{
  "id": "2012.10606",
  "version": "v1",
  "published": "2020-12-19T06:48:51.000Z",
  "updated": "2020-12-19T06:48:51.000Z",
  "title": "How to define your dimension: A discourse on Hausdorff dimension and self-similarity",
  "authors": [
    "Satvik Singh"
  ],
  "comment": "This article aims to provide a light and accessible introduction to the basics of dimension theory and self-similarity",
  "categories": [
    "math.MG",
    "math.HO"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-19T06:48:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hausdorff dimension",
      "self-similarity",
      "arbitrary metric spaces",
      "consistent notion",
      "crucial problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}