fractal_dimension

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Definition

The fractal dimension (or box-counting dimension) of a bounded set $A \subseteq \mathbb{R}^n$ is defined as

$$ \dim(A) = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)} $$

where $N(\epsilon)$ is the minimum number of $\epsilon$-balls (or boxes of side length $\epsilon$) needed to cover $A$, provided this limit exists.

For example, the Cantor_set has fractal dimension $\log 2 / \log 3 \approx 0.631$.