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| [[image:Great Britain Hausdorff.svg|thumb|450px|Estimating the Hausdorff dimension of the [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|coast of Great Britain]]]]
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| In [[mathematics]], the '''Hausdorff dimension''' (also known as the '''Hausdorff–Besicovitch dimension''') is an [[Extended real number line|extended]] non-negative [[real number]] associated with any [[metric space]]. The Hausdorff dimension generalizes the notion of the dimension of a real [[vector space]]. That is, the Hausdorff dimension of an ''n''-dimensional [[inner product space]] equals ''n''. This means, for example, the Hausdorff dimension of a point is zero, the Hausdorff dimension of a line is one, and the Hausdorff dimension of the plane is two. There are, however, many [[fractal|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]].
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| [[Image:Sierpinski deep.svg|thumb|300px|[[Sierpinski triangle]]. A space with fractal dimension log(3)/log(2), which is approximately 1.5849625]]
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| ==Intuition==
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| The intuitive concept of dimension of a geometric object 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 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 take one real number into two both surjectively (so all pairs of numbers are covered) and ''continuously'', so that a one-dimensional object completely fills up a higher dimensional object.
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| 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 [[Lebesgue covering dimension|topological dimension]] explains why. The Lebesgue covering dimension is defined as the minimum number of overlaps that small open balls need to have in order to completely cover the object. When one tries to cover a line by dropping open intervals on it, one always ends up covering some points twice. Covering a plane with disks, one ends up covering some points three times, etc. The topological dimension indicates how many different little balls connect a given point to other points in the space, generically. It indicates how difficult it is to break a geometric object apart into pieces by removing slices.
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| But the topological dimension doesn't determine volumes. 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 as a higher dimensional space. The Hausdorff dimension defines the size notion of dimension, which requires a notion of radius, or ''metric''.
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| Consider the number N(''r'') of [[ball (mathematics)|balls]] of radius at most ''r'' required to cover ''X'' completely. When ''r'' is small, N(''r'') is large. For a "well-behaved" set ''X'', the Hausdorff dimension is the unique number ''d'' such that N(''r'') grows as 1/''r<sup>d</sup>'' as ''r'' approaches zero. The precise definition requires that the dimension "d" so defined is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.
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| 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. But [[Benoît 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 you see around you is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:
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| <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 | authorlink = Benoît Mandelbrot | title = The Fractal Geometry of Nature | publisher = W. H. Freeman | series = Lecture notes in mathematics 1358 | year = 1982 | doi = | isbn = 0-7167-1186-9}}</ref></blockquote>
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| The Hausdorff dimension is a successor to the less sophisticated but in practice very similar [[box-counting dimension]] or [[Minkowski–Bouligand dimension]]. This counts 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. For fractals that occur in nature, the two notions coincide. The [[packing dimension]] is yet another similar notion. These notions (packing dimension, Hausdorff dimension, Minkowski–Bouligand dimension) all give the same value for many shapes, but there are well documented exceptions.
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| ==Formal definitions==
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| ===Hausdorff content===
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| Let ''X'' be a [[metric space]]. If ''S'' ⊂ ''X'' and ''d'' ∈ [0, ∞), the ''d''-dimensional '''Hausdorff content''' of ''S'' is defined by
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| :<math>C_H^d(S):=\inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii }r_i>0\Bigr\}.</math>
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| In other words, <math>C_H^d(S)</math> is the [[infimum]] of the set of numbers δ ≥ 0 such that there is some (indexed) collection of [[metric space|balls]] <math>\{B(x_i,r_i):i\in I\}</math> covering ''S'' with ''r<sub>i</sub>'' > 0 for each ''i'' ∈ ''I'' that satisfies <math>\sum_{i\in I} r_i^d<\delta </math>. (Here, we use the standard convention that [[infimum|inf Ø =∞]].)
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| ===Hausdorff dimension===
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| The '''Hausdorff dimension''' of ''X'' is defined by
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| :<math>\operatorname{dim}_{\operatorname{H}}(X):=\inf\{d\ge 0: C_H^d(X)=0\}.</math>
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| Equivalently, dim<sub>H</sub>(''X'') may be defined as the [[infimum]] of the set of ''d'' ∈ [0, ∞) such that the ''d''-dimensional [[Hausdorff measure]] of ''X'' is zero. This is the same as the supremum of the set of ''d'' ∈ [0, ∞) 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).
