{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "F7G7CJY143"
    },
    {
      "field": "uses",
      "target": "FYNFT90NT2"
    },
    {
      "field": "about",
      "target": "FYNFT90NT2"
    },
    {
      "field": "defines",
      "target": "VYP60TA2EC"
    }
  ],
  "body": "\n## Summary\n\nLet $A$ be a subset of a [[metric space|0VKD1TP7F3]].\nIts Hausdorff dimension is the exponent $s$ at which the $s$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] of $A$ jumps from infinity to zero.\nIt is a real number, not always an integer.\nIt agrees with the usual dimension for smooth curves, surfaces and open sets, and it gives fractals such as the [[Cantor set|0BXNDZFQME]] non-integer dimensions.\n\n## Explanations\n\n### L1 human\n\nA curve has length but no area, and a square has area but infinite length.\nThe Hausdorff dimension finds the exact exponent where a set switches from infinite size to zero size.\nFor a segment this happens at $1$, and for a filled square at $2$.\nThe middle-thirds Cantor set switches at $\\log 2 / \\log 3 \\approx 0.63$: it is too thin to have length but too big to be counted as a few points.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "28A78",
      "28A80"
    ],
    "name": "Hausdorff dimension",
    "same_as": {
      "wikidata": "Q565186",
      "wikipedia": "Hausdorff dimension"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hausdorff_dimension"
      }
    ],
    "tag": "SSNFTHZ6YJ"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/SSNFTHZ6YJ/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/SSNFTHZ6YJ.md",
  "msc": [
    "28A78",
    "28A80"
  ],
  "name": "Hausdorff dimension",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SSN/SSNFTHZ6YJ.md",
  "status": null,
  "tag": "SSNFTHZ6YJ",
  "uri": "m2://SSNFTHZ6YJ",
  "url": "https://math2.tamnd.com/wiki/hausdorff-dimension/",
  "writing": {
    "tag": "SSNFTHZ6YJ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}