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Moved page from wikipedia:en:Hausdorff dimension (history)
此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
{{short description|Invariant}}
[[File:KochFlake.svg|thumb|280px|Example of non-integer dimensions. The first four [[iteration]]s of the [[Koch snowflake|Koch curve]], where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses this scale factor (3) and the number of self-similar objects (4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26.<ref name=CampbellAnnenberg15>MacGregor Campbell, 2013, "5.6 Scaling and the Hausdorff Dimension," at ''Annenberg Learner:MATHematics illuminated'', see [http://www.learner.org/courses/mathilluminated/units/5/textbook/06.php], accessed 5 March 2015.</ref> That is, while the Hausdorff dimension of a single [[point (geometry)|point]] is zero, of a [[line segment]] is 1, of a [[square]] is 2, and of a [[cube]] is 3, for [[fractal]]s such as this, the object can have a non-integer dimension.]]
Example of non-integer dimensions. The first four [[iterations of the Koch curve, where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses this scale factor (3) and the number of self-similar objects (4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26. That is, while the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3, for fractals such as this, the object can have a non-integer dimension.]]
非整数维度示例。前四个[ Koch 曲线的迭代，在每次迭代后，所有原始线段都被替换为四个，每个自相似的复制是原始线段长度的1 / 3。豪斯多夫维数的一个形式使用这个比例因子(3)和自相似物体的数量(4)来计算维度，d，在第一次迭代后为 d (log n) / (log s)(log 4) / (log 3)1.26。也就是说，当一个点的豪斯多夫维数为零，线段为1，正方形为2，立方体为3时，对于像这样的分形，物体可以有一个非整数维度
In [[mathematics]], '''Hausdorff dimension''' is a measure of ''roughness'', or more specifically, [[fractal dimension]], that was first introduced in 1918 by [[mathematician]] [[Felix Hausdorff]].<ref>{{Cite journal |arxiv = 1101.1444|doi = 10.1214/11-STS370|title = Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data|journal = Statistical Science|volume = 27|issue = 2|pages = 247–277|year = 2012|last1 = Gneiting|first1 = Tilmann|last2 = Ševčíková|first2 = Hana|last3 = Percival|first3 = Donald B.}}</ref> For instance, the Hausdorff dimension of a single [[point (geometry)|point]] is zero, of a [[line segment]] is 1, of a [[square]] is 2, and of a [[cube]] is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an [[integer]] agreeing with the usual sense of dimension, also known as the [[Inductive dimension|topological dimension]]. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of [[scaling (geometry)|scaling]] and [[self-similarity]], one is led to the conclusion that particular objects—including [[fractal]]s—have non-integer Hausdorff dimensions. Because of the significant technical advances made by [[Abram Samoilovitch Besicovitch]] allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the ''Hausdorff–Besicovitch dimension.''
In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.
在数学中，豪斯多夫维数是一种粗糙度的度量单位，或者更确切地说，分形维数，是由数学家 Felix Hausdorff 在1918年首次提出的。例如，单点的豪斯多夫维数为零，线段为1，正方形为2，立方体为3。也就是说，对于定义了一个光滑形状或一个有少数几个角的形状---- 传统几何学和科学的形状---- 的点集来说，豪斯多夫维数是一个整数，符合通常的维度意义，也称为拓扑维度。然而，还有一些公式允许计算其他不太简单的物体的维数，其中仅仅根据它们的标度和自相似性质，就可以得出结论，特定的物体ー包括分形ー具有非整数的 Hausdorff 维数。由于阿布拉姆·萨莫伊洛维奇·贝西科维奇的重大技术进步，允许计算高度不规则或“粗糙”集的维度，这个维度也通常被称为 Hausdorff-Besicovitch 维度。
The Hausdorff dimension, more specifically, is a further dimensional number associated with a given set, where the distances between all members of that set are defined. Such a set is termed a [[metric space]]. The dimension is drawn from the [[Extended real number line|extended real numbers]], <math>\overline{\mathbb{R}}</math>, as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.
The Hausdorff dimension, more specifically, is a further dimensional number associated with a given set, where the distances between all members of that set are defined. Such a set is termed a metric space. The dimension is drawn from the extended real numbers, <math>\overline{\mathbb{R}}</math>, as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.
更具体地说，豪斯多夫维数是一个与给定集合相关联的更进一步的维数，其中定义了该集合所有成员之间的距离。这样的集合称为度量空间。维数是从扩展的实数，math overline { mathbb { r } / math，而不是更直观的维数概念，它不与一般的度量空间相关联，只接受非负整数的值。
In mathematical terms, 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 underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that [[fractal|irregular sets]] can have noninteger Hausdorff dimensions. For instance, the [[Koch snowflake]] shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new [[equilateral]] triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4.<ref>Larry Riddle, 2014, "Classic Iterated Function Systems: Koch Snowflake", Agnes Scott College e-Academy (online), see [http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm], accessed 5 March 2015.</ref> That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original.<ref name=CampbellAnnenberg15/> Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=S<sup>D</sup>.<ref name=ClaytonSCTPLS96>Keith Clayton, 1996, "Fractals and the Fractal Dimension," ''Basic Concepts in Nonlinear Dynamics and Chaos'' (workshop), Society for Chaos Theory in Psychology and the Life Sciences annual meeting, June 28, 1996, Berkeley, California, see [http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Workshop.html], accessed 5 March 2015.</ref> This equation is easily solved for D, yielding the ratio of logarithms (or [[natural logarithm]]s) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.
In mathematical terms, 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 underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that irregular sets can have noninteger Hausdorff dimensions. For instance, the Koch snowflake shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new equilateral triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4. That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original. This equation is easily solved for D, yielding the ratio of logarithms (or natural logarithms) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.
用数学术语来说，豪斯多夫维数概括了实向量空间维数的概念。也就是说，n 维内积空间的豪斯多夫维数等于 n。 这就是早期声明的基础，一个点的豪斯多夫维数是零，一条线是一，等等，不规则集可以有非整数的 Hausdorff 维数。例如，右边所示的 Koch 雪花是由一个正三角形构成的; 在每次迭代中，它的组成线段被分成单位长度的3段，新创建的中间线段被用作一个指向外部的新正三角形的基础，然后这个基础线段被删除以保留单位长度4的迭代中的最终对象。也就是说，在第一次迭代之后，每个原始线段都被替换为 n4，其中每个自相似拷贝的长度是原始线段的1 / s 1 / 3。这个方程很容易求解为 d，产生出现在图形中的对数(或自然对数)的比率，并给出ー在 Koch 和其他分形情况下ー这些对象的非整数维数。
The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or [[Minkowski–Bouligand dimension]].
The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension.
豪斯多夫维数是更简单但通常等价的计盒维数或 Minkowski-Bouligand 维数的继承者。
==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.
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 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 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.
