, the Hausdorff dimension
(also known as the Hausdorff–Besicovitch dimension
) is an extended
non-negative real number
associated to any metric space
. The Hausdoff dimension generalizes the notion of the dimension of a real vector space
. In particular, the Hausdorff dimension of a single point is zero, the Hausdoff dimension of a line is one, the Hausdoff dimension of the plane is two, etc. There are however many irregular sets
that have noninteger Hausdorff dimension. The concept 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
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 naive 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, the topological dimension
is always a natural number
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.
To define the Hausdorff dimension for X as non-negative real number (that is a number in the half-closed infinite interval [0, ∞)), 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, as it allows the covering of by balls of different sizes.
For many shapes that are often considered in mathematics, physics and other disciplines, the Hausdorff dimension is an integer. However, sets with non-integer Hausdorff dimension are important and prevalent. Benoît Mandelbrot, a popularizer of fractals, advocates that most shapes found in nature are fractals with non-integer dimension, explaining that "[c]louds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."
There are various closely related notions of possibly fractional dimension. For example box-counting dimension, generalizes 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. (The box-counting dimension is also called the Minkowski-Bouligand dimension).
The packing dimension is yet another notion of dimension admitting fractional values.
These notions (packing dimension, Hausdorff dimension, Minkowski-Bouligand dimension) all give the same value for many shapes, but there are well documented exceptions.
be a metric space. If
-dimensonal Hausdorff content
is defined by
In other words,
is the infimum of the set of numbers
such that there is some (indexed) collection of balls