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| ==Examples==
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| * The [[Euclidean space]] '''R'''<sup>''n''</sup> has Hausdorff dimension ''n''.
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| * The circle '''S'''<sup>1</sup> has Hausdorff dimension 1.
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| * [[Countable set]]s have Hausdorff dimension 0.
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| * [[Fractal]]s often are spaces whose Hausdorff dimension strictly exceeds the [[topological dimension]]. For example, the [[Cantor 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) ≈ 0.63. The [[Sierpinski 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) ≈ 1.58.
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| * [[Space-filling curve]]s like the [[Peano curve|Peano]] and the [[Sierpiński curve]] have the same Hausdorff dimension as the space they fill.
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| * The trajectory of [[Brownian motion]] in dimension 2 and above has Hausdorff dimension 2 [[almost surely]].
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| * An early paper by [[Benoit 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 of [[South Africa]] to 1.25 for the west coast of [[Great 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; however, it does not include all arbitrarily small scales, where measurements would depend on atomic and sub-atomic structures, and are not well defined.
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| * The bond system of an [[amorphous solid]] changes its Hausdorff dimension from Euclidean 3 below [[glass transition temperature]] ''T''<sub>g</sub> (where the amorphous material is solid), to fractal 2.55±0.05 above ''T''<sub>g</sub>, where the amorphous material is liquid.<ref>{{cite journal |author=M.I. Ojovan, W.E. Lee. |title= Topologically disordered systems at the glass transition|journal=J. Phys.: Condensed Matter |volume=18 |issue= 50|pages=11507–20 |year=2006 |url=http://eprints.whiterose.ac.uk/1958/ |doi=10.1088/0953-8984/18/50/007 |bibcode=2006JPCM...1811507O}}</ref>
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| ==Properties of Hausdorff dimension==
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| === Hausdorff dimension and inductive dimension ===
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| Let ''X'' be an arbitrary [[Separable space|separable]] metric space. There is a [[topology|topological]] notion of [[inductive dimension]] for ''X'' which is defined recursively. It is always an integer (or +∞) and is denoted dim<sub>ind</sub>(''X'').
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| '''Theorem'''. Suppose ''X'' is non-empty. Then
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| :<math> \operatorname{dim}_{\mathrm{Haus}}(X) \geq \operatorname{dim}_{\mathrm{ind}}(X). </math>
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| Moreover,
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| :<math> \inf_Y \operatorname{dim}_{\mathrm{Haus}}(Y) =\operatorname{dim}_{\mathrm{ind}}(X), </math>
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| where ''Y'' ranges over metric spaces [[homeomorphic]] to ''X''. In other words, ''X'' and ''Y'' have the same underlying set of points and the metric ''d''<sub>''Y''</sub> of ''Y'' is topologically equivalent to ''d''<sub>''X''</sub>. | |
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| These results were originally established by [[Edward Szpilrajn]] (1907–1976). The treatment in Chapter VII of the Hurewicz and Wallman reference is particularly recommended.
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| === Hausdorff dimension and Minkowski dimension ===
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| 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 number|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.
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| === Hausdorff dimensions and Frostman measures ===
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| If there is a [[measure (mathematics)|measure]] μ defined on [[Borel measure|Borel]] subsets of a metric space ''X'' such that μ(''X'') > 0 and μ(''B''(''x'', ''r'')) ≤ ''r<sup>s</sup>'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then dim<sub>Haus</sub>(''X'') ≥ ''s''. A partial converse is provided by [[Frostman's lemma]]. [[Frostman's lemma|That article]] also discusses another useful characterization of the Hausdorff dimension.
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| === Behaviour under unions and products ===
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| If <math>X=\bigcup_{i\in I}X_i</math> is a finite or countable union, then
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| :<math> \operatorname{dim}_{\mathrm{Haus}}(X) =\sup_{i\in I} \operatorname{dim}_{\mathrm{Haus}}(X_i).</math>
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| This can be verified directly from the definition.
