Hausdorff Dimension — Definition, Formula & Examples

Hausdorff dimension is a way to assign a fractional (non-integer) dimension to a set, capturing how it scales in complexity. It generalizes the familiar notion of dimension so that objects like fractals can have dimensions such as 1.26 or 0.63.

For a metric space $X$ , the Hausdorff dimension is defined as $\dim_H(X) = \inf\{d \geq 0 : H^d(X) = 0\}$ , where $H^d(X)$ is the $d$ -dimensional Hausdorff measure of $X$ . It is the critical value of $d$ at which the Hausdorff measure transitions from infinity to zero.

Key Formula

d = \frac{\ln N}{\ln(1/r)}

Where:

$d$ = Hausdorff (similarity) dimension of the fractal
$N$ = Number of self-similar copies at each iteration
$r$ = Scaling ratio of each copy relative to the whole

How It Works

To find the Hausdorff dimension of a self-similar fractal, you identify how many smaller copies

N

the fractal breaks into and the scaling ratio

r

of each copy. The dimension then satisfies

N \cdot r^d = 1

, giving

d = \frac{\ln N}{\ln(1/r)}

. This formula works directly for strictly self-similar fractals with uniform scaling. For more complex sets, computing Hausdorff dimension requires measure-theoretic arguments involving coverings by small balls.

Worked Example

Problem: Find the Hausdorff dimension of the Sierpiński triangle.

Identify self-similar pieces: At each stage, the Sierpiński triangle is composed of 3 smaller copies of itself, so

N = 3

Determine scaling ratio: Each smaller copy is scaled by a factor of

\frac{1}{2}

relative to the original, so

r = \frac{1}{2}

Apply the formula: Substitute into the self-similarity dimension formula.

d = \frac{\ln 3}{\ln 2} \approx 1.585

Answer: The Hausdorff dimension of the Sierpiński triangle is

\frac{\ln 3}{\ln 2} \approx 1.585

Why It Matters

Hausdorff dimension appears in fractal geometry, dynamical systems, and geometric measure theory. It is used to classify strange attractors in chaos theory and to quantify the roughness of coastlines, turbulence boundaries, and signal noise in applied mathematics and physics.

Common Mistakes

Mistake: Assuming Hausdorff dimension must be an integer.

Correction: Unlike topological dimension, Hausdorff dimension can take any non-negative real value. A set embedded in

\mathbb{R}^2

can have dimension 1.585, for instance.