In many cases, the Hausdorff dimension correctly describes the correction term for a resonator with fractalperimeter in Lorentz's conjecture. However, in general,
the proper dimension to use turns out to be the Minkowski-Bouligand dimension (Schroeder
1991).
Let
be the area traced out by a small circle
with radius following a fractal curve. Then, providing the limit
exists,
(Schroeder 1991). It is conjectured that for all strictly self-similar fractals, the Minkowski-Bouligand dimension is equal to the Hausdorff
dimension ; otherwise .
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80-94, 1939.Lapidus, M. L. and Fleckinger-Pellé, J. "Tambour
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Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H.
Freeman, pp. 41-45, 1991.