Hausdorff dimension

Definition · applies to subsets of $\reals^n$ and metric spaces · also called fractal dimension

High school

Consider a shape, such as a segment, a square, or a cube. Scaling the shape up by a factor of $2$ multiplies its “size” by $2^d$, where $d$ is its dimension: by $2$ for a segment, $4$ for a square, and $8$ for a cube. The Hausdorff dimension makes this precise.

Graduate (metric spaces)

Consider a subset $A$ of a metric space and a number $s \geq 0$. The $s$-dimensional Hausdorff measure of $A$ adds up the diameters of small sets that cover $A$, each raised to the power $s$:

\[\mathcal{H}^s(A) = \lim_{\delta \to 0} \inf \setdef{\sum_{i = 1}^{\infty} \operatorname{diam}(U_i)^s \suchthat A \subseteq \bigcup_{i = 1}^{\infty} U_i, \ \operatorname{diam}(U_i) < \delta}.\]

The Hausdorff dimension of $A$ is the number at which $\mathcal{H}^s(A)$ jumps from $\infty$ to $0$:

\[\dim_H(A) = \inf \setdef{s \geq 0 \suchthat \mathcal{H}^s(A) = 0}.\]

Examples

Worked examples

• The closed unit disk $D = \setdef{(x, y) \suchthat x^2 + y^2 \leq 1}$ : $2$ High school
  The Hausdorff dimension of the closed unit disk is $2$, as for every region of the plane with nonempty interior.
• $[0, 1]$ : $1$ High school
  The Hausdorff dimension of $[0, 1]$ is $1$, as for every segment.
• $\mathbb{Q} \cap [0, 1]$ : $0$ High school
  The Hausdorff dimension of $\mathbb{Q} \cap [0, 1]$ is $0$, as for every countable set.
• The set of real numbers, $\reals$ : $1$ High school
  The Hausdorff dimension of $\reals$ is $1$.
• The set $\setdef{0, 1}$, as a subset of $\reals$ : $0$ High school
  The Hausdorff dimension of $\setdef{0, 1}$ is $0$, as for every countable set.
• The Cantor set : $\log 2 / \log 3 \approx 0.63$ Undergraduate
  The Hausdorff dimension of the Cantor set is $\log 2 / \log 3$, because it consists of $2$ copies of itself, each scaled down by a factor of $3$, so $2 = 3^d$.

Properties

• For a smooth curve, surface, or solid, the Hausdorff dimension is $1$, $2$, or $3$, its usual dimension.
• Every countable set has Hausdorff dimension $0$.
• The Cantor set has Hausdorff dimension $\log 2 / \log 3 \approx 0.63$, because it consists of $2$ copies of itself, each scaled down by a factor of $3$.

Notes

The Hausdorff dimension of a set is a different notion from the dimension of a vector space, which counts the vectors in a basis.

For some shapes, called fractals, the Hausdorff dimension is not a whole number.

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