Finite Euclidean and Non-Euclidean Geometries

TL;DR

This book explores finite Euclidean and non-Euclidean geometries, extending classical concepts to finite fields, and includes new results, applications in finite mechanics, and the role of computational tools in geometric research.

Contribution

It provides a synthetic exposition of finite geometries, presenting new results in quaternionian and projective geometry over arbitrary fields, and discusses applications and computational methods.

Findings

New results in quaternionian geometry

Generalizations of projective geometry over arbitrary fields

Applications of finite geometry in mechanics

Abstract

The purpose of this book is to give an exposition of geometry, from a point of view which complements Klein's Erlangen program. The emphasis is on extending the classical Euclidean geometry to the finite case, but it goes beyond that. After a brief introduction, which gives the main theme, I present the main results, according to a synthetic view of the subject, rather that chronologically. First, I give some variation on the axiomatic treatment of projective geometry, followed by new results on quaternionian geometry, followed by results in geometry over the reals which are generalized over arbitrary fields, then those which depend on properties of finite fields. I then present results in finite mechanics. The role of the computer, which was essential for these inquiries, is briefly surveyed. The methodology to obtain illustrations by drawings is described. I end with a table which enumerates enclosed additional material.