For example we have rule 90 in cellular automaton which can produce Sierpinski triangle, which can be made algorithmically by removing triangles, or also through applying fractal formulas, like what we have in XaoS fractal software's example. So is it impossible, theoretically, to produce any fractal which we can produce from any system, in another one? And the answer must lead to say how we can?
Yes, we can generate fractals from big data, if you bear with me on the new definition fractal: a set or pattern is fractal if there are far more small things than large ones within it.
Jiang B. and Yin J. (2014), Ht-index for quantifying the fractal or scaling structure of geographic features, Annals of the Association of American Geographers, 104(3), 530–541, Preprint: http://arxiv.org/ftp/arxiv/papers/1305/1305.0883.pdf
Jiang B. (2015), Head/tail breaks for visualization of city structure and dynamics, Cities, 43, 69-77, Preprint: http://arxiv.org/ftp/arxiv/papers/1501/1501.03046.pdf
Herewith I show one example below:
Yes, if you apply your right IFS converging to a fractal, you can start with any image including another fractal. Right steps will give the fractal it converges to..
All fractal constructions you speak about are basically substitution fractals. I mean that there is a context free substitution of patterns. You start with any pattern P0, apply substitution and get P1, and your fractal is a limit of Pi. Those fractals which are got by time diagrams of cellular automata are also substitution fractals. This happens because of the strange coincidence that the multi-dimensional sequence in question is p-automatic, for some prime number p. This leads always to an open problem:
- Is any result of a context-free substitution of type (n1, ... nk) also the result of applying limit to a sequence of k-dim. rectangles in the time diagram of some cellular automaton, where the numbers n1 ... nk are not necessarily prime and might be different? Believe me, nobody knows that, because the algebraic nature of n-automatic sequences for not necessarily prime n is unknown... [Substitution of type n1, ...nk means that any k-dimensional unit rectangle is substituted with a rectangle of those edge lenghts..]
On this point we can also mention various Julia sets (complex fractals) which cannot be built with other techniques. For some easier of them, we can build something which is maybe homeomorphic, but not the set itself...
The answer at the question put like this is generally unknown.
``Computability and Julia Sets" by Braverman and Yampolsky may be helpful
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