Editing Hausdorff dimension (section)

==Intuition==
{{refimprove section|date=March 2015}}
The intuitive concept of dimension of a geometric object ''X'' is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the [[cardinality]] of the [[real plane]] is equal to the cardinality of the [[real line]] (this can be seen by an [[Cantor's diagonal argument|argument]] involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a [[space-filling curve]] shows that one can even map the real line to the real plane [[Surjective function|surjectively]] (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and ''continuously'', so that a one-dimensional object completely fills up a higher-dimensional object.

Every space-filling curve hits some points multiple times and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension, also called [[Lebesgue covering dimension]], explains why. This dimension is the greatest integer ''n'' such that in every covering of ''X'' by small open balls there is at least one point where ''n''&nbsp;+&nbsp;1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension&nbsp;''n''&nbsp;=&nbsp;1.

But topological dimension is a very crude measure of the local size of a space (size near a point). A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. A [[fractal]] has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space.

The Hausdorff dimension measures the local size of a space taking into account the distance between points, the [[metric space|metric]]. Consider the number ''N''(''r'') of [[ball (mathematics)|balls]] of radius at most ''r'' required to cover ''X'' completely. When ''r'' is very small, ''N''(''r'') grows polynomially with 1/''r''. For a sufficiently well-behaved  ''X'', the Hausdorff dimension is the unique number ''d'' such that N(''r'') grows as 1/''r<sup>d</sup>'' as ''r'' approaches zero. More precisely, this defines the [[Minkowski–Bouligand dimension|box-counting dimension]], which equals the Hausdorff dimension when the value ''d'' is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.

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. But [[Benoit Mandelbrot]] observed that [[fractal]]s, sets with noninteger Hausdorff dimensions, are found everywhere in nature. He observed that the proper idealization of most rough shapes one sees is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:

<blockquote>Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.<ref name="mandelbrot">{{cite book   | last = Mandelbrot   | first = Benoît   | author-link = Benoit Mandelbrot   | title = The Fractal Geometry of Nature   | publisher = W. H. Freeman   | series = Lecture notes in mathematics 1358   | year = 1982   | isbn = 0-7167-1186-9   | url-access = registration   | url = https://archive.org/details/fractalgeometryo00beno   }}</ref></blockquote>

For fractals that occur in nature, the Hausdorff and [[Minkowski–Bouligand dimension|box-counting dimension]] coincide. The [[packing dimension]] is yet another similar notion which gives the same value for many shapes, but there are well-documented exceptions where all these dimensions differ.{{Example needed|s|date=January 2022}}