ACave_300_S99

ACave_300_S99 - Adam Cave Geometry 300 Classifying Models...

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Adam Cave Geometry 300 Classifying Models of Incidence Geometry up to Isomorphism This paper will focus on classifying all the possibilities of four and five point models of incidence geometry up to isomorphism. The idea for this paper arose while working through a problem concerning whether or not two models of incidence geometry each with exactly four points must be isomorphic. The answer, which we will soon find out to be no, posed more complex issues. For example, although not all of these models are isomorphic, there certainly exist some four-point models that are isomorphic. To break these models down and classify them requires proving which various four-point models exist, and which of these are isomorphic. Taking this idea to the next level deals with the same ideas in relation to five-point models. In more general terms, this idea encompasses what it means for two models to be isomorphic. By definition, isomorphisms of models of incidence geometry are “situations” where “there exists a one-to-one correspondence… between the points of the models and a one-to one correspondence… between the lines of the models such that P lies on l if and only if P’ lies on l’ ” (Greenberg 56). In other words, for two models to be isomorphic, everything about points, lines, and incidence that is true for one must also hold true for the other. Using this, we will first prove exactly which four-point models of incidence geometry exist. Next we will show whether two random four-point models are isomorphic or not. Once accomplished, we will repeat the procedure for models with exactly five points. We will conclude with how these classifications can be applied and possibly expanded. **Note: For the purpose of clarity in this paper, the standard notation of a line will be 1
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altered slightly. The notation of a line will include all points that line is incident with. Example: Line AC will still denote a line incident with points A and C, but it will have the added meaning that line AC is incident only with points A and C. Likewise, line PQRS is incident with only points P, Q, R, and S. Classifying Four-Point Models of Incidence Geometry The first step is to discover and classify four-point models of incidence geometry. We begin with M, a model of incidence geometry with exactly four points. As stated before, since M is a model, we know that all the incidence axioms must hold in M. By the law of excluded middle we know that each line in M must have exactly zero, one, two, three, or four points incident with it. Incidence axiom-1 tells us that a line with zero points or one point can not exist in a model. Similarly, there can be no line in M incident with all four points. This would result in all points being collinear, a violation of incidence axiom-3. This leaves us with the possibility of lines having either two or three points incident with them. These possibilities will be broken down into cases: Case 1: All lines have exactly two points incident on them (no three points are
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ACave_300_S99 - Adam Cave Geometry 300 Classifying Models...

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