论文标题
如何定义您的维度：关于Hausdorff维度和自相似性的论述
How to define your dimension: A discourse on Hausdorff dimension and self-similarity
论文作者
论文摘要
人们经常说，前者是一维的，而后者是两个。但是，对象具有$ d- $尺寸是什么意思？我们可以为任意物体严格地定义一个一致的维度概念，也许是雪花？尺寸必须始终被整数增值吗？在强调了一些关键问题的同时，在定义某些类别的对象的尺寸概念的同时，我们试图通过探索Hausdorff Dimension的概念来回答上述问题 - 将维度分配给任意度量空间的子集的一种了不起的方法。为了正确地提出Hausdorff维度的定义和属性，我们会事先回顾一下关键的措施理论术语。最后，我们讨论了自相似性的概念，并展示了它如何经常违反我们的quotidian直觉，即尺寸必须始终是数字值的。
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.
