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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.]]

[[File:KochFlake.svg|thumb|280px|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。]]

[[File:KochFlake.svg|thumb|280px|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>。MacGregor Campbell, 2013, "5.6 Scaling and the Hausdorff Dimension," 在 ''Annenberg Learner:MATHematics illuminated'', 参见 [http://www.learner.org/courses/mathilluminated/units/5/textbook/06.php], accessed 5 March 2015.</ref> 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。]]