几何物体 x 的直观尺寸概念就是一个人需要多少个独立参数才能挑出一个独特的点。但是，任何由两个参数指定的点都可以由一个参数指定，因为实际平面的基数等于实际行的基数(这可以通过一个参数看到，该参数涉及交织两个数字的数字以产生一个编码相同信息的单个数字)。皮亚诺曲线的例子表明，一个人甚至可以映射实际线到真正的平面满意地(把一个实数转换成一对实数，这样所有的数对都被覆盖)和连续，所以一维物体完全填充了一个高维物体。
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 ''n'' if, in every covering of ''X'' by small open balls, there is at least one point where ''n'' + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension ''n'' = 1.
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 n if, in every covering of X by small open balls, there is at least one point where n + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension n = 1.
每条空间填充曲线都会多次撞击某些点，且不存在连续的倒数。将两个维度以连续和连续可逆的方式映射到一个维度是不可能的。拓扑维度，也被称为拓朴维数，解释了为什么。这个维度是 n，如果在 x 的每个小开球覆盖中，至少有一个点 n + 1个球重叠。例如，当一个点覆盖一条具有短开区间的直线时，某些点必须被覆盖两次，给出维数 n 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.
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.
但是，拓扑维度是对空间局部尺寸(点附近的尺寸)的一个非常粗略的度量。一条几乎是空间填充的曲线仍然可以有一维拓扑，即使它填充了一个区域的大部分面积。分形具有整数的拓扑维数，但就其所占空间的数量而言，它表现得像一个更高维的空间。
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.
The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric. Consider the number N(r) of 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 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.
豪斯多夫维数测量一个空间的局部大小，考虑到点之间的距离，度量。考虑半径最大为 r 的球数 n (r) ，需要完全覆盖 x。当 r 很小时，n (r)以1 / r 增长多项式。对于一个表现足够好的 x，豪斯多夫维数是唯一的数 d，这样当 r 趋近于零时 n (r)增长为1 / r sup d / sup。更确切地说，这定义了盒子计数维度，当值 d 是不足以覆盖空间的增长率和过度充裕的增长率之间的关键边界时，它等于豪斯多夫维数。
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 you see around you is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:
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 fractals, 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:
对于光滑的形状，或者有少量棱角的形状，传统几何和科学的形状，豪斯多夫维数是一个整数，与拓扑维度一致。但是本华·曼德博观察到分形---- 具有非整数 Hausdorff 维数的集合---- 在自然界中随处可见。他观察到，你周围大多数粗糙形状的理想化不是光滑的理想化形状，而是分形理想化形状:
<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 = Benoit 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 | url-access = registration | url = https://archive.org/details/fractalgeometryo00beno }}</ref></blockquote>
<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.</blockquote>
云不是球体，山不是锥体，海岸线不是圆圈，树皮不平滑，闪电也不是直线运动。 / 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.
For fractals that occur in nature, the Hausdorff and 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.
对于自然界中出现的分形，Hausdorff 维数和盒计数维数是一致的。封装尺寸是另一个类似的概念，它给出了许多形状相同的值，但是在所有这些尺寸不同的情况下，有很好的文档说明的例外。
==Formal definitions==
{{unreferenced section|date=March 2015}}
===Hausdorff content===
Let ''X'' be a [[metric space]]. If ''S'' ⊂ ''X'' and ''d'' ∈ [0, ∞), the ''d''-dimensional '''unlimited Hausdorff content''' of ''S'' is defined by
Let X be a metric space. If S ⊂ X and d ∈ [0, ∞), the d-dimensional unlimited Hausdorff content of S is defined by
设 x 是度量空间。若 s something x 和 d ∈[0，∞) ，则 s 的 d 维无限 Hausdorff 内容定义为
:<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>
<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>
数学 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
In other words, <math>C_H^d(S)</math> is the [[infimum]] of the set of numbers <math>\delta \geq 0</math> such that there is some (indexed) collection of [[ball (mathematics)|ball]]s <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 = ∞]].)
In other words, <math>C_H^d(S)</math> is the infimum of the set of numbers <math>\delta \geq 0</math> such that there is some (indexed) collection of 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 inf Ø = ∞.)
换句话说，math c h ^ d (s) / math 是数字集合 math delta geq 0 / math 的下确界，使得在 i / math 中存在一些球集合 math { b (xi，ri) : i 包含 s，对于每个 i ∈ i，r 子 i / sub 0满足 i 中的数学和 i ^ d delta / math。(在这里，我们使用 inf ∞ .)的标准约定
===Hausdorff measurement===
The Hausdorff outer measure is different from the unbounded Hausdorf content in that rather than considering all possible coverings of ''S'', we see what happens when the sizes of the balls go to zero. This is for <math>d \geq 0 </math>, we define the ''d''-dimensional Hausdorff outer measure of ''S'' as
The Hausdorff outer measure is different from the unbounded Hausdorf content in that rather than considering all possible coverings of S, we see what happens when the sizes of the balls go to zero. This is for <math>d \geq 0 </math>, we define the d-dimensional Hausdorff outer measure of S as
豪斯多夫外测度不同于无界的豪斯多夫内容，因为我们不考虑 s 的所有可能覆盖，我们看到当球的大小为零时会发生什么。这是为了数学 d geq 0 / math，我们定义了 s 的 d 维 Hausdorff 外测度为
:<math> \mathcal{H}^d(S):=\lim_{r \to 0} \inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii } 0 < r_i < r\Bigr\}.</math>
<math> \mathcal{H}^d(S):=\lim_{r \to 0} \inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii } 0 < r_i < r\Bigr\}.</math>
数学函数 h ^ d (s) : lim { r to 0} inf Bigl sum i r i ^ d: text {有一个{ s text by balls with radii }0 r i r Bigr }的封面。 数学
===Hausdorff dimension===
The '''Hausdorff dimension''' of ''X'' is defined by
The Hausdorff dimension of X is defined by
X 的豪斯多夫维数定义为
:<math>\dim_{\operatorname{H}}(X):=\inf\{d\ge 0: \mathcal{H}^d(X)=0\}.</math>
<math>\dim_{\operatorname{H}}(X):=\inf\{d\ge 0: \mathcal{H}^d(X)=0\}.</math>
Math operatorname { h }(x) : inf ge 0: mathcal { h } ^ d (x)0} . / math
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).
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).
等价地，dim 子 h / sub (x)可定义为 d ∈[0，∞)集的下确界，使得 x 的 d 维 Hausdorff 测度为零。这与 d ∈[0，∞)的集合的上确界相同，因此 x 的 d 维 Hausdorff 测度是无限的(除非后一个集合 d 是空的，豪斯多夫维数为零)。
==Examples==
[[Image:Sierpinski deep.svg|thumb|250px|Dimension of a further [[fractal]] example. The [[Sierpinski triangle]], an object with Hausdorff dimension of log(3)/log(2)≈1.58.<ref name=ClaytonSCTPLS96/>]]
Dimension of a further [[fractal example. The Sierpinski triangle, an object with Hausdorff dimension of log(3)/log(2)≈1.58.]]
