Math Question, "Ostrom/Wagner Theorem"

[B Dids]

OK, so my grandfather is the "Ostrom" of this theorem, but because he hogged all the math brains in the family- none of us have any clue what the hell it means.

On the off chance that any of your Skalansky esque Math folks have a clue- what the hell is this all about. So far all I've been able to gather is something about finite and infinte planes.

[Mayhap]

This is really simple.
On finite planes, two football teams will complete their game and one team will win.
On infinite planes, the same two teams will grow old and die before the game finishes.
/M

[andyfox]

Try the probability forum, or PM Bruze Z., he'd be your man for this, one would think.

[Mayhap]

Google returns pretty rich results on this as well.
/M

[B Dids]

Google didn't give me much. Nothing at least that I could understand.

[pzhon]

A projective plane is an abstract geometry defined by a set of points and a set of lines. There doesn't have to be any notion of angles, or of a point being between two other points on a line. In fact, there may be only finitely many points on each line.

Points: {A,B,C}
Lines: {{A,B},{B,C},{C,A}}
You can draw this as a triangle with vertices A, B, C. The sides of the triangle represent the lines, but only the vertices count as points.

This geometry satisfies two other conditions that justify calling it a "projective plane."
/images/graemlins/diamond.gifEvery pair of points is on a unique line.
/images/graemlins/diamond.gifEvery pair of lines intersects in a unique point.

Here is another, more complicated example.

Points: {0,1,2,3,4,5,6}
Lines: {{0,1,3},{1,2,4},{2,3,5},{3,4,6},{4,5,0},{5,6,1},{ 6,0,2}}

You may see a 7-fold symmetry: The permutation 0->1->2->3->4->5->6->0 sends lines to lines, e.g., the line {2,3,5} is sent to {3,4,6}, also a line. In fact, the set of symmetries is much larger; there are 168 symmetries. Any 3 noncollinear points can be sent to any other 3 noncollinear points by some symmetry.

The Ostrom-Wagner Theorem says that any finite projective plane so symmetric that any two points can be sent to any other two points by some symmetry must have an algebraic set of coordinates. (I hope I got that right. I didn't see a complete proof anywhere.)

In particular, the projective plane with 7 points can be rewritten as follows:

Points: {(0,0,1),(0,1,0),(1,0,0),(1,1,0),(1,0,1),(0,1,1),( 1,1,1)}
Lines: Any 3 points that add up to (0,0,0), mod 2, i.e., their sum has all even coordinates. (0,0,1)+(0,1,1)+(0,1,0)=(0,2,2), so {(0,0,1),(0,1,1),(0,1,0)} is a line.

This is the projective plane over the field with 2 elements. It's a bit harder to describe the field with 9 elements, but for any power q of a prime, there is a unique associated field and an associated projective plane based on triples of elements of the field, with q^2+q+1 points and q^2+q+1 lines. I believe it is still an open problem whether there are any projective planes other than these.

For a card game based on finite geometry, try Set (http://www.setgame.com/set/). This game with 81 cards challenges you to find triples of cards called "Sets." These are lines in affine 4-space over the field with 3 elements, but you don't have to know that to play.

[daryn]

set is a fun game. it's interesting to see how different people perform playing the game. think you're "smarter" than your friend? play set with him and get destroyed!

[Utah]

The little online puzzle was fun. I didnt do very well. It took me 7:30 to complete.

[B Dids]

Thanks!!

I'm still struggling to make total sense of that- but that puts me a lot closer than I was before.

[pzhon]

[ QUOTE ]
I believe it is still an open problem whether there are any projective planes other than these [projective planes over finite fields].

[/ QUOTE ]

Correction: There are projective planes not directly based on finite fields of order q. The open question is whether all finite projective planes have q+1 points on each line for some prime power q. In particular, it is an open problem whether there is a projective plane with 12+1 points on each line. The two smaller non-prime powers 6 and 10 have been ruled out.

[jdl22]

The thing about set is that people like us are at a disadvantage. If you try to use a system you are screwed compared to people that are able to naturally just look and figure it out.

I get killed because I tend to take two cards and see if there is a third and when there isn't move on to the next two. Takes forever.