Exact self-similarity?

  • #1
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Does self-similarity of fractals ever represent an exact, albeit scaled down, reproduction?
 

Answers and Replies

  • #2
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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].
 
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  • #3
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and cantor set :D
 
  • #4
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Can one define a "simplicity limit" beyond which exact self-similarity does not occur?
 
  • #5
CRGreathouse
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Can one define a "simplicity limit" beyond which exact self-similarity does not occur?
Care to be more specific? Otherwise, I could define the simplicity limit as "the fractal is exactly self-similar" and get the desired result.
 
  • #6
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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.
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?
 
  • #7
CRGreathouse
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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?
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.
 
  • #8
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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?
 
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