Finite Projective Planes

1. Finite Projective Planes By Aaron Walker Wagner Geometry Finite Geometry Projective Planes Finite Projective Planes

2. 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.

3. Projective Plane Cont. • It contains the following properties • Given any two distinct points, there is exactly one line incident with both of them. • Give any two distinct lines, there is exactly one point incident with them both. • There are four points such that no line is incident with more than two of them.

4. 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 from your vision. There are many projective planes both finite and infinite. Today we are only going to focus on the finite projective planes.

5. 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.

6. 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.

7. N = 3

8. 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.

9. 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.

10. N = 4

11. N = 5

12. 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.

13. Others • N = 7 works • N = 8 works • N = 9 works • N = 11 works

14. 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.

15. 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.

16. Sources • http://mathworld.wolfram.com/ProjectivePlane.html • http://www.math.msu.edu/~dhand/MTH482SS09/Lect18.pdf • http://kahrstrom.com/mathematics/documents/OnProjectivePlanes.pdf • http://mathworld.wolfram.com/Bruck-Ryser-ChowlaTheorem.html