{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "C5VXQVNNH2"
    }
  ],
  "body": "\n## Summary\n\nThe covering density of a covering of space by bodies measures, on average, how many times each point is covered.\nIt is at least $1$, and the covering problem asks for the smallest possible value, the covering density of the body.\nIt is the counterpart of the [[packing density|4P5PSWADJD]], with overlaps allowed and gaps forbidden.\nFor discs in the plane the optimum is $2\\pi/\\sqrt{27} \\approx 1.2092$, given by the hexagonal lattice arrangement.\n\n## Explanations\n\n### L1 human\n\nSuppose you must light a large square with lamps that each light a disc, and every spot must be lit.\nThe discs must overlap somewhere, and the covering density measures how much light is wasted on average.\nFor equal discs, the best is to place the centres in a triangular grid, so that each disc is cut to a regular hexagon by its neighbours.\nEven then about $21\\%$ of the light is spent on overlaps.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "52C17",
      "52C15",
      "11H31"
    ],
    "name": "Covering density",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Covering_problems"
      }
    ],
    "tag": "BWSN2JAMTK"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/BWSN2JAMTK/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/BWSN2JAMTK.md",
  "msc": [
    "52C17",
    "52C15",
    "11H31"
  ],
  "name": "Covering density",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BWS/BWSN2JAMTK.md",
  "status": null,
  "tag": "BWSN2JAMTK",
  "uri": "m2://BWSN2JAMTK",
  "url": "https://tamnd.github.io/math2/wiki/covering-density/",
  "writing": {
    "tag": "BWSN2JAMTK",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}