# Exact self-similarity?

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

## Answers and Replies

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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and cantor set :D

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

CRGreathouse
Homework Helper
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.

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
Homework Helper
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.

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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