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Hausdorff dimension

The Hausdorff 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 dimension on well-behaved sets, but it is applicable to many more sets and is not always a natural number.

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 typically have fractional Hausdorff dimension, whence the name. For example, the Cantor set 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).



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