For a set $F \subset \mathbb{R}^n$ and $\delta > 0$, define:

$H^\delta_s(F) = \inf\left\{\sum_{i} r_i^s : F \subset \bigcup_i B_i, \text{ diam}(B_i) \leq \delta\right\}$

The Hausdorff $s$-measure is $H^s(F) = \lim_{\delta \to 0} H^\delta_s(F)$. The Hausdorff dimension is:

$\dim_H(F) = \inf\{s : H^s(F) = 0\} = \sup\{s : H^s(F) = \infty\}$

This dimension generalizes integer dimensions: smooth curves have $\dim_H = 1$, surfaces have $\dim_H = 2$, but fractals can have non-integer values like $\dim_H = \log 3/\log 2$ for the Cantor set.

Cover a set $F$ with boxes of side length $\epsilon$. Let $N(\epsilon)$ be the minimum number needed. The box-counting dimension (or Minkowski dimension) is:

$\dim_B(F) = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$

Equivalently, $N(\epsilon) \sim \epsilon^{-d}$ where $d = \dim_B(F)$. This dimension is often easier to compute numerically than Hausdorff dimension.

A set $F$ is self-similar if it is the union of scaled, rotated, and translated copies of itself. Formally, $F$ satisfies:

$F = \bigcup_{i=1}^k S_i(F)$

where $\{S_i\}$ are contracting similarity transformations. The collection $\{S_i\}$ forms an iterated function system (IFS).

For self-similar sets with non-overlapping pieces, the dimension satisfies:

$\sum_{i=1}^k r_i^d = 1$

where $r_i$ is the contraction ratio of $S_i$ and $d = \dim_H(F)$.