📂Dynamics

Similarity Dimension

Definition

Let’s say a set $\displaystyle A := \lim_{n \to \infty} A_{n}$ is a self-similar set. When a subset of $A_{0}$ similar to $A_{0}$ is called a copy of $A_{0}$, let’s call $r$, which when multiplied by the volume of the copy of $A_{0}$ equates to the volume of $A_{0}$, the scale factor. If $A_{1}$ has $m$ mutually disjoint copies of $A_{0}$, then $d$ defined as follows is called the similarity dimension¹. $$ d := {\frac{ \log m }{ \log r }} $$ Here, volume refers to length, area, volume, etc.

Explanation

Similarity dimension is a type of fractal dimension that is naturally defined in a geometric sense. An intuitively understandable example of the concept can be imagined by dividing each side of a square into $n$ parts and drawing a new line segment.

If you divide the side length into $2$ parts, the new smaller square has a side length of the original $1/2$, and there will be $4$ such smaller squares. Similarly, dividing the side length into $3$ parts yields $9$ squares with a side length of the original $1/3$. The scale factor in the definition of similarity dimension can be considered as $r = n$, the reciprocal of the reduction in length, and it’s not hard to check that the number of copies is $m = n^{2}$. According to this, the similarity dimension of a square is calculated as $$ d = {\frac{ \log m }{ \log r }} = {\frac{ \log n^{2} }{ \log n }} = 2 $$ We can say that the similarity dimension of a square is $2$, which aligns with the common understanding that we consider a square as a $2$-dimensional shape. Unsurprisingly, this consistently holds true for general hypercubes $[0, 1]^{d}$.

Cantor Set

In the Cantor set, instead of the segment length reducing to $1/3$, such segments now appear $2$ times. Given $r = 3$ and $m = 2$, the similarity dimension of the Cantor set is computed as follows. $$ d = {\frac{ \log m }{ \log r }} = {\frac{ \log 2 }{ \log 3 }} \approx 0.63 $$ This inspires the notion that the Cantor set, while having a total length of $0$ and being an uncountable set, yet not a complete segment, might possess a dimension somewhere between $0$ and $1$ dimensional.

von Koch Curve

In the von Koch curve, instead of the segment length reducing to $1/3$, such segments appear $4$ times. Given $r = 3$ and $m = 4$, the similarity dimension of the von Koch curve is computed as follows. $$ d = {\frac{ \log m }{ \log r }} = {\frac{ \log 4 }{ \log 3 }} \approx 1.26 $$ Even though the von Koch curve’s length is infinite, there is no reason for it to create an area at those extreme folding points. This result intuitively shows that the von Koch curve is positioned in a dimension greater than $1$ dimensional but smaller than $2$ dimensional.

Limitations

It would be beneficial if examining examples of similarity dimension improves your grasp of fractals, but unfortunately, you will seldom encounter similarity dimension again in your life. While it’s advantageous that precise values can be calculated without the aid of computers, it can only be mentioned for self-similar sets with clear rules, as exact definitions are inherently difficult to establish. For geometric elements presented as data in the real world, such rules are unknown, thereby confining similarity dimension to a textbook concept.

See Also

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