- Hausdorff dimension
In

mathematics , the**Hausdorff dimension**(also known as the**Hausdorff–Besicovitch dimension**) is an extended non-negativereal number associated to anymetric space . The Hausdoff dimension generalizes the notion of the dimension of a realvector 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 themathematician Felix Hausdorff . Many of the technical developments used to compute the Hausdorff dimension for highly irregular sets were obtained byAbram Samoilovitch Besicovitch .**Informal discussion**Intuitively, the dimension of a set (for example, a

subset ofEuclidean 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 (theCartesian coordinate s of the point), so in this sense, the plane is two-dimensional. As one would expect, thetopological dimension is always anatural number .However, topological dimension behaves in quite unexpected ways on certain highly irregular sets such as

fractal s. For example, theCantor 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/"r"^{"d"}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 $X$ 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 offractal s, 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." cite book

last = Mandelbrot

first = Benoît

authorlink = Benoît Mandelbrot

title = The Fractal Geometry of Nature

publisher = W. H. Freeman

series = Lecture notes in mathematics 1358

year = 1982

doi =

isbn = 0716711869]There are various closely related notions of possibly fractional dimension. For example

box-counting dimension , generalizes the idea of counting the squares ofgraph 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 theMinkowski-Bouligand dimension ).Thepacking 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.**Formal definition**Let $X$ be a metric space. If $Ssubset\; X$ and $din\; [0,infty)$, the $d$-dimensonal

**Hausdorff content**of $S$ is defined by:$C\_H^d(S):=infBigl\{sum\_i\; r\_i^d:\; ext\{\; there\; is\; a\; cover\; of\; \}\; S\; ext\{\; by\; balls\; with\; radii\; \}r\_i>0Bigr\}.$In other words, $C\_H^d(S)$ is the infimum of the set of numbers $deltage\; 0$ such that there is some (indexed) collection of balls $\{B(x\_i,r\_i):iin\; I\}$ with $r\_i>0$ for each $iin\; I$ which satisfies $sum\_\{iin\; I\}r\_i^dmath>.(One\; can\; assume,\; with\; no\; loss\; of\; generality,\; that\; the\; index\; set$ I$is\; the\; natural\; numbers$ mathbb\; N$.)\; Here,\; we\; use\; the\; standard\; convention\; thatinf\; \xd8\; =\infty .\; The$**Hausdorff dimension**of $X$ is defined by:$operatorname\{dim\}\_\{operatorname\{H(X):=inf\{dge\; 0:\; C\_H^d(X)=0\}.$Equivalently, $operatorname\{dim\}\_\{operatorname\{H(X)$ may be defined as the infimum of the set of $din\; [0,infty)$ such that the $d$-dimensional

Hausdorff measure of $X$ is zero. This is the same as the supremum of the set of $din\; [0,infty)$ such that the $d$-dimensional Hausdorff measure of $X$ is infinite (except that when this latter set of numbers $d$ is empty the Hausdorff dimension is zero).**Examples*** The

Euclidean space **R**^{"n"}has Hausdorff dimension "n".

* The circle S^{1}has Hausdorff dimension 1.

*Countable set s have Hausdorff dimension 0.

*Fractal s often are spaces whose Hausdorff dimension strictly exceeds thetopological dimension . For example, theCantor set (a zero-dimensional topological space) is a union of two copies of itself, each copy shrunk by a factor 1/3; this fact can be used to prove that its Hausdorff dimension is $ln\; 2/ln\; 3,$ which is approximately $0\{.\}63$ (seenatural logarithm ). TheSierpinski triangle is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of $ln\; 3/\; ln\; 2$, which is approximately $1\{.\}58$.

*Space-filling curve s like the Peano and theSierpiński curve have the same Hausdorff dimension as the space they fill.

* The trajectory ofBrownian motion in dimension 2 and above has Hausdorff dimension 2almost surely .

