Twenty-five point geometry.

Abstract

The development of a mathematical system must follow a rigid set of rules. There is, however, one rule or process that may be included at different times in the development of the system. That is the introduction of a model for the system. In most cases, the system is developed first and then a model is constructed. In this paper a few axioms and theorems are introduced and then the system is expanded after the examination of two isomorphic models. This process is used to examine a geometry of 25 points. When considering a geometry, it is a common process to compare the system to Euclidean geometry. Any discussion of Euclidean geometry leads to a consideration of Fuclids fifth postulate or one of several other statements equivalent to it. The two statements which are discussed in detail in this paper are Playfair's axiom and the Pythagorean Theorem. The thesis then consists of a partial development of a 25 point geometry, considerable discussion in chapters two, three, and five of models for the ~eometry, and a comparison of the 25-point geometry ,"ith Euclidean geometry.