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,

$$D_M\equiv \lim_{r\to 0} {\ln F(r)\over -\ln r}+2$$

(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

References

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.