* An early paper byBenoit Mandelbrot entitled "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension " and subsequent work by other authors have claimed that the Hausdorff dimension of many coastlines can be estimated. Their results have varied from 1.02 for the coastline ofSouth Africa to 1.25 for the west coast ofGreat Britain . However, 'fractal dimensions' of coastlines and many other natural phenomena are largely heuristic and cannot be regarded rigorously as a Hausdorff dimension. It is based on scaling properties of coastlines at a large range of scales, but which does not however include all arbitrarily small scales, where measurements would depend on atomic and sub-atomic structures, and are not well defined.**Properties of Hausdorff dimension****Hausdorff dimension and inductive dimension**Let "X" be an arbitrary separable metric space. There is a topological notion of

inductive dimension for "X" which is defined recursively. It is always an integer (or +∞) and is denoted dim_{ind}("X").**Theorem**. Suppose "X" is non-empty. Then :$operatorname\{dim\}\_\{mathrm\{Haus(X)\; geq\; operatorname\{dim\}\_\{mathrm\{ind(X)$Moreover:$inf\_Y\; operatorname\{dim\}\_\{mathrm\{Haus(Y)\; =operatorname\{dim\}\_\{mathrm\{ind(X)$where "Y" ranges over metric spaceshomeomorphic to "X". In other words, "X" and "Y" have the same underlying set of points and the metric "d"_{"Y"}of "Y" is topologically equivalent to "d"_{"X"}.These results were originally established by

Edward Szpilrajn (1907-1976). The treatment in Chapter VIII of the Hurewicz and Wallman reference is particularly recommended.**Hausdorff dimension and Minkowski dimension**The

Minkowski dimension is similar to the Hausdorff dimension, except that it is not associated with a measure. The Minkowski dimension of a set is at least as large as the Hausdorff dimension. In many situations, they are equal. However, the set of rational points in $[0,1]$ has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.**Hausdorff dimensions and Frostman measures**If there is a measure $mu$ defined on Borel subsets of a metric space $X$ such that $mu(X)>0$ and $mu(B(x,r))le\; r^s$ holds for some constant $s>0$ and for every ball $B(x,r)$ in $X$, then $operatorname\{dim\}\_\{mathrm\{Haus(X)\; geq\; s$. A partial converse is provided by

Frostman's lemma . That article also discusses another useful characterization of the Hausdorff dimension.**Behaviour under unions and products**If $X=igcup\_\{iin\; I\}X\_i$ is a finite or countable union, then :$operatorname\{dim\}\_\{mathrm\{Haus(X)\; =sup\_\{iin\; I\}\; operatorname\{dim\}\_\{mathrm\{Haus(X\_i).$This can be verified directly from the definition.

If $X$ and $Y$ are metric spaces, then the Hausdorff dimension of their product satisfies:$operatorname\{dim\}\_\{mathrm\{Haus(X\; imes\; Y)ge\; operatorname\{dim\}\_\{mathrm\{Haus(X)+\; operatorname\{dim\}\_\{mathrm\{Haus(Y).$An example in which the inequality is strict has been constructed by J. M. Marstrand [

*Marstrand, J. M. The dimension of Cartesian product sets. Proc. Cambridge Philos. Soc. 50, (1954). 198--202.*] . It is known that when $X$ and $Y$ are Borel subsets of $R^n$, the Hausdorff dimension of $X\; imes\; Y$ is bounded from above by the Hausdorff dimension of $X$ plus the upper packing dimension of $Y$. These facts are discussed in Mattila (1995).**elf-similar sets**Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set "E" is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ("E") = "E", although the exact definition is given below.

