Do All Fractals Exhibit Exact Self-Similarity?

[Loren Booda]

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

 

[rasmhop]

Unless I misunderstand what you mean, yes many simple fractals are of this type. See for instance http://en.wikipedia.org/wiki/Sierpinski_triangle" .

 

[boboYO]

and cantor set :D

 

[Loren Booda]

Can one define a "simplicity limit" beyond which exact self-similarity does not occur?

 

[CRGreathouse]

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.

 

[Loren Booda]

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?

 

[CRGreathouse]

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.

 

[Loren Booda]

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?