进一步的维数[分形的例子。谢尔宾斯基三角形，一个豪斯多夫维数为3 / log (2)≈1.58的物体]
* [[Countable set]]s have Hausdorff dimension 0.<ref name="schleicher">{{cite journal |last1=Schleicher |first1=Dierk |title=Hausdorff Dimension, Its Properties, and Its Surprises |journal=The American Mathematical Monthly |date=June 2007 |volume=114 |issue=6 |pages=509–528 |doi=10.1080/00029890.2007.11920440 |language=en |issn=0002-9890|arxiv=math/0505099 }}</ref>
* The [[Euclidean space]] ℝ<sup>''n''</sup> has Hausdorff dimension ''n'', and the circle '''S'''<sup>1</sup> has Hausdorff dimension 1.<ref name="schleicher" />
* [[Fractal]]s often are spaces whose Hausdorff dimension strictly exceeds the [[topological dimension]].<ref name="mandelbrot" /> 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; hence, it can be shown that its Hausdorff dimension is ln(2)/ln(3) ≈ 0.63.<ref>{{cite book | last=Falconer | first = Kenneth |title=Fractal Geometry: Mathematical Foundations and Applications | publisher=[[John Wiley and Sons]] | edition=2nd | year=2003}}</ref> 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.<ref name=CampbellAnnenberg15/> These Hausdorff dimensions are related to the "critical exponent" of the [[Master theorem (analysis of algorithms)|Master theorem]] for solving [[Recurrence relation|recurrence relations]] in the [[analysis of algorithms]].
* [[Space-filling curve]]s like the [[Peano curve]] have the same Hausdorff dimension as the space they fill.
* The trajectory of [[Brownian motion]] in dimension 2 and above is conjectured to be Hausdorff dimension 2.<ref>{{cite book | last=Morters | first=Peres | title= Brownian Motion | publisher=[[Cambridge University Press]] | year=2010 }}</ref>
[[image:Great Britain Hausdorff.svg|thumb|250px|Estimating the Hausdorff dimension of the [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|coast of Great Britain]]]]
coast of Great Britain]]
大不列颠海岸]
* [[Lewis Fry Richardson]] has performed detailed experiments to measure the approximate Hausdorff dimension for various coastlines. His results have varied from 1.02 for the coastline of [[South Africa]] to 1.25 for the west coast of [[Great Britain]].<ref name="mandelbrot" />
==Properties of Hausdorff dimension==
{{refimprove section|date=March 2015}}
=== Hausdorff dimension and inductive dimension ===
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'').
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<sub>ind</sub>(X).
设 x 是任意可分度量空间。对于 x 有一个递归定义的归纳维数拓扑概念。它总是一个整数(或 + ∞) ，并且表示 dim sub ind / sub (x)。
'''Theorem'''. Suppose ''X'' is non-empty. Then
Theorem. Suppose X is non-empty. Then
定理。假设 x 是非空的。然后
:<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). </math>
<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). </math>
<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X).数学
Moreover,
Moreover,
此外,
:<math> \inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(X), </math>
<math> \inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(X), </math>
(y) dim { operatorname { Haus }(y) dim { operatorname { ind }(x) ，/ math
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>.
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>.
其中 y 是度量空间同胚到 x 的范围。换句话说，x 和 y 具有相同的基本点集，y 的度规 d 子 y / 子拓扑等价于 d 子 x / 子。
These results were originally established by [[Edward Szpilrajn]] (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.{{full citation needed|date=March 2015}}
These results were originally established by Edward Szpilrajn (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.
这些结果最初是由 Edward Szpilrajn (1907-1976)建立的，例如，见 Hurewicz 和 Wallman，第七章。
=== Hausdorff dimension and Minkowski dimension ===
The [[Minkowski dimension]] is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. 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.
The Minkowski dimension is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. 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.
闵可夫斯基维度与豪斯多夫维数相似，至少和它一样大，而且在许多情况下是相等的。然而，[0,1]中有理点集的豪斯多夫维数为0，Minkowski 维数为1。还有一些紧集的 Minkowski 维数严格大于豪斯多夫维数。
=== Hausdorff dimensions and Frostman measures ===
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]].{{citation needed|date=March 2015}}<ref>This Wikipedia article also discusses further useful characterizations of the Hausdorff dimension.{{clarify|date=March 2015}}</ref>
If there is a measure μ defined on 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.
如果在度量空间 x 的 Borel 子集上定义一个测度，使得(x)0和(b (x，r))≤ rsup s / sup 对于某个常数 s0和 x 中的每个球 b (x，r)成立，则 dim sub Haus / sub (x)≥ s。 部分逆向由霜人引理提供。
=== Behaviour under unions and products ===
If <math>X=\bigcup_{i\in I}X_i</math> is a finite or countable union, then
If <math>X=\bigcup_{i\in I}X_i</math> is a finite or countable union, then
如果 math x x i / math 是一个有限或可数的联合，则
:<math> \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i).</math>
<math> \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i).</math>
(x) sup { i }{ operatorname { Haus }(xi) . / math
This can be verified directly from the definition.
This can be verified directly from the definition.
这可以直接从定义中验证。
If ''X'' and ''Y'' are non-empty 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>
If X and Y are non-empty metric spaces, then the Hausdorff dimension of their product satisfies
如果 x 和 y 是非空度量空间，那么它们产品的豪斯多夫维数满足
:<math> \dim_{\operatorname{Haus}}(X\times Y)\ge \dim_{\operatorname{Haus}}(X)+ \dim_{\operatorname{Haus}}(Y).</math>
<math> \dim_{\operatorname{Haus}}(X\times Y)\ge \dim_{\operatorname{Haus}}(X)+ \dim_{\operatorname{Haus}}(Y).</math>
(x 乘以 y) ge dim { operatorname { Haus }(x) + dim { operatorname { Haus }(y) . / math
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).
This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1. 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 upper packing dimension of Y. These facts are discussed in Mattila (1995).
这种不平等可以是严格的。有可能找到两个维数为0的集合，其乘积的维数为1。在相反的方向上，我们知道当 x 和 y 是 r sup n / sup 的 Borel 子集时，x y 的豪斯多夫维数从上面以 x 的豪斯多夫维数加上 y 的填充维数为界。Mattila (1995)讨论了这些事实。
==Self-similar sets==
{{refimprove section|date=March 2015}}
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.
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.
许多由自相似条件定义的集合具有可以显式确定的维数。粗略地说，集合 e 是自相似的，如果它是集值变换的不动点，即(e) e，尽管下面给出了确切的定义。
<blockquote>'''Theorem'''. Suppose
<blockquote>Theorem. Suppose
块引定理。假设
:<math> \psi_i: \mathbf{R}^n \rightarrow \mathbf{R}^n, \quad i=1, \ldots , m </math>
<math> \psi_i: \mathbf{R}^n \rightarrow \mathbf{R}^n, \quad i=1, \ldots , m </math>
Math psi i: mathbf { r } ^ n right tarrow mathbf { r } ^ n， quad i 1， ldots，m / math
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
are 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
Rsup n / sup 上的压缩常数 r 子 j / 子1的压缩映射。然后有一个唯一的非空紧集
:<math> A = \bigcup_{i=1}^m \psi_i (A). </math>
<math> A = \bigcup_{i=1}^m \psi_i (A). </math>
数学 a bigcup { i 1} ^ m psi i (a)。数学
</blockquote>
</blockquote>
/ blockquote
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>
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<sup>n</sup> with the Hausdorff distance.
