如何定义你的维度：关于豪斯多夫维数和自相似性的讨论
How to define your dimension: A discourse on Hausdorff dimension and self-similarity
摘要：
人们常把直线和平面区别开来，说前者是一维的，而后者是二维的。但是，一个物体有$d-$维数意味着什么呢？我们能否为任意物体（比如雪花）严格定义一个一致的维数概念？维数必须总是整数值吗？在强调了一些关键的问题，一个人遇到的同时定义一个合理的概念，为某类对象的尺寸，我们试图回答上述问题，探索的概念豪斯多夫维数-一个显着的方法分配维度的子集的任意度量空间。为了正确地表述豪斯多夫维数的定义和性质，我们事先回顾了临界测度论术语。最后，我们讨论了自相似性的概念，并展示了它如何经常违背我们的直觉，维度必须始终是整数值。
英文摘要：
One often distinguishes between a line and a plane by saying that the former is one-dimensional while the latter is two. But, what does it mean for an object to have $d-$dimensions? Can we define a consistent notion of dimension rigorously for arbitrary objects, say a snowflake, perhaps? And must the dimension always be integer-valued? After highlighting some crucial problems that one encounters while defining a sensible notion of dimension for a certain class of objects, we attempt to answer the above questions by exploring the concept of Hausdorff dimension -- a remarkable method of assigning dimension to subsets of arbitrary metric spaces. In order to properly formulate the definition and properties of the Hausdorff dimension, we review the critical measure-theoretic terminology beforehand. Finally, we discuss the notion of self-similarity and show how it often defies our quotidian intuition that dimension must always be integer-valued.
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