{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "WSKWY54JDN"
    }
  ],
  "body": "\n## Example\n\nLet $S_0 = [0, 1]^2$.\nDivide each square of side $3^{-k}$ in $S_k$ into $9$ equal squares of side $3^{-k-1}$.\nRemove the interior of the middle one; the result is $S_{k+1}$.\nThe *Sierpiński carpet* is $S = \\bigcap_{k \\ge 0} S_k$.\n\nThe set $S$ is a [[continuum|NJQ196M1WR]].\nEach $S_k$ is a finite union of closed squares, so it is [[compact|0TVJZYB6SH]].\nIt is [[connected|0V032VWTHD]], since the $8$ kept squares in each divided square form a ring in which neighbours share an edge.\nThe intersection of a decreasing sequence of nonempty compact connected sets in the plane is nonempty, compact and connected.\n\n## Remarks\n\nThe set $S_k$ consists of $8^k$ squares of area $9^{-k}$, so $S$ has area $\\lim_k (8/9)^k = 0$.\nIts Hausdorff dimension is $\\log 8 / \\log 3 \\approx 1.893$.\nIts [[Lebesgue covering dimension|A3W7RGXXK9]] is $1$.\nSierpiński showed that every compact subset of the plane of dimension at most $1$ embeds in the carpet, so it is a universal plane curve.\n\n## Explanations\n\n### L1 human\n\nTake a square, cut it into a three by three grid, and throw away the middle piece.\nDo the same with each of the eight pieces left, and keep going forever.\nWhat remains is a lacy set with holes at every scale.\nIt has no area, yet it is all in one piece, and it contains a copy of every curve that can be drawn in the plane.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "instance_of": [
      "NJQ196M1WR"
    ],
    "kind": "example",
    "msc": [
      "54F15",
      "28A80",
      "54F50"
    ],
    "name": "Sierpiński carpet",
    "same_as": {
      "wikidata": "Q280081",
      "wikipedia": "Sierpiński carpet"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Sierpi%C5%84ski_carpet"
      }
    ],
    "tag": "PCM3H6J3EE",
    "uses": [
      "0TVJZYB6SH",
      "0V032VWTHD",
      "A3W7RGXXK9"
    ]
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/PCM3H6J3EE/",
  "links": [
    {
      "field": "instance_of",
      "target": "NJQ196M1WR"
    },
    {
      "field": "uses",
      "target": "0TVJZYB6SH"
    },
    {
      "field": "uses",
      "target": "0V032VWTHD"
    },
    {
      "field": "uses",
      "target": "A3W7RGXXK9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/PCM3H6J3EE.md",
  "msc": [
    "54F15",
    "28A80",
    "54F50"
  ],
  "name": "Sierpiński carpet",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PCM/PCM3H6J3EE.md",
  "status": null,
  "tag": "PCM3H6J3EE",
  "uri": "m2://PCM3H6J3EE",
  "url": "https://math2.tamnd.com/wiki/sierpinski-carpet/",
  "writing": {
    "tag": "PCM3H6J3EE",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}