这个定理来源于 Stefan Banach 的压缩映射不动点定理，该定理应用于 r sup n / sup 的非空紧子集的完备空间豪斯多夫距离。
===The open set condition===
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>.
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>.
为了确定自相似集 a 的维数(在某些情况下) ，我们需要一个称为开集条件(OSC)的技术条件。
There is a relatively compact open set ''V'' such that
There is a relatively compact open set V such that
有一个相对紧的开集 v
:<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
数学大杯{ i } ^ m psi (v) subseteq v，/ math
where the sets in union on the left are pairwise [[disjoint sets|disjoint]].
where the sets in union on the left are pairwise disjoint.
左边的集合是成对不相交的。
The open set condition is a separation condition that ensures the images ψ<sub>''i''</sub>(''V'') do not overlap "too much".
The open set condition is a separation condition that ensures the images ψ<sub>i</sub>(V) do not overlap "too much".
开集条件是保证图像子 i / 子(v)不重叠“太多”的分离条件。
'''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 | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>
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 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
定理。假设开集条件成立，并且每个子 i / sub 是一个相似量，这是一个等距和围绕某一点的膨胀的合成。那么唯一的不动点是一个集合，它的豪斯多夫维数是 s，其中 s 是 s 的唯一解
:<math> \sum_{i=1}^m r_i^s = 1. </math>
<math> \sum_{i=1}^m r_i^s = 1. </math>
数学1 ^ m r i ^ s 1。数学
The contraction coefficient of a similitude is the magnitude of the dilation.
The contraction coefficient of a similitude is the magnitude of the dilation.
相似物的收缩系数就是膨胀的大小。
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
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
我们可以使用这个定理来计算豪斯多夫维数谢尔宾斯基三角形的密封圈(或者有时候叫做 Sierpinski 垫圈)。考虑平面上的三个非共线点，一个子1 / 子，一个子2 / 子，一个子3 / 子。对应映射的唯一非空不动点是一个 Sierpinski 垫片，其维数 s 是对应映射的唯一解
:<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>
<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>
数学左(压缩{1}{2}右) ^ s + 左(压缩{1}右) ^ s + 左(压缩{1}右) ^ s 3左(压缩{1}右) ^ s 1。数学
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
Taking natural logarithms 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
取上述方程两边的自然对数，我们可以求出 s，即: s ln (3) / ln (2)。该密封垫具有自相似性，满足 OSC 要求。一般来说，集合 e 是一个映射的不动点
: <math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>
<math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>
数学 a 映射 / 压缩 / 压缩 / 压缩
is self-similar if and only if the intersections
is self-similar if and only if the intersections
是自相似的当且仅当
:<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, </math>
<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, </math>
数学 h ^ s 左( psi i (e) cap psi j (e)右)0，/ math
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:
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:
其中 s 是 e 的豪斯多夫维数，h sup s / sup 表示 Hausdorff 测度。对于 Sierpinski 垫圈(交叉点就是点)来说，这一点很明显，但更普遍的情况是:
'''Theorem'''. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.
Theorem. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.
定理。在与前一定理相同的条件下，其唯一不动点是自相似的。
==See also==
* [[List of fractals by Hausdorff dimension]] Examples of deterministic fractals, random and natural fractals.
* [[Assouad dimension]], another variation of fractal dimension that, like Hausdorff dimension, is defined using coverings by balls
* [[Intrinsic dimension]]
* [[Packing dimension]]
* [[Fractal dimension]]
==References==
{{reflist}}
==Further reading==
* {{cite book |last1=Dodson |first1=M. Maurice |title=Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot |volume=72 |issue=1 |pages=305–347 |last2=Kristensen |first2=Simon |chapter=Hausdorff Dimension and Diophantine Approximation |date=June 12, 2003 |arxiv=math/0305399 |bibcode = 2003math......5399D |doi=10.1090/pspum/072.1/2112110|series=Proceedings of Symposia in Pure Mathematics |isbn=9780821836378 }}
* {{cite book |last1=Hurewicz |first1=Witold |authorlink1=Witold Hurewicz |last2=Wallman |first2=Henry |authorlink2=Henry Wallman |title=Dimension Theory |url=https://archive.org/details/in.ernet.dli.2015.84609 |publisher=Princeton University Press |year=1948 }}
* {{cite journal |author=E. Szpilrajn |authorlink=Edward Marczewski |title=La dimension et la mesure |journal=Fundamenta Mathematicae |volume=28 |pages=81–9 |year=1937 }}
* {{cite journal
| last1=Marstrand
| last1=Marstrand
最后一站马斯特兰德
| first1=J. M. | title=The dimension of cartesian product sets | year=1954 | journal=Proc. Cambridge Philos. Soc.
| first1=J. M. | title=The dimension of cartesian product sets | year=1954 | journal=Proc. Cambridge Philos. Soc.
首先1 j。笛卡儿积集的尺寸 | 1954年 | 期刊 Proc。Cambridge Philos.Soc.
| volume=50
| volume=50
第50卷
| issue=3
| issue=3
第三期
| pages=198–202
| pages=198–202
第198-202页
| doi=10.1017/S0305004100029236|bibcode = 1954PCPS...50..198M }}
| doi=10.1017/S0305004100029236|bibcode = 1954PCPS...50..198M }}
10.1017 / S0305004100029236 | bibcode 1954PCPS... 50. . 198 m }
* {{Cite book
| last1=Mattila
| last1=Mattila
最后1个 Mattila
| first1=Pertti | author1-link=Pertti Mattila| title=Geometry of sets and measures in Euclidean spaces | publisher=[[Cambridge University Press]]
| first1=Pertti | author1-link=Pertti Mattila| title=Geometry of sets and measures in Euclidean spaces | publisher=Cambridge University Press
| first1 Pertti | author1-link Pertti Mattila | title 欧几里德空间中集合和度量的几何 | 出版商剑桥大学出版社
| isbn=978-0-521-65595-8 | year=1995}}
| isbn=978-0-521-65595-8 | year=1995}}
| isbn 978-0-521-65595-8 | year 1995}
* {{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}}
* {{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
* {{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|hdl=10338.dmlcz/100363 |url=http://dml.cz/bitstream/handle/10338.dmlcz/100363/CzechMathJ_09-1959-3_5.pdf }}
* {{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}
*{{cite book | last=Falconer | first = Kenneth |title=Fractal Geometry: Mathematical Foundations and Applications | publisher=[[John Wiley and Sons]] | edition=2nd | year=2003}}
==External links==
* [https://www.encyclopediaofmath.org/index.php/Hausdorff_dimension Hausdorff dimension] at [https://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
* [https://www.encyclopediaofmath.org/index.php/Hausdorff_measure Hausdorff measure] at [https://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
{{DEFAULTSORT:Hausdorff Dimension}}
[[Category:Fractals]]
Category:Fractals
分类: 分形
[[Category:Metric geometry]]
Category:Metric geometry
分类: 米制几何
[[Category:Dimension theory]]
Category:Dimension theory
范畴: 维度理论
<noinclude>
<small>This page was moved from [[wikipedia:en:Hausdorff dimension]]. Its edit history can be viewed at [[分形维数/edithistory]]</small></noinclude>
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{{short description|Invariant}}
[[File:KochFlake.svg|thumb|280px|Example of non-integer dimensions. The first four [[iteration]]s of the [[Koch snowflake|Koch curve]], where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses this scale factor (3) and the number of self-similar objects (4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26.<ref name=CampbellAnnenberg15>MacGregor Campbell, 2013, "5.6 Scaling and the Hausdorff Dimension," at ''Annenberg Learner:MATHematics illuminated'', see [http://www.learner.org/courses/mathilluminated/units/5/textbook/06.php], accessed 5 March 2015.</ref> That is, while the Hausdorff dimension of a single [[point (geometry)|point]] is zero, of a [[line segment]] is 1, of a [[square]] is 2, and of a [[cube]] is 3, for [[fractal]]s such as this, the object can have a non-integer dimension.]]
