Hausdorff dimension

The Hausdorff dimension 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 H^d(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 or the d-th powers of these diameters is less than or equal to m.

The Hausdorff dimension d(M) is then defined to be the infimum of all d > 0 such that H^d(M) = 0. The Hausdorff dimension is well-defined for any metric space M and we always have 0 ≤ d(M) ≤ ∞.

Examples

• The Euclidean space Rⁿ has Hausdorff dimension n.
• 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).