Hausdorff Dimension
Version 1.0.0 · Updated 2026-07-28
CORE DEFINITION
Hausdorff dimension, also known as Hausdorff-Besicovitch dimension or fractal dimension, was introduced by the German mathematician Felix Hausdorff in 1918. It allows defining the dimension of any subset of a metric space, including complex sets such as fractals. For simple geometric shapes like lines, rectangles, and cuboids, the Hausdorff dimension equals their usual geometric or topological dimension. In general, the Hausdorff dimension of an object, unlike topological dimension, is not necessarily a natural number but may be a non-integer rational or irrational number.
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Hausdorff dimension, also known as Hausdorff-Besicovitch dimension or fractal dimension, was introduced by the German mathematician Felix Hausdorff in 1918. It allows defining the dimension of any subset of a metric space, including complex sets such as fractals. For simple geometric shapes like lines, rectangles, and cuboids, the Hausdorff dimension equals their usual geometric or topological dimension.
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The dimension can be fractional, describing the rate at which detail increases when an object is magnified (roughness); the coastline dimension is about 1.26, lying between a one-dimensional line and a two-dimensional surface.
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- zh.wikipedia.orghttps://zh.wikipedia.org/wiki/%E8%B1%AA%E6%96%AF%E5%A4%9A%E5%A4%AB%E7%BB%B4%E6%95%B0verified
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