Let $\Theta$ be a bounded subset of $\R^n$ let $N_{\Theta}(\epsilon)$ be the minimum number of balls of radius $\epsilon$ required to cover $\Theta$ . Then define the Hausdorff dimension $d_H$ of $\Theta$ to be $$ d_H (\Theta):= - \lim_{\epsilon \rightarrow 0} \frac{\log N_{\Theta}(\epsilon)}{\log \epsilon} $$

Hausdorff dimension is easy to calculate for simple objects like the Sierpinski gasket or a Koch curve. Each of these may be covered with a collection of scaled-down copies of itself. In fact, in the case of the Sierpinski gasket, one can take the individual triangles in each approximation as balls in the covering. At stage $n$ , there are $3^n$ triangles of radius $\frac{1}{2^n}$ , and so the Hausdorff dimension of the Sierpinski triangle is at most $- \frac{n \log 3}{n \log 1/2} = \frac{\log 3}{\log 2}$ , and it can be shown that it is equal to $\frac{\log 3}{\log 2}$ .

From some notes from Koro

Define the diameter $|C|$ of a bounded subset $C$ of $X$ to be $\sup_{x,y\in C} d(x,y)$ .

Define a countable $r$ -cover of $X$ to be a collection of subsets $C_i$ of $X$ indexed by some countable set $I$ , such that $|C_i| < r$ and $X = \cup_{i\in I} C_i$ .

We also define the function $$H^D_r (X) = \inf \sum_{i\in I} |C_i|^D$$ where the infimum is over all countable $r$ -covers of $X$ . The Hausdorff dimension of $X$ may then be defined as $$d_H (X)=\inf \{ D\mid \lim_{r\rightarrow 0} H^D_r(X)=0 \}.$$ When $X$ is a subset of $\R^n$ with any restricted norm-induced metric, then this definition reduces to that given above.