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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.]]

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

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 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.

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。

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豪斯多夫维数。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。

豪斯多夫维数是更简单但通常等价的计盒维数或闵可夫斯基维数Minkowski-Bouligand的继承者。

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.

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。

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，一个人甚至可以完美和连续地映射实际线到真正的平面(把一个实数转换成一对实数，这样所有的数对都被覆盖)，由此一维物体完全填充了一个高维物体。

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^d 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.

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^d 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。

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的度量。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 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:

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:

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 豪斯多夫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.</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

<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 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.

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

设 x 是度量空间。若 s something x 和 d ∈[0，∞) ，则 s 的 d 维无限 豪斯多夫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>

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

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，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 维豪斯多夫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>

Equivalently, dim_H(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&nbsp;∈&nbsp;[0,&nbsp;∞) 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_H(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&nbsp;∈&nbsp;[0,&nbsp;∞) 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)。

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 维豪斯多夫 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)。

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

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>