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Fano's Geometry

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  1. Jun 17, 2017 #1
    1. The problem statement, all variables and given/known data
    In Fano's Geometry, we have the following axioms a. There exists at least one line b. Every line has exactly three points on it c. Not all points are on the same line d. For two distinct points, there exists exactly one line on both of them e. Each two lines have at least one point on both of them

    We were asked to create a model, and I was able to do that with no problem. Where I ran into problems was in the second part. The second part was to prove that the 4th axiom(part d) was independent. I am stuck on how to make this happen.

    2. Relevant equations
    There are no equations.

    3. The attempt at a solution
    When attempting the problem I think I need to remove one of the lines off of my model. I am not sure if this is a step in the right direction. If it is, I am not sure where to go from there.
     
  2. jcsd
  3. Jun 18, 2017 #2
    I never studied Fano's Geometry, although I think I saw it in a proof that there's, up to isomorphism, a unique simple group of order 168.
    But let me try to help you.
    Can't you create a model that does not satisfy d) but satisfied all the other axioms?
    I was thinking about it, the axiom d says "exactly one". So I think that we can keep adding lines.
    I thought the following, I hope I'm not wrong.
    We create a circumferencee with 2n points on it, equidistants. For example, n=4. Also we add the center as a point. Lines are then, diameters of the circle that touch a pair of antipodal points.
    Doesn't this satisfy all the axioms but d.?
    a. There are n lines, take n>0 and you're done check
    b. Everyline has exactly 3 points. check
    c. Not all points are on the same line. Take n>1 and you're done.
    e. The center is common to all lines. check
    In which subject did you found out this? I find it very interesting, sadly I never had anything similar in my college.
    Also, it doens't satisfy d, as there isn't a line that joins two points in the circle that are not antipodal (remember n>2, so there are at least 2 points that are not antipodal).
     
    Last edited: Jun 18, 2017
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