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Triangulation proof

  1. Oct 4, 2009 #1
    1. The problem statement, all variables and given/known data

    A triangulation of a convex polygon is a decomposition of the polygon into
    triangles whose interiors do not overlap and whose vertices lie at vertices of
    the polygon. Prove that there are Cn−2 ways to triangulate an n-sided convex
    polygon

    2. Relevant equations

    cn= (1/2)(2n choose n)

    3. The attempt at a solution

    I tried a proof by induction:

    let n=3.
    there is only one way to triangulate a triangle, and
    cn-2=c1=(1/2)(2 choose 1)=1. So the statement holds for n=3

    now assume the statement holds for n=k
    So the number of ways to triangulate a k-sided polygon is Ck-2=(1/2)(2(k-2) choose k-2)

    Now let n=k+1

    This is where I get stuck and I dont' know what to do.... any help/hints would be appreciated.
     
  2. jcsd
  3. Oct 8, 2009 #2

    tiny-tim

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    Hi dancergirlie! :smile:

    (try using the X2 and X2 tags just above the Reply box :wink:)

    I don't think that formula is right …

    For n = 4, there are only 2 solutions, but C4-2 = C2 = (1/2)4C2 = 3. :confused:
     
  4. Oct 9, 2009 #3

    tiny-tim

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    Perhaps I'm misunderstanding the question, but i'm getting …

    For n = 5, there are 5 solutions, all "3-fans".

    For n = 6, there are 13 solutions, 6 "4-fans", 6 "pairs-of-2-fans", and 1 with a central triangle.

    If I'm counting correctly (and of course I may not be :redface:), I don't see how 13 can come out of a "choose" function.
     
  5. Oct 9, 2009 #4

    tiny-tim

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    Catalan numbers

    oops … I did miss one :blushing: … there are 2 with a central triangle, making 14.

    So the number of triangulations of an n-sided polygon for n = 3 4 5 and 6 are 1 2 5 and 14, which look like the Catalan numbers, see http://en.wikipedia.org/wiki/Catalan_number

    They're 2nCn/(n+1), not 2nCn/2 as in the original question. :rolleyes:

    They're generated by Cn+1 = ∑i=0n CiCn-i, which is fairly easy to prove for triangulations of a polygon. :smile:

    (there's a proof of the more direct equation (n+2)Cn+1 = (4n+2)Cn in the wikipedia article, but I'm afraid I don't understand it at all :redface:)
     
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