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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建模是使用这个比例因子(3)和自相似物体的数量(4)来计算维度，设在第一次迭代后为 D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26.<ref name=CampbellAnnenberg15>。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。

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.

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。

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 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。

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。 这就是早期假设的基础，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，然后人们删除这个基础线段用来保留单位长度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 N=4，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——在 Koch 和其他分形情况下——这些对象的非整数维数。