Hausdorff Dimension

Definition

For a self-similar set composed of $N$ copies of itself scaled by a factor $r$, the Hausdorff (similarity) dimension is:

Classical Examples

Fractal	$N$	$r$	Dimension
Cantor set	2	1/3	$\log 2 / \log 3 \approx 0.631$
Sierpinski triangle	3	1/2	$\log 3 / \log 2 \approx 1.585$
Koch curve	4	1/3	$\log 4 / \log 3 \approx 1.262$
Sierpinski carpet	8	1/3	$\log 8 / \log 3 \approx 1.893$
Menger sponge	20	1/3	$\log 20 / \log 3 \approx 2.727$

Box-Counting Method

In practice, the Hausdorff dimension can be estimated by the box-counting dimension: cover the set with boxes of side length $\epsilon$ and count the number $N(\epsilon)$ of boxes needed. Then:

Connections

The similarity dimension formula is used to compute the dimension of attractors of Iterated Function SystemsIterated Function Systems$$A = \bigcup_{i=1}^{N} f_i(A)$$Constructing fractals via contractive affine transformationsRead more →. The boundary of the Mandelbrot SetMandelbrot Set$$z_{n+1} = z_n^2 + c$$Interactive visualization of z_{n+1} = z_n^2 + c and its connectednessRead more → famously has Hausdorff dimension 2.