Main Page | See live article | Alphabetical index 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) ≤ ∞. Examples 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). This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.