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The box counting method of measuring the fractal dimension of an object is $$D = \lim_{\epsilon \rightarrow 0}{ {\log N( \epsilon)} \over {\log { {1}\over{ \epsilon }}}},$$ where the classic example is calculating fractal dimension of the coast of England.

But let us imagine a scenario where the coast goes off to infinity in the north and south dimension. Let us fix $\epsilon$, and let $N$ be the independent variable, and let $R_N$ be radius of the smallest circle that contains $N$ boxes. Can we not write

$$D = \lim_{N \rightarrow \infty}{{\log { R_N}}\over {\log N} }.$$

It seems to me this is a perfectly valid method, but I cannot find it anywhere.

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  • $\begingroup$ The box counting dimension is fundamentally only applicable to bounded sets. Falconer discusses a notion of "modified box counting dimension" for unbounded sets. I would check his texts for details. $\endgroup$
    – Xander Henderson ♦
    Commented Sep 10, 2023 at 22:38
  • $\begingroup$ Thanks for the tip. $\endgroup$
    – Chris
    Commented Sep 10, 2023 at 22:46
  • $\begingroup$ Upon further reflection, the definition you propose does not make any sense to me. You fix some $\varepsilon$, but it is never used in the definition. I also don't know what it means for $R_N$ to be the radius of the smallest ball which contains $N$ boxes---$N$ boxes which intersect the set? are these boxes non-overlapping? What do you mean by a "circle"? Typically, "circle" just means the boundary of a disk. Do you mean a disk? or more generally, a ball (in $\mathbb{R}^n$)? $\endgroup$
    – Xander Henderson ♦
    Commented Sep 11, 2023 at 2:44
  • $\begingroup$ You may be right that my definition does not make sense. I have not tried to write it down rigorously. It does make intuitive sense, but maybe there is a flaw in my intuition. I'll try to find the time to work it out more rigorously. Do you think it makes sense in 2D? $\endgroup$
    – Chris
    Commented Sep 11, 2023 at 2:57
  • $\begingroup$ if the $N(\epsilon)$ (non-overlapping, on a grid) boxes of size $\epsilon$ are all in a straight line, $R_{N(\epsilon)}$ will be larger than when the boxes are clustered more compactly, which makes the size of $D$ counter-intuitive versus usual concept of dimension $\endgroup$
    – Claude
    Commented Sep 11, 2023 at 7:25

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