Hausdorff dimension

In mathematics, the Hausdorff dimension is an extended non-negative real number (that is a number in the closed infinite interval ) associated to any metric space . It was introduced in 1918 by the mathematician Felix Hausdorff. Many of the technical developments used to compute the Hausdorff dimension for highly irregular sets were obtained by Abram Samoilovitch Besicovitch. For this reason, Hausdorff dimension is sometimes referred to as Hausdorff-Besicovitch dimension. It is also less frequently called the capacity dimension or fractal dimension.

Related Topics:
Mathematics - Real number - Metric space - 1918 - Mathematician - Felix Hausdorff - Abram Samoilovitch Besicovitch

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Intuitively, the dimension of a set (for example, a subset of Euclidean space) is the number of independent parameters needed to describe a point in the set. One mathematical concept which closely models this naïve idea is that of topological dimension of a set. For example a point in the plane is described by two independent parameters (the cartesian coordinates of the point), so in this sense, the plane is two-dimensional. As one would expect, topological dimension is always a natural number.

Related Topics:
Subset - Euclidean space - Topological dimension - Cartesian coordinate - Natural number

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However, topological dimension behaves in quite unexpected ways on certain highly irregular sets such as fractals. For example the Cantor set has topological dimension zero, but in some sense it behaves as a higher dimensional space. Hausdorff dimension gives another way to define dimension, which takes the metric into account.

Related Topics:
Fractal - Cantor set

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To define the Hausdorff dimension for X, we first consider the number N(r) of balls of radius at most r required to cover X completely. Clearly, as r gets smaller N(r) gets larger. Very roughly, if N(r) grows in the same way as 1/rd as r is squeezed down towards zero, then we say X has dimension d. In fact the rigorous definition of Hausdorff dimension is somewhat roundabout, since it first defines an entire family of covering measures for X. It turns out that Hausdorff dimension refines the concept of topological dimension and also relates it to other properties of the space such as area or volume.

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It should be noted that there are various closely related notions of possibly fractional dimension. For example box-counting dimension, generalises the idea of counting the squares of graph paper in which a point of X can be found, as the size of the squares is made smaller and smaller. In many cases these notions coincide, but the relation between them is highly technical.

Related Topics:
Box-counting dimension - Graph paper

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