{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "J8STHVJGDS"
    },
    {
      "field": "uses",
      "target": "P2V0XS4F63"
    }
  ],
  "body": "\n## Summary\n\nA generalized quadrangle is an [[incidence structure|FGKNAFNHA3]] of points and lines with no triangles.\nIn it, a point off a line is joined to exactly one point of that line.\nIt is the case $n = 4$ of a [[generalized polygon|PCT2HY9AAZ]].\nExamples include the polar spaces of rank $2$ attached to quadrics and to symplectic and Hermitian forms.\nThey are basic building blocks in finite geometry.\n\n## Explanations\n\n### L1 human\n\nIn a generalized quadrangle, two points share at most one line, and there are no triangles.\nInstead, given a point and a line not through it, there is exactly one way to walk from the point along a line to the given line.\nA simple picture is a grid: the points of a square grid with its rows and columns as lines.\nA point not on a given row lies on exactly one column, and that column meets the row in exactly one point.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "51E12",
      "05B25"
    ],
    "name": "Generalized quadrangle",
    "same_as": {
      "wikidata": "Q2754558",
      "wikipedia": "Generalized quadrangle"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Generalized_quadrangle"
      }
    ],
    "specializes": [
      "PCT2HY9AAZ"
    ],
    "tag": "GQJCPFSYVK"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/GQJCPFSYVK/",
  "links": [
    {
      "field": "specializes",
      "target": "PCT2HY9AAZ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/GQJCPFSYVK.md",
  "msc": [
    "51E12",
    "05B25"
  ],
  "name": "Generalized quadrangle",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GQJ/GQJCPFSYVK.md",
  "status": null,
  "tag": "GQJCPFSYVK",
  "uri": "m2://GQJCPFSYVK",
  "url": "https://tamnd.github.io/math2/wiki/generalized-quadrangle/",
  "writing": {
    "tag": "GQJCPFSYVK",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}