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When studying fractals, one of the properties named by Benoit Mandelbrot is the self-similarity (and it's variations) of the fractal objects.

From Wikipedia:

In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales.[2] Self-similarity is a typical property of artificial fractals.

This can be easily seen in pictures: https://upload.wikimedia.org/wikipedia/commons/4/4b/Fractal_fern_explained.png https://en.wikipedia.org/wiki/Fractal

My question is, is there any way to demonstrate this property in a 'pure mathematical way' ? I know Mandelbrot's purpose was exactly the opposite, to link and demonstrate all this theories in a visual way, but I think any kind of pure demonstration would be great to understand this objects. I haven't been able to find any research about this so I don't know where I could start.

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  • $\begingroup$ What do you mean by "demonstration" exactly? $\endgroup$ – Travis Jul 6 '17 at 11:23
  • $\begingroup$ @Travis I'm sorry, english is not my native language so I may not express correctly. I mean something like proof or statement. In this case, a way to 'confirm' self-similarity. $\endgroup$ – Pedro Javier Fernández Jul 6 '17 at 11:25
  • $\begingroup$ For an example, see en.wikipedia.org/wiki/Cantor_set#Self-similarity. $\endgroup$ – lhf Jul 6 '17 at 11:52
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You can only prove selfsimilarity for fractal sets $K$ that are defined mathematically, e.g., the Cantor set, the Sierpinsky carpet, Koch's curve. In these cases $K$ is not only similar to a part of itself, but $K$ is in fact the union of (more or less disjoint) similar copies of itself: $$K=\bigcup_{i=1}^m f_i(K)\ ,$$ whereby the $f_i$ are similarities. In the above examples this is evident by inspection. All these examples go under the heading Iterated Function System. See the literature quoted in the linked Wikipedia entry, notably Barnsley and Falconer.

If you require only that $K$ is similar to a part of itself (as in your quote from Wikipedia) then there are examples which you might not consider "fractal", e.g., the logarithmic spiral $$r=e^\phi\qquad(-\infty<\phi\leq0)\ .$$

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  • $\begingroup$ Christian is there any paper or book where this formula is used? $\endgroup$ – Pedro Javier Fernández Aug 2 '17 at 11:56
  • $\begingroup$ Thanks for the edit. I'm going to check and I will accept the answer. $\endgroup$ – Pedro Javier Fernández Aug 15 '17 at 10:06
  • $\begingroup$ I don't think that all mathematically defined fractals are self-similar. Here is a basic example. Take a random walk. Replace each unit length segment of that with a random walk that fixes the end points. Repeat ad infinitum. $\endgroup$ – j0equ1nn Jun 1 '18 at 3:37
  • $\begingroup$ Also, if you require only that $K$ is similar to a part of itself, you don't need to go to far to find non-fractal examples. Take a line. $\endgroup$ – j0equ1nn Jun 1 '18 at 3:46
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There is no way to demonstrate that property mathematically because the property does not hold. Not all fractals are self-similar and not all self-similar objects are fractals. The rigorous definition of a fractal is an object having a Hausdorff dimension that is less than its topological dimension.

This post asks a similar question and the selected answer gives more detail: Why must fractals be self-referential?

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