Minkowski-Bouligand Dimension

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).

Let F(r) be the area traced out by a small circle with radius r 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 D; otherwise D_M>D.

See also

Hausdorff Dimension, Minkowski Cover, Minkowski Sausage

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Berry, M. V. "Diffractals." J. Phys. A12, 781-797, 1979.Hunt, F. V.; Beranek, L. L.; and Maa, D. Y. "Analysis of Sound Decay in Rectangular Rooms." J. Acoust. Soc. Amer. 11, 80-94, 1939.Lapidus, M. L. and Fleckinger-Pellé, J. "Tambour fractal: vers une résolution de la conjecture de Weyl-Berry pour les valeurs propres du laplacien." Compt. Rend. Acad. Sci. Paris Math. Sér 1 306, 171-175, 1988.Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, pp. 41-45, 1991.

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Minkowski-Bouligand Dimension

Cite this as:

Weisstein, Eric W. "Minkowski-Bouligand Dimension." From MathWorld--A Wolfram Web Resource.

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