Example of non-integer dimensions. The first four [[iterations of the Koch curve, where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses this scale factor (3) and the number of self-similar objects (4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26. That is, while the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3, for fractals such as this, the object can have a non-integer dimension.]]
非整数维度示例。前四个[ Koch 曲线的迭代，在每次迭代后，所有原始线段都被替换为四个，每个自相似的复制是原始线段长度的1 / 3。豪斯多夫维数的一个形式使用这个比例因子(3)和自相似物体的数量(4)来计算维度，d，在第一次迭代后为 d (log n) / (log s)(log 4) / (log 3)1.26。也就是说，当一个点的豪斯多夫维数为零，线段为1，正方形为2，立方体为3时，对于像这样的分形，物体可以有一个非整数维度
In [[mathematics]], '''Hausdorff dimension''' is a measure of ''roughness'', or more specifically, [[fractal dimension]], that was first introduced in 1918 by [[mathematician]] [[Felix Hausdorff]].<ref>{{Cite journal |arxiv = 1101.1444|doi = 10.1214/11-STS370|title = Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data|journal = Statistical Science|volume = 27|issue = 2|pages = 247–277|year = 2012|last1 = Gneiting|first1 = Tilmann|last2 = Ševčíková|first2 = Hana|last3 = Percival|first3 = Donald B.}}</ref> For instance, the Hausdorff dimension of a single [[point (geometry)|point]] is zero, of a [[line segment]] is 1, of a [[square]] is 2, and of a [[cube]] is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an [[integer]] agreeing with the usual sense of dimension, also known as the [[Inductive dimension|topological dimension]]. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of [[scaling (geometry)|scaling]] and [[self-similarity]], one is led to the conclusion that particular objects—including [[fractal]]s—have non-integer Hausdorff dimensions. Because of the significant technical advances made by [[Abram Samoilovitch Besicovitch]] allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the ''Hausdorff–Besicovitch dimension.''
In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.
在数学中，豪斯多夫维数是一种粗糙度的度量单位，或者更确切地说，分形维数，是由数学家 Felix Hausdorff 在1918年首次提出的。例如，单点的豪斯多夫维数为零，线段为1，正方形为2，立方体为3。也就是说，对于定义了一个光滑形状或一个有少数几个角的形状---- 传统几何学和科学的形状---- 的点集来说，豪斯多夫维数是一个整数，符合通常的维度意义，也称为拓扑维度。然而，还有一些公式允许计算其他不太简单的物体的维数，其中仅仅根据它们的标度和自相似性质，就可以得出结论，特定的物体ー包括分形ー具有非整数的 Hausdorff 维数。由于阿布拉姆·萨莫伊洛维奇·贝西科维奇的重大技术进步，允许计算高度不规则或“粗糙”集的维度，这个维度也通常被称为 Hausdorff-Besicovitch 维度。
The Hausdorff dimension, more specifically, is a further dimensional number associated with a given set, where the distances between all members of that set are defined. Such a set is termed a [[metric space]]. The dimension is drawn from the [[Extended real number line|extended real numbers]], <math>\overline{\mathbb{R}}</math>, as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.
The Hausdorff dimension, more specifically, is a further dimensional number associated with a given set, where the distances between all members of that set are defined. Such a set is termed a metric space. The dimension is drawn from the extended real numbers, <math>\overline{\mathbb{R}}</math>, as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.
更具体地说，豪斯多夫维数是一个与给定集合相关联的更进一步的维数，其中定义了该集合所有成员之间的距离。这样的集合称为度量空间。维数是从扩展的实数，math overline { mathbb { r } / math，而不是更直观的维数概念，它不与一般的度量空间相关联，只接受非负整数的值。
In mathematical terms, 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 underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that [[fractal|irregular sets]] can have noninteger Hausdorff dimensions. For instance, the [[Koch snowflake]] shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new [[equilateral]] triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4.<ref>Larry Riddle, 2014, "Classic Iterated Function Systems: Koch Snowflake", Agnes Scott College e-Academy (online), see [http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm], accessed 5 March 2015.</ref> That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original.<ref name=CampbellAnnenberg15/> Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=S<sup>D</sup>.<ref name=ClaytonSCTPLS96>Keith Clayton, 1996, "Fractals and the Fractal Dimension," ''Basic Concepts in Nonlinear Dynamics and Chaos'' (workshop), Society for Chaos Theory in Psychology and the Life Sciences annual meeting, June 28, 1996, Berkeley, California, see [http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Workshop.html], accessed 5 March 2015.</ref> This equation is easily solved for D, yielding the ratio of logarithms (or [[natural logarithm]]s) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.
In mathematical terms, 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 underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that irregular sets can have noninteger Hausdorff dimensions. For instance, the Koch snowflake shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new equilateral triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4. That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original. This equation is easily solved for D, yielding the ratio of logarithms (or natural logarithms) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.
用数学术语来说，豪斯多夫维数概括了实向量空间维数的概念。也就是说，n 维内积空间的豪斯多夫维数等于 n。 这就是早期声明的基础，一个点的豪斯多夫维数是零，一条线是一，等等，不规则集可以有非整数的 Hausdorff 维数。例如，右边所示的 Koch 雪花是由一个正三角形构成的; 在每次迭代中，它的组成线段被分成单位长度的3段，新创建的中间线段被用作一个指向外部的新正三角形的基础，然后这个基础线段被删除以保留单位长度4的迭代中的最终对象。也就是说，在第一次迭代之后，每个原始线段都被替换为 n4，其中每个自相似拷贝的长度是原始线段的1 / s 1 / 3。这个方程很容易求解为 d，产生出现在图形中的对数(或自然对数)的比率，并给出ー在 Koch 和其他分形情况下ー这些对象的非整数维数。
The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or [[Minkowski–Bouligand dimension]].
The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension.
豪斯多夫维数是更简单但通常等价的计盒维数或 Minkowski-Bouligand 维数的继承者。
==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.
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 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 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.
