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Exact self-similarity?

  1. Jan 16, 2010 #1
    Does self-similarity of fractals ever represent an exact, albeit scaled down, reproduction?
  2. jcsd
  3. Jan 16, 2010 #2
    Unless I misunderstand what you mean, yes many simple fractals are of this type. See for instance http://en.wikipedia.org/wiki/Sierpinski_triangle" [Broken].
    Last edited by a moderator: May 4, 2017
  4. Jan 16, 2010 #3
    and cantor set :D
  5. Jan 17, 2010 #4
    Can one define a "simplicity limit" beyond which exact self-similarity does not occur?
  6. Jan 17, 2010 #5


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    Care to be more specific? Otherwise, I could define the simplicity limit as "the fractal is exactly self-similar" and get the desired result.
  7. Jan 17, 2010 #6
    Here's a first attempt at specificity:

    Does fractal self-similarity scale by rational numbers, and fractals that are not self-similar scale by irrational numbers?
  8. Jan 18, 2010 #7


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    No, since the fractal generated as the limit of

    F0 = triangle with unit sides
    Fn = F(n-1) plus triangle with sides of length x^n
    where all triangles are oriented similarly and share a common point

    is exactly self-similar but scales by x which can be chosen to be irrational.
  9. Jan 18, 2010 #8
    Does a Mandelbrot set ever have exact self-similarity, and if not, is there a measure of how close the set comes to it?

    My original question should have been: do all fractals have some presence of exact self-similarity?
    Last edited: Jan 18, 2010
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