Hausdorff dimension

Introduced in 1918 by mathematician Felix Hausdorff.

Hausdorff dimension (plural Hausdorff dimensions)

1. (mathematical analysis) A type of fractal dimension, a real-valued measure of a geometric object that assigns 1 to a line segment, 2 to a square and 3 to a cube. Formally, given a metric space X and a subset of X labeled S, the Hausdorff dimension of S is the infimum of all real-valued d for which the d-dimensional Hausdorff content of S is zero.
   • If S is nonempty then if the d-dimensional Hausdorff content of S is zero then d is larger than the Hausdorff dimension of S, and if the d-dimensional Hausdorff content of S is infinite then d is smaller or equal to the Hausdorff dimension of S. If the d-dimensional Hausdorff content of S is finite and positive then d is equal to the Hausdorff dimension of S.

• fractal dimension