and Angle Trisectors Brocard Points Part 4: Geometry of Quadrilaterals Quadrilaterals or Tetragons Cyclic Quadrilaterals Part 5: Geometry of Pentagons and higher Polygons Polygons Polygrams Regular Polygons Part 6: Geometry of Curves Curves Ellipses Spirals Part 8: Solid Geometry Spatial Coordinates

After a long delay I have at last got round to converting some of my extensive notes on Geometry into html form that can be displayed here on my website. The delay was mainly due to wishing to try to improve the diagrams, which were drawn using the drawing programme in Lotus WordPro. However I have still not mastered a better geometry drawing program, so have decided to go ahead anyway and hope to improve the diagrams later. More seriously there are also basic problems with geometric concepts that need to be resolved to make them concordant with my empirical or finitist philosophy.

## 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.

## Some Sources Consulted

Euclid, The Elements, The First Six Books, Robson, Levey and Franklin (undated) (based on the Simson edition).

John Sturgeon Mackay, The Elements of Euclid (Books I to VI and parts of XI and XII), W & R Chambers, Edinburgh 1906.

D. Hilbert, The Foundations of Geometry, Translated by E.J.Townsend, Open Court Publishing Co, La Salle, Illinois, reprint 1950 (first published 1902).

H. G. Forder, The Foundations of Euclidean Geometry, Dover Publications, New York, 1958 (reprint of original Cambridge University Press edition 1927).

N. A. Court, College Geometry, Barnes and Noble, 1964 reprint of 1952 second edition (first edition 1925).

H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons Inc, second edition 1969 (first published 1961).

Robert Dixon, Mathographics, Basil Blackwell Limited 1987.