Hausdorff Dimension

Informally, self-similar objects with parameters  and  are described by a power law such as

where

is the "dimension" of the scaling law, known as the Hausdorff dimension.

Formally, let  be a subset of a metric space . Then the Hausdorff dimension  of  is the infimum of  such that the -dimensional Hausdorff measure of  is 0 (which need not be an integer).

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).

REFERENCES:

Duvall, P.; Keesling, J.; and Vince, A. "The Hausdorff Dimension of the Boundary of a Self-Similar Tile." J. London Math. Soc. 61, 649-760, 2000.

Federer, H. Geometric Measure Theory. New York: Springer-Verlag, 1969.

Harris, J. W. and Stocker, H. "Hausdorff Dimension." §4.11.3 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 113-114, 1998.

Hausdorff, F. "Dimension und äußeres Maß." Math. Ann. 79, 157-179, 1919.

Ott, E. "Appendix: Hausdorff Dimension." Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 100-103, 1993.

Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, pp. 41-45, 1991.