📂Dynamics

Box-Counting Dimension

Definition ^{1 2}

Suppose there is a bounded set $S \subset \mathbb{R}^{n}$. Define the minimum number of hypercubes with a side length of $\varepsilon$ needed to cover $S$ as $N \left( \varepsilon \right)$. If $d = d \left( S \right)$, defined as follows, exists, then we call $d$ the box-counting dimension of $S$. $$d \left( S \right) := \lim_{\varepsilon \to \infty} {\frac{ \log N \left( \varepsilon \right) }{ \log \left( 1 / \varepsilon \right) }}$$

Explanation

The box-counting dimension, or box dimension, while also defined in a limit sense, can be seen as a slightly less rigorous relaxation of the concept of the Hausdorff dimension, depending on the perspective. It primarily emerges to describe fractals, but first, let’s check if it is also an appropriate definition for lines and surfaces, which we intuitively understand.

Surface

Covering a surface with an area of $A$ using squares with a side length of $\varepsilon$, as shown in the image above, means the sum of the areas of $N \left( \varepsilon \right)$ squares, each of area $\varepsilon^{2}$, is slightly larger than $A$. Precisely comparing the numbers is meaningless, and this approximation becomes more accurate as $\varepsilon \to 0$. $$A \approx \varepsilon^{2} N \left( \varepsilon \right)$$ Here, regardless of $A$, $N \left( \varepsilon \right)$ increases as $\varepsilon$ decreases, meaning $N \left( \varepsilon \right)$ can be described by the following proportional equation. $$N \left( \varepsilon \right) \propto {\frac{ 1 }{ \varepsilon^{2} }}$$ As previously mentioned, this is a universally derived phenomenon independent of $A$, so if we take the logarithm on both sides of $\left( {\frac{ 1 }{ \varepsilon }} \right)^{2} \propto N \left( \varepsilon \right)$ with $A$ omitted, we obtain the following. $$\begin{align*} & \log \left( {\frac{ 1 }{ \varepsilon }} \right)^{2} \propto \log N \left( \varepsilon \right) \\ \implies & 2 \propto {\frac{ \log N \left( \varepsilon \right) }{ \log \left( 1 / \varepsilon \right) }} \end{align*}$$

Curve

As with surfaces, requiring $N \left( \varepsilon \right)$ squares to cover a curve with a length of $L$ implies $\varepsilon N \left( \varepsilon \right) \approx L$. Of course, since the diagonal of a square is $\sqrt{2} \varepsilon$, fewer squares might be needed, but as noted earlier, in the limit sense, minor scalar multiples are insignificant. Similarly, $\left( 1 / \varepsilon \right)^{1} \propto N \left( \varepsilon \right)$ for curves leads to the following observation. $$\begin{align*} & \log \left( {\frac{ 1 }{ \varepsilon }} \right)^{1} \propto \log N \left( \varepsilon \right) \\ \implies & 1 \propto {\frac{ \log N \left( \varepsilon \right) }{ \log \left( 1 / \varepsilon \right) }} \end{align*}$$

Fractals

The Cantor set $C$ requires at least $2$ intervals to cover it using an interval with a length of $1/3$, one on the left and one on the right. To cover it with an interval of length $1/3^{2}$, $2^{2}$ intervals are needed this time. The Cantor set continues infinitely, and to cover the Cantor set $C$ with an interval of length $1/3^{n}$, $2^{n}$ intervals are needed. Re-expressing this in equation form, for $\varepsilon = \left( 1 / 3 \right)^{n}$, $N \left( \varepsilon \right) = 2^{n}$, and its Cantor dimension is computed as follows. $$d = \lim_{\varepsilon \to 0} {\frac{ \log N \left( \varepsilon \right) }{ \log \left( 1 / \varepsilon \right) }} = {\frac{ log 2^{n} }{ \log 3^{n} }} = {\frac{ \log 2 }{ \log 3 }} \approx 0.6309$$ This value coincides with that calculated using another fractal dimension, the similarity dimension.

Interestingly, this ‘dimension’ doesn’t resolve neatly to an integer, but appears as a fraction between $0$ and $1$. Although the Cantor set is an infinite set, even an uncountable set, and thus cannot be said to lie in dimension $0$, the length being $0$ makes the conclusion that it has a dimension between $0$ and $1$ possibly correct.

See also

• Fractal
• Hausdorff dimension
• Similarity dimension
• Box-counting dimension
• Correlation dimension