几何物体 x 的直观尺寸概念就是一个人需要多少个独立参数才能挑出一个独特的点。但是，任何由两个参数指定的点都可以由一个参数指定，因为实际平面的基数等于实际行的基数(这可以通过一个参数看到，该参数涉及交织两个数字的数字以产生一个编码相同信息的单个数字)。皮亚诺曲线的例子表明，一个人甚至可以映射实际线到真正的平面满意地(把一个实数转换成一对实数，这样所有的数对都被覆盖)和连续，所以一维物体完全填充了一个高维物体。
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 ''n'' if, in every covering of ''X'' by small open balls, there is at least one point where ''n'' + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension ''n'' = 1.
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 n if, in every covering of X by small open balls, there is at least one point where n + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension n = 1.
每条空间填充曲线都会多次撞击某些点，且不存在连续的倒数。将两个维度以连续和连续可逆的方式映射到一个维度是不可能的。拓扑维度，也被称为拓朴维数，解释了为什么。这个维度是 n，如果在 x 的每个小开球覆盖中，至少有一个点 n + 1个球重叠。例如，当一个点覆盖一条具有短开区间的直线时，某些点必须被覆盖两次，给出维数 n 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.
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.
但是，拓扑维度是对空间局部尺寸(点附近的尺寸)的一个非常粗略的度量。一条几乎是空间填充的曲线仍然可以有一维拓扑，即使它填充了一个区域的大部分面积。分形具有整数的拓扑维数，但就其所占空间的数量而言，它表现得像一个更高维的空间。
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.
The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric. Consider the number N(r) of 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 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.
豪斯多夫维数测量一个空间的局部大小，考虑到点之间的距离，度量。考虑半径最大为 r 的球数 n (r) ，需要完全覆盖 x。当 r 很小时，n (r)以1 / r 增长多项式。对于一个表现足够好的 x，豪斯多夫维数是唯一的数 d，这样当 r 趋近于零时 n (r)增长为1 / r sup d / sup。更确切地说，这定义了盒子计数维度，当值 d 是不足以覆盖空间的增长率和过度充裕的增长率之间的关键边界时，它等于豪斯多夫维数。
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 you see around you is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:
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 fractals, 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:
对于光滑的形状，或者有少量棱角的形状，传统几何和科学的形状，豪斯多夫维数是一个整数，与拓扑维度一致。但是本华·曼德博观察到分形---- 具有非整数 Hausdorff 维数的集合---- 在自然界中随处可见。他观察到，你周围大多数粗糙形状的理想化不是光滑的理想化形状，而是分形理想化形状:
<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 = Benoit 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 | url-access = registration | url = https://archive.org/details/fractalgeometryo00beno }}</ref></blockquote>
<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.</blockquote>
云不是球体，山不是锥体，海岸线不是圆圈，树皮不平滑，闪电也不是直线运动。 / 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.
For fractals that occur in nature, the Hausdorff and 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.
对于自然界中出现的分形，Hausdorff 维数和盒计数维数是一致的。封装尺寸是另一个类似的概念，它给出了许多形状相同的值，但是在所有这些尺寸不同的情况下，有很好的文档说明的例外。
==Formal definitions==
{{unreferenced section|date=March 2015}}
===Hausdorff content===
Let ''X'' be a [[metric space]]. If ''S'' ⊂ ''X'' and ''d'' ∈ [0, ∞), the ''d''-dimensional '''unlimited Hausdorff content''' of ''S'' is defined by
Let X be a metric space. If S ⊂ X and d ∈ [0, ∞), the d-dimensional unlimited Hausdorff content of S is defined by
设 x 是度量空间。若 s something x 和 d ∈[0，∞) ，则 s 的 d 维无限 Hausdorff 内容定义为
:<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>
<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>
数学 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
In other words, <math>C_H^d(S)</math> is the [[infimum]] of the set of numbers <math>\delta \geq 0</math> such that there is some (indexed) collection of [[ball (mathematics)|ball]]s <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 = ∞]].)
In other words, <math>C_H^d(S)</math> is the infimum of the set of numbers <math>\delta \geq 0</math> such that there is some (indexed) collection of 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 inf Ø = ∞.)
换句话说，math c h ^ d (s) / math 是数字集合 math delta geq 0 / math 的下确界，使得在 i / math 中存在一些球集合 math { b (xi，ri) : i 包含 s，对于每个 i ∈ i，r 子 i / sub 0满足 i 中的数学和 i ^ d delta / math。(在这里，我们使用 inf ∞ .)的标准约定
===Hausdorff measurement===
The Hausdorff outer measure is different from the unbounded Hausdorf content in that rather than considering all possible coverings of ''S'', we see what happens when the sizes of the balls go to zero. This is for <math>d \geq 0 </math>, we define the ''d''-dimensional Hausdorff outer measure of ''S'' as
The Hausdorff outer measure is different from the unbounded Hausdorf content in that rather than considering all possible coverings of S, we see what happens when the sizes of the balls go to zero. This is for <math>d \geq 0 </math>, we define the d-dimensional Hausdorff outer measure of S as
豪斯多夫外测度不同于无界的豪斯多夫内容，因为我们不考虑 s 的所有可能覆盖，我们看到当球的大小为零时会发生什么。这是为了数学 d geq 0 / math，我们定义了 s 的 d 维 Hausdorff 外测度为
:<math> \mathcal{H}^d(S):=\lim_{r \to 0} \inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii } 0 < r_i < r\Bigr\}.</math>
<math> \mathcal{H}^d(S):=\lim_{r \to 0} \inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii } 0 < r_i < r\Bigr\}.</math>
数学函数 h ^ d (s) : lim { r to 0} inf Bigl sum i r i ^ d: text {有一个{ s text by balls with radii }0 r i r Bigr }的封面。 数学
===Hausdorff dimension===
The '''Hausdorff dimension''' of ''X'' is defined by
The Hausdorff dimension of X is defined by
X 的豪斯多夫维数定义为
:<math>\dim_{\operatorname{H}}(X):=\inf\{d\ge 0: \mathcal{H}^d(X)=0\}.</math>
<math>\dim_{\operatorname{H}}(X):=\inf\{d\ge 0: \mathcal{H}^d(X)=0\}.</math>
Math operatorname { h }(x) : inf ge 0: mathcal { h } ^ d (x)0} . / math
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).
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).
等价地，dim 子 h / sub (x)可定义为 d ∈[0，∞)集的下确界，使得 x 的 d 维 Hausdorff 测度为零。这与 d ∈[0，∞)的集合的上确界相同，因此 x 的 d 维 Hausdorff 测度是无限的(除非后一个集合 d 是空的，豪斯多夫维数为零)。
==Examples==
[[Image:Sierpinski deep.svg|thumb|250px|Dimension of a further [[fractal]] example. The [[Sierpinski triangle]], an object with Hausdorff dimension of log(3)/log(2)≈1.58.<ref name=ClaytonSCTPLS96/>]]
Dimension of a further [[fractal example. The Sierpinski triangle, an object with Hausdorff dimension of log(3)/log(2)≈1.58.]]
