# non-Euclidean -help on proof

• Jun 8th 2006, 01:12 PM
Judi
non-Euclidean -help on proof
Let M be a finte projective plane so that all lines in M have the same number of points lying on them, call this number N+1

Prove:the toall number of poins in M is N^2+N+1 and total number of lines
in M is N^2+N+1.
• Jul 7th 2006, 12:06 PM
Rebesques
Projective stuff... Great :confused:

From what I can recall, a line passes through every two points.

For the total number of lines, define a_ij to indicate the line passing through points i and j. All lines have N+1 points; So the cardinality of (a_ij) is (N+1)^2. But also, the diagonal (a_ii) does not define a line; So exclude N lines from this list, to get a total of (N+1)^2-N=N^2+N+1 lines.

For the number of points, I think there should be more than just N^2+N+1... Check again plz.
• Jul 8th 2006, 02:27 PM
Quick
I have seen this question posted before, but I can't find the post :eek:
• Jul 8th 2006, 03:00 PM
rgep
Choose a point P and line L with P not lying on L (if you cannot do this it is not a plane but a line).

The lines through P are in 1-1 correspondence with the points on L. Any line through P meets in in just one point, and any point on L defines a line through P. So there are N+1 lines, each of which has N points other than P. So there are N(N+1)+1 points in the plane.

Now there are (N+1)(N^2+N+1) pairs of the form (P,T) with P a point and T a line through P. Since each line T has N+1 points P, each line occurs in this list (N+1) times. So the total number of lines is (N^2+N+1).

There's another way of seeing the latter. Use duality to observe that every property of the plane remains true if you interchange the words point and plane. Since there are N+1 points on each line, and we showed N+1 lines through each point, there are the same number of points as lines by duality.
• Jul 8th 2006, 08:32 PM
Judi
thanks, guys
• Jul 15th 2006, 12:13 PM
Rebesques
Quote:

every property of the plane remains true if you interchange the words point and plane.
Oops! How could I forget something so important??? :o :o

Maybe besides reputation points we also need dumbness points, I would score high :D :D :D
• Jul 15th 2006, 07:24 PM
ThePerfectHacker
Quote:

Originally Posted by Rebesques
Oops! How could I forget something so important??? :o :o

Maybe besides reputation points we also need dumbness points, I would score high :D :D :D

There is, is called negative rep.
Just click on rep and say "do not approve".