HaussdorffDimension

The Hausdorff dimension is a numeric invariant that measures the size or roughness of a subset of a metric space, generalizing the ordinary notion of dimension to non‑integer values. It is central to fractal geometry and geometric measure theory.

For a subset A of a metric space (X, d), and a nonnegative real s, the s-dimensional Hausdorff measure H^s(A) is defined by taking covers of A with sets U_i of small diameter and summing diam(U_i)^s, then letting the maximal diameter go to zero and taking a limit. The Hausdorff dimension of A is defined as the critical value dim_H(A) = inf{ s ≥ 0 : H^s(A) = 0 } = sup{ s ≥ 0 : H^s(A) = ∞ }. In many cases the threshold is sharp: lower s yield infinite measure, higher s yield zero measure, and at the dimension value the measure is finite and nonzero (though this is not universal).

Key properties include invariance under isometries and, more generally, under bi-Lipschitz mappings. The dimension is monotone with respect to set inclusion and is unchanged by taking closures. Scaling a set by a factor preserves its Hausdorff dimension.

Common examples illustrate its variety: a point has dim_H = 0; a line segment has dim_H = 1; the Cantor middle-thirds set has dim_H = log 2 / log 3 ≈ 0.6309; the Koch curve has dim_H = log 4 / log 3 ≈ 1.2619; the Sierpinski triangle has dim_H = log 3 / log 2 ≈ 1.585.

In practice, dim_H often complements Hausdorff measures and is used to distinguish sets beyond intuitive topological dimensions. It is particularly tractable for self-similar sets satisfying the open set condition, where the dimension solves a simple balance equation involving contraction ratios. See also Hausdorff measure.