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Hausdorff Dimension


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

 N=s^d,

where

 d=(lnN)/(lns)

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

Formally, let A be a subset of a metric space X. Then the Hausdorff dimension D(A) of A is the infimum of d>=0 such that the d-dimensional Hausdorff measure of A 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).


See also

Capacity Dimension, Fractal, Fractal Dimension, Minkowski-Bouligand Dimension, Self-Similarity

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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.

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Hausdorff Dimension

Cite this as:

Weisstein, Eric W. "Hausdorff Dimension." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HausdorffDimension.html

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