Results 1 to 6 of 6

Math Help - Incidence geometry

  1. #1
    Member
    Joined
    Nov 2007
    Posts
    108

    Incidence geometry

    This seems like hard problem. Anyone has any suggestion how to tackle it?
    Show that in a finite incidence geometry, the number of lines is greater than or equal to the number of points.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by namelessguy View Post
    This seems like hard problem. Anyone has any suggestion how to tackle it?
    Show that in a finite incidence geometry, the number of lines is greater than or equal to the number of points.
    Between any two distinct points there exists exactly one line. This is one of the axioms of incidence geometry. Now if there are n points (and say n\geq 2) then the number of pairs that we can form is equal to {n\choose 2} while the number of lines is n.

    EDIT: Mistake
    Last edited by ThePerfectHacker; February 22nd 2009 at 01:27 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2007
    Posts
    108
    Quote Originally Posted by ThePerfectHacker View Post
    Between any two distinct points there exists exactly one line. This is one of the axioms of incidence geometry. Now if there are n points (and say n\geq 2) then the number of pairs that we can form is equal to {n\choose 2} while the number of lines is n.
    Did you mean that the number pairs we can form is equal to {n\choose 2} which is the number lines, while the number of points is only n? I did learn the axiom, and thanks a lot for your help TPH. It seems much easier.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Nov 2007
    Posts
    108
    @TPH, actually, I tried to draw some examples for some n points, but it appears that the number lines can be less that {n\choose2}. I guess it could be that {n\choose2} is the maximum number of lines that we can get from n points. If this is the case, then I'm stuck again
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,639
    Thanks
    1592
    Awards
    1
    Quote Originally Posted by namelessguy View Post
    I tried to draw some examples for some n points, but it appears that the number lines can be less that {n\choose2}. I guess it could be that {n\choose2} is the maximum number of lines that we can get from n points.
    That is truly the case.
    Example: The Pappus finite geometry. In this geometry there are nine points and nine lines with 3 points on each line and 3 lines on each point.

    Thus this problem depends on the set of axioms in use.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Mar 2009
    Posts
    1

    Incidence geometry: #lines >= #points

    In an incidence geometry, two points are connected by exactly one line, there are (at least) three noncollinear points, and every line contains at least two points. In this context the de Bruijn-Erdos theorem (1948) says that there are at least as many lines as points. There's a proof in the text Combinatorics of Finite Geometries by Lynn Margaret Batten. One can also find proofs on the net by searching for "de Bruijn-Erdos theorem".

    George
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Incidence Geometry-Elliptical parallel property
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: March 24th 2010, 09:12 AM
  2. Give an example of a finite incidence geometry
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 3rd 2010, 07:01 AM
  3. Prove this is not an incidence geometry
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: August 30th 2009, 03:12 PM
  4. incidence geometry
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: February 16th 2008, 05:32 PM
  5. Incidence Geometry and Integer lattice
    Posted in the Advanced Math Topics Forum
    Replies: 7
    Last Post: January 30th 2008, 06:51 PM

Search Tags


/mathhelpforum @mathhelpforum