Hausdorff Dimension
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).
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