{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nA *non-Desarguesian plane* is a [[projective plane|DANFNN8C4Y]] in which [[Desargues's theorem|5Z19Y02MME]] does not hold.\nA concrete one is the projective completion of the [[Moulton plane|3V9GYYXM1H]].\nCheck: the Moulton plane has triangles $ABC$ and $A'B'C'$ in perspective from a point $O$.\nIn it $AB \\parallel A'B'$ and $AC \\parallel A'C'$, but $BC$ is not parallel to $B'C'$.\nAdd the line at infinity.\nThen $AB \\cap A'B'$ and $AC \\cap A'C'$ lie on the line at infinity, but $BC \\cap B'C'$ does not, so the three points are not collinear.\n\nSo this plane is not a [[Desarguesian plane|HPX85PA8NX]], and it is not isomorphic to $\\mathbb{P}^2(D)$ for any division ring $D$.\n\n## Remarks\n\nEvery projective plane of order at most $8$ is Desarguesian.\nThe smallest non-Desarguesian planes have order $9$.\nThere are exactly four projective planes of order $9$, and three of them are non-Desarguesian: the Hall plane, its dual and the Hughes plane.\nThe [[octonionic projective plane|S26GYRVDR5]] is a non-Desarguesian plane that is still a [[Moufang plane|7F4HSNJJH8]].\nEvery projective plane can be coordinatized by a [[planar ternary ring|Q0NXXFADT9]], which in a non-Desarguesian plane does not come from a division ring.\n\n## Explanations\n\n### L1 human\n\nA projective plane only has to obey a few rules about points and lines.\nOne might expect that these rules force the plane to look like the usual plane built from numbers.\nNon-Desarguesian planes show that this is false: they obey the rules, yet Desargues's theorem on triangles fails in them.\nSo no system of coordinates from a field, or even from a noncommutative division ring, can describe them.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "DANFNN8C4Y"
    ],
    "kind": "example",
    "msc": [
      "51A35",
      "51E15"
    ],
    "name": "Non-Desarguesian plane",
    "not_instance_of": [
      {
        "note": "Desargues's theorem fails for two triangles in perspective",
        "target": "HPX85PA8NX"
      }
    ],
    "same_as": {
      "wikidata": "Q3391357",
      "wikipedia": "Non-Desarguesian plane"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Non-Desarguesian_plane"
      }
    ],
    "tag": "HYE5FT4M34",
    "uses": [
      "5Z19Y02MME",
      "3V9GYYXM1H"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/HYE5FT4M34/",
  "links": [
    {
      "field": "instance_of",
      "target": "DANFNN8C4Y"
    },
    {
      "field": "not_instance_of",
      "target": "HPX85PA8NX",
      "note": "note: Desargues's theorem fails for two triangles in perspective."
    },
    {
      "field": "uses",
      "target": "5Z19Y02MME"
    },
    {
      "field": "uses",
      "target": "3V9GYYXM1H"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/HYE5FT4M34.md",
  "msc": [
    "51A35",
    "51E15"
  ],
  "name": "Non-Desarguesian plane",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HYE/HYE5FT4M34.md",
  "status": null,
  "tag": "HYE5FT4M34",
  "uri": "m2://HYE5FT4M34",
  "url": "https://tamnd.github.io/math2/wiki/non-desarguesian-plane/",
  "writing": {
    "tag": "HYE5FT4M34",
    "kind": "example",
    "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
      }
    ]
  }
}