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第43行： − + 第51行：
第51行： − + 第75行：
第75行： − 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。+ 第83行：
第83行： − 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:+ 第101行：
第101行： − −
→Intuition
==Intuition==
==Intuition概念==
{{refimprove section|date=March 2015}}
{{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 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。
几何物体 x 的直观尺寸概念就是一个物体需要多少个独立参数才能找到一个独特的点。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<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.
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时，会考虑到点之间距离的度量。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。但是伯努·曼德布洛特观察到分形---- 具有非整数豪斯多夫维数的集合---- 在自然界中随处可见。他观察到，我们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:
对于自然界中出现的分形，豪斯多夫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。
对于自然界中出现的分形，豪斯多夫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==
==Formal definitions==