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| If ''X'' and ''Y'' are metric spaces, then the Hausdorff dimension of their product satisfies<ref>{{cite journal |author=Marstrand, J. M. |title=The dimension of Cartesian product sets |journal=Proc. Cambridge Philos. Soc. |volume=50 |issue=3 |pages=198–202 |year=1954 |doi=10.1017/S0305004100029236 |bibcode = 1954PCPS...50..198M }}</ref>
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| :<math> \operatorname{dim}_{\mathrm{Haus}}(X\times Y)\ge \operatorname{dim}_{\mathrm{Haus}}(X)+ \operatorname{dim}_{\mathrm{Haus}}(Y).</math>
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| This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1.<ref>{{cite book | last = Falconer | first = Kenneth J. | title = Fractal geometry. Mathematical foundations and applications | publisher = John Wiley & Sons, Inc., Hoboken, New Jersey | year = 2003 | doi = | isbn = }}</ref> In the opposite direction, it is known that when ''X'' and ''Y'' are Borel subsets of '''R'''<sup>''n''</sup>, the Hausdorff dimension of ''X'' × ''Y'' is bounded from above by the Hausdorff dimension of ''X'' plus the [[packing dimension|upper packing dimension]] of ''Y''. These facts are discussed in Mattila (1995).
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| ==Self-similar sets==
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| 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.
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| <blockquote>'''Theorem'''. Suppose
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| :<math> \psi_i: \mathbf{R}^n \rightarrow \mathbf{R}^n, \quad i=1, \ldots , m </math>
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| are [[Contraction mapping|contractive]] mappings on '''R'''<sup>''n''</sup> with contraction constant ''r<sub>j</sub>'' < 1. Then there is a unique ''non-empty'' compact set ''A'' such that
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| :<math> A = \bigcup_{i=1}^m \psi_i (A). </math>
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| </blockquote>
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| The theorem follows from [[Stefan Banach]]'s [[Contractive mapping theorem|contractive mapping fixed point theorem]] applied to the complete metric space of non-empty compact subsets of '''R'''<sup>''n''</sup> with the [[Hausdorff distance]].<ref>{{cite book |author=Falconer, K. J. |title=The Geometry of Fractal Sets |publisher=Cambridge University Press |location=Cambridge, UK |year=1985 |isbn=0-521-25694-1 |chapter=Theorem 8.3}}</ref>
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| ===The open set condition===
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| To determine the dimension of the self-similar set ''A'' (in certain cases), we need a technical condition called the ''open set condition'' (OSC) on the sequence of contractions ψ<sub>''i''</sub>.
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| There is a relatively compact open set ''V'' such that
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| :<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math> | |
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| where the sets in union on the left are pairwise [[disjoint sets|disjoint]].
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| The open set condition is a separation condition that ensures the images ψ<sub>''i''</sub>(''V'') do not overlap "too much".
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| '''Theorem'''. Suppose the open set condition holds and each ψ<sub>''i''</sub> is a similitude, that is a composition of an [[isometry]] and a [[dilation (metric space)|dilation]] 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<ref>{{cite journal |author=Tsang, K. Y. |title=Dimensionality of Strange Attractors Determined Analytically |journal=Phys. Rev. Lett. |volume=57|pages=1390–1393 |year=1986 |pmid=10033437 |issue=12 |url=http://prl.aps.org/abstract/PRL/v57/i12/p1390_1 |doi=10.1103/PhysRevLett.57.1390|bibcode = 1986PhRvL..57.1390T }}</ref>
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| :<math> \sum_{i=1}^m r_i^s = 1. </math>
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| The contraction coefficient of a similitude is the magnitude of the dilation.
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| 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''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub> in the plane '''R'''<sup>2</sup> and let ψ<sub>''i''</sub> be the dilation of ratio 1/2 around ''a<sub>i</sub>''. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket and the dimension ''s'' is the unique solution of
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| :<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>
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| Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC. In general a set ''E'' which is a fixed point of a mapping
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| : <math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>
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| is self-similar if and only if the intersections
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| :<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, </math>
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| where ''s'' is the Hausdorff dimension of ''E'' and ''H<sup>s</sup>'' denotes [[Hausdorff measure]]. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally:
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| '''Theorem'''. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.
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| ==Existence of Sets with Prescribed Hausdorff Dimension==
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| By generalizing the construction of the [[Cantor Set]], it is easy to construct subsets of Euclidean spaces with arbitrary Hausdorff dimension (see, for example, [http://classes.yale.edu/fractals/FracAndDim/cantorDims/CantorDims.html this discussion] for dimension 0 < ''r'' < 1, and [http://www.rose-hulman.edu/mathjournal/archives/2006/vol7-n1/paper9/v7n1-9pd.pdf this paper]<ref>Soltanifar, M.(2006) ''On A Sequence of Cantor Fractals'', Rose Hulman Undergraduate Mathematics Journal, Vol 7, No 1, paper 9.</ref> for a discussion of lifting the construction to higher dimension).
