The Hausdorff dimension
(also: Hausdorff-Besicovitch dimension
, capacity dimension
and fractal dimension
), introduced by Felix Hausdorff
, gives a way to accurately measure the dimension
of complicated sets such as fractals
. The Hausdorff dimension agrees with the ordinary (topological) dimension on "well-behaved sets", but it is applicable to many more sets and is not always a natural number
. The Hausdorff dimension should not be confused with the (similar) box-counting dimension.
If M is a metric space, and d > 0 is a real number, then the d-dimensional Hausdorff measure Hd(M) is defined to be the infimum of all m > 0 such that for all r > 0, M can be covered by countably many closed sets of diameter < r and the sum of the d-th powers of these diameters is less than or equal to m.
It turns out that for most values of d, this measure Hd(M) is either 0 or ∞. If d is smaller than the "true dimension" of M, then Hd(M) = ∞; if it is bigger then Hd(M) = 0.
The Hausdorff dimension d(M) is then defined to be the "cutoff point", i.e. the infimum of all d > 0 such that Hd(M) = 0. The Hausdorff dimension is a well-defined real number for any metric space M and we always have 0 ≤ d(M) ≤ ∞.
- The Euclidean space Rn has Hausdorff dimension n.
- The circle S1 has Hausdorff dimension 1.
- Countable sets have Hausdorff dimension 0.
- Fractals are defined to be sets whose Hausdorff dimension strictly exceeds the topological dimension. For example, the Cantor set (a zerodimensional topological space) is a union of two copies of itself, each copy shrunk by a factor 1/3; this fact can be used to prove that its Hausdorff dimension is ln(2)/ln(3) (see natural logarithm). The Sierpinski triangle is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln(3)/ln(2).