**Theorem**. Suppose:$psi\_i:\; mathbb\{R\}^n\; ightarrow\; mathbb\{R\}^n,\; quad\; i=1,\; ldots\; ,\; m$

are contractive mappings on

**R**^{"n"}with contraction constant "r"_{"j"}< 1. Then there is a unique "non-empty" compact set "A" such that: $A\; =\; igcup\_\{i=1\}^m\; psi\_i\; (A).$

The theorem follows from

Stefan Banach 's contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets of**R**^{"n"}with theHausdorff distance [*K. J. Falconer, "The Geometry of Fractal Sets", Cambridge University Press, 1985 Theorem 8.3*] .To determine the dimension of the self-similar set "A" (in certain cases), we need a technical condition called the "open set condition" on the sequence of contractions ψ

_{"i"}which is stated as follows: There is a relatively compact open set "V" such that:$igcup\_\{i=1\}^mpsi\_i\; (V)\; subseteq\; V$

where the sets in union on the left are pairwise

disjoint .**Theorem**. Suppose the open set condition holds and each ψ_{"i"}is a similitude, that is a composition of anisometry and adilation around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is "s" where "s" is the unique solution of:$sum\_\{i=1\}^m\; r\_i^s\; =\; 1.$

Note that the contraction coefficient of a similitude is the magnitude of the dilation.

We can use this theorem to compute the Hausdorff dimension of the Sierpinski triangle (or sometimes called Sierpinski gasket). Consider three

non-collinear points "a"_{1}, "a"_{2}, "a"_{3}in the plane**R**² and let ψ_{"i"}be the dilation of ratio 1/2 around "a"_{"i"}. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket and the dimension "s" is the unique solution of:$left(frac\{1\}\{2\}\; ight)^s+left(frac\{1\}\{2\}\; ight)^s+left(frac\{1\}\{2\}\; ight)^s\; =\; 3\; left(frac\{1\}\{2\}\; ight)^s\; =1.$

Taking natural logarithms of both sides of the above equation, we can solve for "s", that is:

:$s\; =\; frac\{ln\; 3\}\{ln\; 2\}.$

The Sierpinski gasket is self-similar. In general a set "E" which is a fixed point of a mapping

: $A\; mapsto\; psi(A)\; =\; igcup\_\{i=1\}^m\; psi\_i(A)$

is self-similar if and only if the intersections

:$H^sleft(psi\_i(E)\; cap\; psi\_j(E)\; ight)\; =0$

where "s" is the Hausdorff dimension of "E" and $H^s$ denotes

Hausdorff measure . This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally:**Theorem**. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.**ee also***

List of fractals by Hausdorff dimension , some examples of deterministic fractals, random and natural fractals**Historical references***

A. S. Besicovitch , "On Linear Sets of Points of Fractional Dimensions",Mathematische Annalen **101**(1929).

*A. S. Besicovitch andH. D. Ursell , "Sets of Fractional Dimensions", Journal of the London Mathematical Society, v12 (1937). Several selections from this volume are reprinted in "Classics on Fractals",ed. Gerald A. Edgar, Addison-Wesley (1993) ISBN 0-201-58701-7 See chapters 9,10,11.

*F. Hausdorff , "Dimension und äußeres Maß", Mathematische Annalen**79**(1–2) (March 1919) pp. 157–179.**Notes****References*** M. Maurice Dodson and Simon Kristensen, [

*http://arxiv.org/abs/math/0305399 "Hausdorff Dimension and Diophantine Approximation"*] (June 12, 2003).

* W. Hurewicz and H. Wallman, "Dimension Theory", Princeton University Press, 1948.

* E. Szpilrajn, "La dimension et la mesure", Fundamenta Mathematica 28, 1937, pp 81-89.

* Citation

last1=Marstrand

first1=J. M. | title=The dimension of cartesian product sets | year=1954 | journal=Proc. Cambridge Philos. Soc.

volume=50

issue=3

pages=198–202

* Citation

last1=Mattila

first1=Pertti | title=Geometry of sets and measures in Euclidean spaces | publisher=Cambridge University Press

isbn=978-0-521-65595-8 | year=1995

*Wikimedia Foundation.
2010.*

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