Informally, self-similar objects with parameters and are described by a power law such as
where
is the "dimension" of the scaling law, known as the Hausdorff dimension.
Formally, let be a subset of a metric space . Then the Hausdorff dimension of is the infimum of such that the -dimensional Hausdorff measure of is 0 (which need not be an integer).
In many cases, the Hausdorff dimension correctly describes the correction term for a resonator with fractal perimeter in Lorentz's conjecture. However, in general, the proper dimension to use turns out to be the Minkowski-Bouligand dimension (Schroeder 1991).