Hausdorff dimension

English

Etymology

Introduced in 1918 by mathematician Felix Hausdorff.

Noun

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.

Hypernyms

• fractal dimension

Translations

type of fractal dimension

• Chinese:

  Mandarin: 豪斯多夫維數／豪斯多夫维数 (Háosīduōfū wéishù)

• Czech: Hausdorffova dimenze
• Finnish: Hausdorffin dimensio
• French: dimension de Hausdorff (fr) f
• Greek: διάσταση Χάουσντορφ f (diástasi Cháousntorf)
• Russian: разме́рность Хаусдорфа f (razmérnostʹ Xausdorfa)
• Spanish: dimensión de Hausdorff f