Hausdorff dimension

Noun

Meaning

Hausdorff dimension (plural Hausdorff dimensions)

(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.

Examples

However, 'fractal dimensions' of coastlines and many other natural phenomena are largely heuristic and cannot be regarded rigorously as a Hausdorff dimension. Source: Internet

For example, Shishikura proves that, for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has Hausdorff dimension two, and then transfers this information to the parameter plane. citation. Source: Internet

For a sufficiently well-behaved X, the Hausdorff dimension is the unique number d such that N(r) grows as 1/r d as r approaches zero. Source: Internet

For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer agreeing with the topological dimension. Source: Internet

Hausdorff dimension and Minkowski dimension The Minkowski dimension is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. Source: Internet

If X and Y are non-empty metric spaces, then the Hausdorff dimension of their product satisfies citation : This inequality can be strict. Source: Internet

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