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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 fourpoint models that are isomorphic.
To
break these models down and classify them requires proving which various fourpoint
models exist, and which of these are isomorphic.
Taking this idea to the next level deals
with the same ideas in relation to fivepoint 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 onetoone correspondence… between the points of the models and
a oneto 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 fourpoint models of incidence
geometry exist.
Next we will show whether two random fourpoint 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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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 FourPoint Models of Incidence Geometry
The first step is to discover and classify fourpoint 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 axiom1 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 axiom3.
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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 Spring '07
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