= Hausdorff dimension
{c}
{title2=$\dim_HE$}
{wiki}
The Hausdorff dimension is the critical exponent at which <Hausdorff measure> changes from infinite to zero:
$$
\dim_HE=\inf\{s\geq0:\mathcal H^s(E)=0\}
=\sup\{s\geq0:\mathcal H^s(E)=\infty\},
$$
with the standard endpoint convention for an empty supremum. This measures fine-scale geometric size rather than the number of coordinates used to describe the <set>. A line segment has <Hausdorff dimension> one; a square has dimension two; the middle-third <Cantor set> has dimension $\log2/\log3\approx0.63093$.
Why can a dimension be fractional? At a scale reduced by a factor of three, the <Cantor set> needs twice as many construction intervals. The balance $2(1/3)^d=1$ gives $d=\log2/\log3$, between the growth rates of finitely many points and a whole interval. Making this argument rigorous requires a <cover> upper bound and the <mass distribution principle> lower bound, as in <Hausdorff dimension of the Cantor set>.
At the critical exponent itself, <Hausdorff measure> can be zero, finite and positive, or infinite; dimension does not specify that endpoint value. For example, <finite sets> and <countably infinite sets> both have dimension zero, although their <counting measures> differ. Dimension is also different from a finite-resolution estimate: <box-counting dimension> is another asymptotic definition, and fitting a scaling slope over a handful of observed scales does not establish either dimension.
The metric matters. Multiplying every distance by a fixed positive constant multiplies $\mathcal H^s$ by that constant to the power $s$ and preserves <Hausdorff dimension>. A <Lipschitz map> cannot increase it, because mapped <cover> <diameters> increase by at most a fixed factor. A <bi-Lipschitz map> therefore preserves it by applying this fact in both directions. These properties make <Hausdorff dimension> useful for comparing geometric models while controlling how their measurements change.
Back to article page
