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How to find fractal dimension of Gosper Island

  1. Feb 26, 2012 #1
    Hi, I'm not sure if this is the right place for this....if it isn't if I could be redirected/if a moderator could move my post to the right place I would greatly appreciate it.

    In any case, I am trying to understand fractal dimensions. I read through wikipedia's description and I believe I kind of understand it. I understand how one gets the fraction dimension for Koch snowflake or Sierpinski triangle. However, I cannot figure out the same thing for the Gosper Island. http://www.wolframalpha.com/input/?i=Gosper+Island&a=*C.Gosper+Island-_*Formula.dflt-&f2=2&x=12&y=10&f=GosperIsland.n_2 . It seems to me that it should be N = 7 and epsilon = 1/3 where N and epsilon are discussed here http://en.wikipedia.org/wiki/Fractal_dimension. However apparently Hausdorff dimension gives log(9/7) which suggests to me that N = 9 and epsilon = 1/7, which confuses me greatly.

    If someone could help better understand fractal dimensions and what I am missing here I would greatly appreciate it. Thanks in advance!
     
  2. jcsd
  3. Feb 27, 2012 #2
    this may be incorrect, but this is how i interpreted the explanations.

    at iteration zero start with a side of unit length. it has one cover of diameter length one. N=1, L=1

    at iteration one, we have replaced the side with three sides each of length 3/7. equivalently, there are three coverings each of diameter 3/7
    N=3, L=3/7

    at iteration two, we have nine sides, each now scaled by (3/7)th's of (3/7)th's.
    N=3^2, L=(3/7)^2

    continuing...
    N=3^n, L=(3/7)^n

    so using the equation
    Dimension = - log(N^n)/log(L^n) = - log(3)/log(3/7) = 1.296?
     
  4. Feb 28, 2012 #3

    Bacle2

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    Notice that the dimension given in Wolfram is the dimension after only two iterations; after 4 iterations it was 81/49.
     
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