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| It is also easy to see that if <math> A \subset \mathbb{R}^n</math> has Hausdorff dimension ''r'', then any translation, rotation, or dilation of ''A'' has the same Hausdorff dimension ''r''. So from a single bounded set of dimension ''r'', we immediately generate continuum-many sets of dimension ''r'' by any of these operations (since the only translation or dilation fixing a bounded set ''A'' is the identity).
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| But it is also possible to show, for any given ''r'' ≥ 0, and integer ''n '' ≥ ''r'', that there are ''at least'' continuum-many subsets of <math> A \subset \mathbb{R}^n</math> of Hausdorff dimension ''r'' which ''cannot'' be obtained from one another by translations, rotations, and dilations. For example, this should be true of the above-cited constructions using generalized Cantor sets, although the cited sources do not verify this.
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| ==See also==
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| * [[List of fractals by Hausdorff dimension]] Examples of deterministic fractals, random and natural fractals.
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| * [[Intrinsic dimension]]
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| * [[Packing dimension]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{cite journal |last1=Dodson |first1=M. Maurice |last2=Kristensen |first2=Simon |title=Hausdorff Dimension and Diophantine Approximation |date=June 12, 2003 |volume=1 |issue=305 |journal=Fractal geometry and applications: a jubilee of Beno\^it Mandelbrot. Part, --347, Proc. Sympos. Pure Math., 72, Part , Amer. Math. Soc., Providence, RI, . |arxiv=math/0305399 |bibcode = 2003math......5399D }}
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| * {{cite book |last1=Hurewicz |first1=Witold |authorlink1=Witold Hurewicz |last2=Wallman |first2=Henry |authorlink2=Henry Wallman |title=Dimension Theory |publisher=Princeton University Press |year=1948 }}
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| * {{cite journal |author=E. Szpilrajn |authorlink=Edward Marczewski |title=La dimension et la mesure |journal=Fundamenta Mathematica |volume=28 |pages=81–9 |year=1937 }}
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| * {{cite journal
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| | last1=Marstrand
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| | first1=J. M. | title=The dimension of cartesian product sets | year=1954 | journal=Proc. Cambridge Philos. Soc.
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| | volume=50
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| | issue=3
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| | pages=198–202
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| | doi=10.1017/S0305004100029236|bibcode = 1954PCPS...50..198M }}
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| * {{Cite book
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| | last1=Mattila
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| | first1=Pertti | title=Geometry of sets and measures in Euclidean spaces | publisher=[[Cambridge University Press]]
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| | isbn=978-0-521-65595-8 | year=1995}}
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| ===Historical===
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| * {{cite journal |author=A. S. Besicovitch |authorlink=A. S. Besicovitch |title=On Linear Sets of Points of Fractional Dimensions |journal=[[Mathematische Annalen]] |volume=101 |year=1929 | doi=10.1007/BF01454831| issue=1 |pages= 161–193}}
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| * {{cite journal |author1=A. S. Besicovitch |authorlink1=A. S. Besicovitch |author2=H. D. Ursell |authorlink2=H. D. Ursell |title=Sets of Fractional Dimensions |journal=Journal of the London Mathematical Society |volume=12 |year=1937 | issue=1 | doi=10.1112/jlms/s1-12.45.18 | pages=18–25 }}<br/>Several selections from this volume are reprinted in {{cite book |author=Edgar, Gerald A. |title=Classics on fractals |publisher=Addison-Wesley |location=Boston |year=1993 |isbn=0-201-58701-7}} See chapters 9,10,11
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| * {{cite journal |author=F. Hausdorff |authorlink=F. Hausdorff |title=Dimension und äußeres Maß |journal=Mathematische Annalen |volume=79 |issue=1–2 |pages=157–179 |date=March 1919 |doi=10.1007/BF01457179}}
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Hausdorff_dimension Hausdorff dimension] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| * [http://www.encyclopediaofmath.org/index.php/Hausdorff_measure Hausdorff measure] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| {{DEFAULTSORT:Hausdorff Dimension}}
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| [[Category:Fractals]]
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| [[Category:Metric geometry]]
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| [[Category:Dimension theory]]
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