### Aaron

```Finite Projective
Planes
By Aaron Walker Wagner
Geometry
Finite Geometry
Projective Planes
Finite Projective Planes
Definitions
 Projective Plane
 It is a geometric structure
 It contains a set of lines (not necessarily straight), a set of
points, and a relation between the lines and points called
incidence

Incident – When two objects touch each other.
Projective Plane Cont.
 It contains the following properties
1.
Given any two distinct points, there is exactly one line incident
with both of them.
2.
Give any two distinct lines, there is exactly one point incident
with them both.
3.
There are four points such that no line is incident with more
than two of them.
In Euclidean Geometry two Parallel
lines will never meet, but in a projective
plan parallel lines will intersect.
Therefore, any two lines in a
projective plane intersect at one
and only one point.
Think of it as these railroad
tracks that look like they will
intersect as they go farther away
There are many projective
planes both finite and infinite.
Today we are only going to
focus on the finite projective
planes.
Finite Projective Plane
 A finite projective plane of order N is defined as a set of
N2 + N + 1 points, where N ≥ 2, with the properties that:
 Any two points determine a line,
 Any two lines determine a point,
 Every point has N + 1 lines on it, and
 Every line contains N + 1 points.
The case of N = 2
 So N2 + N + 1
 7 points and 7 lines
 As you can see one line is not straight, which is okay in
finite projective planes.
 All results isomorphic with PG(2,2)
 At every line there are 3 points
incident with it.
 At every point there are 3 lines
incident with it.
N=3
N=3
 As you can see there are 13 points, 13 lines.
 Every point has 4 lines incident with it.
 Every line has 4 points incident with it.
 So when the order is 3, all results are isomorphic with
PG(2,3)
 As you can see this gets very complicated very fast.
Others
 Some people say a finite projective plane exists when the
order is a prime or a power of a prime.
 It is conjectured that these are the only finite projective
planes, but this has yet to be proven.
N=4
N=5
N=6
 Not possible as an order of a projected plane. This was
proved by Gaston Tarry in 1901, who was working on a
mathematical puzzle proposed by Euler.
 In 1949 Bruck & Ryser created the theorm that follows:
If a finite projective plane of order q exists and q is
congruent to 1 or 2 mod 4, then q must be the sum of
two squares. This rules out 6 and 14.
Others
 N = 7 works
 N = 8 works
 N = 9 works
 N = 11 works
N = 10
 This is impossible as an order of a projective plane. This
has been proved using heavy computer calculation.
 Bruck & Ryser’s theorem did not rule this out, instead it
was proven using a combination of coded theory and a
large scale computer search.
Others
 N = 12: This is conjectured to be impossible, but no one
really knows for sure. It is currently an open question.
 N > 12: No one has any idea. There is constant study of
if they would work, and what they would look like, but
there is currently no proof that there can be a finite
projective plane for N > 12.
 As I said earlier many believe that all primes and the
primes raised to power are the only finite projective
planes, but that has yet to be shown. Proving this has
become one of the most important unsolved problems in
combinatorics.
Sources
 http://mathworld.wolfram.com/ProjectivePlane.html
 http://www.math.msu.edu/~dhand/MTH482SS09/Lect
18.pdf
 http://kahrstrom.com/mathematics/documents/OnProj
ectivePlanes.pdf
 http://mathworld.wolfram.com/Bruck-RyserChowlaTheorem.html
```