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 .
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 1306,
171-175, 1988.Schroeder, M. Fractals,
Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H.
Freeman, pp. 41-45, 1991.
be the area traced out by a small circle
with radius 
following a fractal curve. Then, providing the limit
exists,
; otherwise 
.
