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I have a finite set of points, and plot the graph log(N) vs. log(e). I see a polygonal chain (the final slope, starting at some size of e, is zero, of course). If the set represents some physical property of a complex system, is there any interpretation for that kind of series of slopes? Any literature around? And, generally - if we do that kind of fractal analysis, what does the overall shape of logN vs log(e) tell us (if anything)?

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  • $\begingroup$ Please explain more precisely what you mean. What are $N$ and $e$? If it's a finite set, it's certainly not a fractal. $\endgroup$
    – Robert Israel
    Commented Mar 4, 2013 at 21:05
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    $\begingroup$ Sure - I am calculating number (N) of segments of equal length needed to cover the set as function of size of the segment (e). (explained here: en.wikipedia.org/wiki/Fractal_dimension) Then I plot it in double-log coordinates. Theoretically, slope of that curve as it approaches 0 is the fractal dimension of the set. But I have a finite set, and the overall curve is a polygonal chain. Basically, what I am asking is - what could be the intuition behind that? Any literature on such applications of fractal analysis? $\endgroup$
    – mt_christo
    Commented Mar 5, 2013 at 1:30
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    $\begingroup$ It sounds a bit like a multi-fractal, (en.wikipedia.org/wiki/Multifractal_system) which has "mixed" fractal dimensions. If your data do not have sufficient resolution, it might be an artifact that it eventually becomes zero. Now, some DLA-systems (en.wikipedia.org/wiki/Diffusion-limited_aggregation) have something similar happening in them, if I recall correctly. $\endgroup$
    – Per Alexandersson
    Commented Mar 5, 2013 at 10:56
  • $\begingroup$ Perhaps begin with Mandelbot's paper, "How Long is the Coast of Britain?" $\endgroup$
    – Gerald Edgar
    Commented Mar 5, 2013 at 13:38

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The canonical reference for this material (at least with the people I hang out with) is Ken Falconer's Fractal Geometry Mathematical Foundations and Applications.

For most sets that are fully self-similar with infinite levels of geometry such as the Cantor set the function $\log(N)/\log(e)$ will be a similar polygonal chain far away from 0. It will get smoother as you approach zero and there will be a non-zero slope there. What you have however is a finite set of points so for an $e$ below a certain threshold $N$ stays constant. That is the function will have slope zero and you have a set of fractal dimension zero. In the book there is some discussion of how to interpret situations like there where you are approximating something that will have a non-zero dimension. But that is not something I can say much about.

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