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Dimension of a further [[fractal example. The Sierpinski triangle, an object with Hausdorff dimension of log(3)/log(2)≈1.58.]]
Dimension of a further [[fractal example. The Sierpinski triangle, an object with Hausdorff dimension of log(3)/log(2)≈1.58.]]
[[Image:谢尔宾斯基三角形 deep.svg|thumb|250px|进一步的[[分形维数]]的例子，进一步的分形维数的例子是谢尔宾斯基三角形，它是一个豪斯多夫维数为log(3)/log(2)≈1.58.<ref name=ClaytonSCTPLS96/>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>
* [[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>
*[[可数集]]拥有豪斯多夫维数0.<ref name="schleicher">{{cite journal |last1=Schleicher |first1=Dierk |title=豪斯多夫维数, Its Properties, and Its Surprises |journal=The American Mathematical Monthly美国数学期刊 |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" />
* 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" />
*[[教育空间]] ℝ<sup>''n''</sup> 有豪斯多夫维数 ''n''，循环'''S'''<sup>1</sup> 拥有豪斯多夫维数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]].
* [[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.
* [[Space-filling curve]]s like the [[Peano curve]] have the same Hausdorff dimension as the space they fill.
*[[空间填充曲线]]拥有和他们填充空间同样的豪斯多夫维数，如[[皮亚诺曲线]]。
* 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>
* 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>
coast of Great Britain]]
coast of Great Britain]]
大不列颠海岸]
[[大不列颠海岸
* 刘易斯·弗赖伊·理查森[[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" />
* 刘易斯·弗赖伊·理查森[[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" />
*刘易斯·弗赖伊·理查森（Lewis Fry Richardson）已经通过豪斯多夫维数去测量了很多海岸线。它的结果涵盖从1.02的南非海岸线到1.25的大英帝国西海岸模型。
*刘易斯·弗赖伊·理查森（Lewis Fry Richardson）已经通过豪斯多夫维数去测量了很多海岸线。它的结果涵盖从1.02的南非海岸线到1.25的大英帝国西海岸模型。]]
==Properties of Hausdorff dimension豪斯多夫维数特性==
==Properties of Hausdorff dimension豪斯多夫维数特性==