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Math Help - non-Euclidean -help on proof

  1. #1
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    Smile 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.
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  2. #2
    Super Member Rebesques's Avatar
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    Projective stuff... Great

    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.
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  3. #3
    MHF Contributor Quick's Avatar
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    I have seen this question posted before, but I can't find the post
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  4. #4
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    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.
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  5. #5
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    thanks, guys
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  6. #6
    Super Member Rebesques's Avatar
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    every property of the plane remains true if you interchange the words point and plane.
    Oops! How could I forget something so important???

    Maybe besides reputation points we also need dumbness points, I would score high
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  7. #7
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    Quote Originally Posted by Rebesques
    Oops! How could I forget something so important???

    Maybe besides reputation points we also need dumbness points, I would score high
    There is, is called negative rep.
    Just click on rep and say "do not approve".
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