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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.

Xander Henderson
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Chris
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  • 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. – Xander Henderson Sep 10 '23 at 22:38
  • Thanks for the tip. – Chris Sep 10 '23 at 22:46
  • 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 $\mathbb{R}^n$)? – Xander Henderson Sep 11 '23 at 02:44
  • 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? – Chris Sep 11 '23 at 02:57
  • 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 – Claude Sep 11 '23 at 07:25
  • I'm not sure I get your point. You aren't taking into account that the boxes must efficiently cover the underlying fractal object. – Chris Sep 11 '23 at 15:24
  • If the fractal object is a straight line, then $R_N(\epsilon) = \epsilon N(\epsilon)$. if the fractal object fills in 2D (e.g. a Hilbert-like curve), then $R_N(\epsilon) = \sqrt{2} \epsilon \sqrt{N(\epsilon)}$. In the first case, $D = 1$ as expected, but in the second case $D = \lim_{N(\epsilon) \to \infty} \frac{ \frac{1}{2} \log{2} + \log{\epsilon} + \frac{1}{2} \log{N(\epsilon)}}{\log{N(\epsilon)}} = \frac{1}{2}$ which is less. Maybe taking the reciprocal of your $D$ would work, in this case giving the expected dimension $2$ ? – Claude Sep 12 '23 at 07:30
  • I probably should have written $R_N(\epsilon) \propto \epsilon N(\epsilon)$ and $R_N(\epsilon) \propto \epsilon \sqrt{N(\epsilon)}$, instead of equalities, but the end result is the same. – Claude Sep 12 '23 at 08:00
  • https://tglad.blogspot.com/2017/08/reframing-geometry-to-include-negative.html https://tglad.blogspot.com/2017/09/complex-dimensional-geometry.html – Claude Sep 12 '23 at 08:53
  • @Claude If I require that the object extends to infinity in two directions, then doesn't that rule out a space filling object? In my mind, I'm picturing a coast of England where the coast goes forever. – Chris Sep 12 '23 at 15:44

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