MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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)?

share|cite|improve this question
Please explain more precisely what you mean. What are $N$ and $e$? If it's a finite set, it's certainly not a fractal. – Robert Israel Mar 4 '13 at 21:05
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: 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? – mt_christo Mar 5 '13 at 1:30
It sounds a bit like a multi-fractal, ( 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 ( have something similar happening in them, if I recall correctly. – Per Alexandersson Mar 5 '13 at 10:56
Perhaps begin with Mandelbot's paper, "How Long is the Coast of Britain?" – Gerald Edgar Mar 5 '13 at 13:38
up vote 2 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.