进一步的维数[分形的例子。谢尔宾斯基三角形，一个豪斯多夫维数为3 / log (2)≈1.58的物体]
* [[Countable set]]s have Hausdorff dimension 0.<ref name="schleicher">{{cite journal |last1=Schleicher |first1=Dierk |title=Hausdorff Dimension, Its Properties, and Its Surprises |journal=The American Mathematical Monthly |date=June 2007 |volume=114 |issue=6 |pages=509–528 |doi=10.1080/00029890.2007.11920440 |language=en |issn=0002-9890|arxiv=math/0505099 }}</ref>
* The [[Euclidean space]] ℝ<sup>''n''</sup> has Hausdorff dimension ''n'', and the circle '''S'''<sup>1</sup> has Hausdorff dimension 1.<ref name="schleicher" />
* [[Fractal]]s often are spaces whose Hausdorff dimension strictly exceeds the [[topological dimension]].<ref name="mandelbrot" /> 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; hence, it can be shown that its Hausdorff dimension is ln(2)/ln(3) ≈ 0.63.<ref>{{cite book | last=Falconer | first = Kenneth |title=Fractal Geometry: Mathematical Foundations and Applications | publisher=[[John Wiley and Sons]] | edition=2nd | year=2003}}</ref> 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.<ref name=CampbellAnnenberg15/> These Hausdorff dimensions are related to the "critical exponent" of the [[Master theorem (analysis of algorithms)|Master theorem]] for solving [[Recurrence relation|recurrence relations]] in the [[analysis of algorithms]].
* [[Space-filling curve]]s like the [[Peano curve]] have the same Hausdorff dimension as the space they fill.
* The trajectory of [[Brownian motion]] in dimension 2 and above is conjectured to be Hausdorff dimension 2.<ref>{{cite book | last=Morters | first=Peres | title= Brownian Motion | publisher=[[Cambridge University Press]] | year=2010 }}</ref>
[[image:Great Britain Hausdorff.svg|thumb|250px|Estimating the Hausdorff dimension of the [[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|coast of Great Britain]]]]
coast of Great Britain]]
大不列颠海岸]
* [[Lewis Fry Richardson]] has performed detailed experiments to measure the approximate Hausdorff dimension for various coastlines. His results have varied from 1.02 for the coastline of [[South Africa]] to 1.25 for the west coast of [[Great Britain]].<ref name="mandelbrot" />
==Properties of Hausdorff dimension==
{{refimprove section|date=March 2015}}
=== Hausdorff dimension and inductive dimension ===
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'').
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<sub>ind</sub>(X).
设 x 是任意可分度量空间。对于 x 有一个递归定义的归纳维数拓扑概念。它总是一个整数(或 + ∞) ，并且表示 dim sub ind / sub (x)。
'''Theorem'''. Suppose ''X'' is non-empty. Then
Theorem. Suppose X is non-empty. Then
定理。假设 x 是非空的。然后
:<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). </math>
<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). </math>
<math> \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X).数学
Moreover,
Moreover,
此外,
:<math> \inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(X), </math>
<math> \inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(X), </math>
(y) dim { operatorname { Haus }(y) dim { operatorname { ind }(x) ，/ math
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>.
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>.
其中 y 是度量空间同胚到 x 的范围。换句话说，x 和 y 具有相同的基本点集，y 的度规 d 子 y / 子拓扑等价于 d 子 x / 子。
These results were originally established by [[Edward Szpilrajn]] (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.{{full citation needed|date=March 2015}}
These results were originally established by Edward Szpilrajn (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.
这些结果最初是由 Edward Szpilrajn (1907-1976)建立的，例如，见 Hurewicz 和 Wallman，第七章。
=== Hausdorff dimension and Minkowski dimension ===
The [[Minkowski dimension]] is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. 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.
The Minkowski dimension is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. 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.
闵可夫斯基维度与豪斯多夫维数相似，至少和它一样大，而且在许多情况下是相等的。然而，[0,1]中有理点集的豪斯多夫维数为0，Minkowski 维数为1。还有一些紧集的 Minkowski 维数严格大于豪斯多夫维数。
=== Hausdorff dimensions and Frostman measures ===
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]].{{citation needed|date=March 2015}}<ref>This Wikipedia article also discusses further useful characterizations of the Hausdorff dimension.{{clarify|date=March 2015}}</ref>
If there is a measure μ defined on 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.
如果在度量空间 x 的 Borel 子集上定义一个测度，使得(x)0和(b (x，r))≤ rsup s / sup 对于某个常数 s0和 x 中的每个球 b (x，r)成立，则 dim sub Haus / sub (x)≥ s。 部分逆向由霜人引理提供。
=== Behaviour under unions and products ===
If <math>X=\bigcup_{i\in I}X_i</math> is a finite or countable union, then
If <math>X=\bigcup_{i\in I}X_i</math> is a finite or countable union, then
如果 math x x i / math 是一个有限或可数的联合，则
:<math> \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i).</math>
<math> \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i).</math>
(x) sup { i }{ operatorname { Haus }(xi) . / math
This can be verified directly from the definition.
This can be verified directly from the definition.
这可以直接从定义中验证。
If ''X'' and ''Y'' are non-empty 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>
If X and Y are non-empty metric spaces, then the Hausdorff dimension of their product satisfies
如果 x 和 y 是非空度量空间，那么它们产品的豪斯多夫维数满足
:<math> \dim_{\operatorname{Haus}}(X\times Y)\ge \dim_{\operatorname{Haus}}(X)+ \dim_{\operatorname{Haus}}(Y).</math>
<math> \dim_{\operatorname{Haus}}(X\times Y)\ge \dim_{\operatorname{Haus}}(X)+ \dim_{\operatorname{Haus}}(Y).</math>
(x 乘以 y) ge dim { operatorname { Haus }(x) + dim { operatorname { Haus }(y) . / math
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).
This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1. 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 upper packing dimension of Y. These facts are discussed in Mattila (1995).
这种不平等可以是严格的。有可能找到两个维数为0的集合，其乘积的维数为1。在相反的方向上，我们知道当 x 和 y 是 r sup n / sup 的 Borel 子集时，x y 的豪斯多夫维数从上面以 x 的豪斯多夫维数加上 y 的填充维数为界。Mattila (1995)讨论了这些事实。
==Self-similar sets==
{{refimprove section|date=March 2015}}
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.
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.
许多由自相似条件定义的集合具有可以显式确定的维数。粗略地说，集合 e 是自相似的，如果它是集值变换的不动点，即(e) e，尽管下面给出了确切的定义。
<blockquote>'''Theorem'''. Suppose
<blockquote>Theorem. Suppose
块引定理。假设
:<math> \psi_i: \mathbf{R}^n \rightarrow \mathbf{R}^n, \quad i=1, \ldots , m </math>
<math> \psi_i: \mathbf{R}^n \rightarrow \mathbf{R}^n, \quad i=1, \ldots , m </math>
Math psi i: mathbf { r } ^ n right tarrow mathbf { r } ^ n， quad i 1， ldots，m / math
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
are 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
Rsup n / sup 上的压缩常数 r 子 j / 子1的压缩映射。然后有一个唯一的非空紧集
:<math> A = \bigcup_{i=1}^m \psi_i (A). </math>
<math> A = \bigcup_{i=1}^m \psi_i (A). </math>
数学 a bigcup { i 1} ^ m psi i (a)。数学
</blockquote>
</blockquote>
/ blockquote
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>
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<sup>n</sup> with the Hausdorff distance.
