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 be the Area traced out by a small Circle with Radius following a fractal curve. Then,
providing the Limit exists,
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.