# Novel approach to the box counting calculation of fractal dimension

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.

• 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. Commented Sep 10, 2023 at 22:38
• Thanks for the tip. Commented Sep 10, 2023 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 $\mathbb{R}^n$)? Commented Sep 11, 2023 at 2:44
• 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? Commented Sep 11, 2023 at 2:57
• 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 Commented Sep 11, 2023 at 7:25