这个定理来源于 Stefan Banach 的压缩映射不动点定理，该定理应用于 r sup n / sup 的非空紧子集的完备空间豪斯多夫距离。
===The open set condition===
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>.
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>.
为了确定自相似集 a 的维数(在某些情况下) ，我们需要一个称为开集条件(OSC)的技术条件。
There is a relatively compact open set ''V'' such that
There is a relatively compact open set V such that
有一个相对紧的开集 v
:<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>
数学大杯{ i } ^ m psi (v) subseteq v，/ math
where the sets in union on the left are pairwise [[disjoint sets|disjoint]].
where the sets in union on the left are pairwise disjoint.
左边的集合是成对不相交的。
The open set condition is a separation condition that ensures the images ψ<sub>''i''</sub>(''V'') do not overlap "too much".
The open set condition is a separation condition that ensures the images ψ<sub>i</sub>(V) do not overlap "too much".
开集条件是保证图像子 i / 子(v)不重叠“太多”的分离条件。
'''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 | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>
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 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
定理。假设开集条件成立，并且每个子 i / sub 是一个相似量，这是一个等距和围绕某一点的膨胀的合成。那么唯一的不动点是一个集合，它的豪斯多夫维数是 s，其中 s 是 s 的唯一解
:<math> \sum_{i=1}^m r_i^s = 1. </math>
<math> \sum_{i=1}^m r_i^s = 1. </math>
数学1 ^ m r i ^ s 1。数学
The contraction coefficient of a similitude is the magnitude of the dilation.
The contraction coefficient of a similitude is the magnitude of the dilation.
相似物的收缩系数就是膨胀的大小。
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
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
我们可以使用这个定理来计算豪斯多夫维数谢尔宾斯基三角形的密封圈(或者有时候叫做 Sierpinski 垫圈)。考虑平面上的三个非共线点，一个子1 / 子，一个子2 / 子，一个子3 / 子。对应映射的唯一非空不动点是一个 Sierpinski 垫片，其维数 s 是对应映射的唯一解
:<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>
<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>
数学左(压缩{1}{2}右) ^ s + 左(压缩{1}右) ^ s + 左(压缩{1}右) ^ s 3左(压缩{1}右) ^ s 1。数学
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
Taking natural logarithms 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
取上述方程两边的自然对数，我们可以求出 s，即: s ln (3) / ln (2)。该密封垫具有自相似性，满足 OSC 要求。一般来说，集合 e 是一个映射的不动点
: <math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>
<math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>
数学 a 映射 / 压缩 / 压缩 / 压缩
is self-similar if and only if the intersections
is self-similar if and only if the intersections
是自相似的当且仅当
:<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, </math>
<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, </math>
数学 h ^ s 左( psi i (e) cap psi j (e)右)0，/ math
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:
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:
其中 s 是 e 的豪斯多夫维数，h sup s / sup 表示 Hausdorff 测度。对于 Sierpinski 垫圈(交叉点就是点)来说，这一点很明显，但更普遍的情况是:
'''Theorem'''. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.
Theorem. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.
定理。在与前一定理相同的条件下，其唯一不动点是自相似的。
==See also==
* [[List of fractals by Hausdorff dimension]] Examples of deterministic fractals, random and natural fractals.
* [[Assouad dimension]], another variation of fractal dimension that, like Hausdorff dimension, is defined using coverings by balls
* [[Intrinsic dimension]]
* [[Packing dimension]]
* [[Fractal dimension]]
==References==
{{reflist}}
==Further reading==
* {{cite book |last1=Dodson |first1=M. Maurice |title=Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot |volume=72 |issue=1 |pages=305–347 |last2=Kristensen |first2=Simon |chapter=Hausdorff Dimension and Diophantine Approximation |date=June 12, 2003 |arxiv=math/0305399 |bibcode = 2003math......5399D |doi=10.1090/pspum/072.1/2112110|series=Proceedings of Symposia in Pure Mathematics |isbn=9780821836378 }}
* {{cite book |last1=Hurewicz |first1=Witold |authorlink1=Witold Hurewicz |last2=Wallman |first2=Henry |authorlink2=Henry Wallman |title=Dimension Theory |url=https://archive.org/details/in.ernet.dli.2015.84609 |publisher=Princeton University Press |year=1948 }}
* {{cite journal |author=E. Szpilrajn |authorlink=Edward Marczewski |title=La dimension et la mesure |journal=Fundamenta Mathematicae |volume=28 |pages=81–9 |year=1937 }}
* {{cite journal
| last1=Marstrand
| last1=Marstrand
最后一站马斯特兰德
| first1=J. M. | title=The dimension of cartesian product sets | year=1954 | journal=Proc. Cambridge Philos. Soc.
| first1=J. M. | title=The dimension of cartesian product sets | year=1954 | journal=Proc. Cambridge Philos. Soc.
首先1 j。笛卡儿积集的尺寸 | 1954年 | 期刊 Proc。Cambridge Philos.Soc.
| volume=50
| volume=50
第50卷
| issue=3
| issue=3
第三期
| pages=198–202
| pages=198–202
第198-202页
| doi=10.1017/S0305004100029236|bibcode = 1954PCPS...50..198M }}
| doi=10.1017/S0305004100029236|bibcode = 1954PCPS...50..198M }}
10.1017 / S0305004100029236 | bibcode 1954PCPS... 50. . 198 m }
* {{Cite book
| last1=Mattila
| last1=Mattila
最后1个 Mattila
| first1=Pertti | author1-link=Pertti Mattila| title=Geometry of sets and measures in Euclidean spaces | publisher=[[Cambridge University Press]]
| first1=Pertti | author1-link=Pertti Mattila| title=Geometry of sets and measures in Euclidean spaces | publisher=Cambridge University Press
| first1 Pertti | author1-link Pertti Mattila | title 欧几里德空间中集合和度量的几何 | 出版商剑桥大学出版社
| isbn=978-0-521-65595-8 | year=1995}}
| isbn=978-0-521-65595-8 | year=1995}}
| isbn 978-0-521-65595-8 | year 1995}
* {{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}}
* {{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
* {{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|hdl=10338.dmlcz/100363 |url=http://dml.cz/bitstream/handle/10338.dmlcz/100363/CzechMathJ_09-1959-3_5.pdf }}
* {{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}
*{{cite book | last=Falconer | first = Kenneth |title=Fractal Geometry: Mathematical Foundations and Applications | publisher=[[John Wiley and Sons]] | edition=2nd | year=2003}}
==External links==
* [https://www.encyclopediaofmath.org/index.php/Hausdorff_dimension Hausdorff dimension] at [https://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
* [https://www.encyclopediaofmath.org/index.php/Hausdorff_measure Hausdorff measure] at [https://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
{{DEFAULTSORT:Hausdorff Dimension}}
[[Category:Fractals]]
Category:Fractals
分类: 分形
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分类: 米制几何
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Category:Dimension theory
范畴: 维度理论
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