{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\Pi$ be a set of points with a family of subsets called lines and a distance function $d \\colon \\Pi \\times \\Pi \\to \\R$.\nFor a point $O$, the rays from $O$ carry a measure: $\\angle AOB$ denotes the angle of the rays $OA$ and $OB$.\n*Birkhoff's axioms* are the following four postulates.\n\n1. Line measure: the points of each [[line|0MP9BCF6DP]] $\\ell$ can be put in bijection with the [[real numbers|08J2M1JYAG]], $A \\mapsto x_A$, so that $\\abs{x_A - x_B} = d(A, B)$ for all $A, B \\in \\ell$.\n2. Point and line: through two distinct points there is exactly one line.\n3. Angle measure: the rays from each point $O$ can be put in bijection with $\\R / 2\\pi\\Z$. Then $\\angle AOB$ is the difference of the numbers of $OA$ and $OB$. Moreover, as $B$ moves continuously on a line not through $O$, the number of the ray $OB$ varies continuously.\n4. Similarity: let $ABC$ and $A'B'C'$ be [[triangles|0CTNFGYMM2]] and $k \u003e 0$ such that $d(A', B') = k\\, d(A, B)$, $d(A', C') = k\\, d(A, C)$ and $\\angle B'A'C' = \\pm \\angle BAC$. Then $d(B', C') = k\\, d(B, C)$, $\\angle C'B'A' = \\pm \\angle CBA$ and $\\angle A'C'B' = \\pm \\angle ACB$.\n\n## Remarks\n\nExample: $\\R^2$ with the usual distance and the polar angle of rays satisfies all four postulates.\nThe similarity postulate holds there because a similarity of ratio $k$ maps one triangle onto the other.\nNon-example: the sphere with great circles as lines and the great-circle distance fails postulate 2, since there are infinitely many great circles through two antipodal points.\nThe axioms of [[angles|0GJAJC6TNZ]] are those of a protractor, and the axioms of lines are those of a ruler.\nThe four postulates determine the Euclidean plane up to isometry.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "6BB0KD6752",
    "kind": "definition",
    "msc": [
      "51M05",
      "51F05"
    ],
    "name": "Birkhoff's axioms",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Birkhoff%27s_axioms"
      }
    ],
    "tag": "A67HV2KCHR",
    "uses": [
      "08J2M1JYAG",
      "0MP9BCF6DP",
      "0GJAJC6TNZ",
      "0CTNFGYMM2"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/A67HV2KCHR/",
  "links": [
    {
      "field": "defines",
      "target": "6BB0KD6752"
    },
    {
      "field": "uses",
      "target": "08J2M1JYAG"
    },
    {
      "field": "uses",
      "target": "0MP9BCF6DP"
    },
    {
      "field": "uses",
      "target": "0GJAJC6TNZ"
    },
    {
      "field": "uses",
      "target": "0CTNFGYMM2"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/A67HV2KCHR.md",
  "msc": [
    "51M05",
    "51F05"
  ],
  "name": "Birkhoff's axioms",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/A67/A67HV2KCHR.md",
  "status": null,
  "tag": "A67HV2KCHR",
  "uri": "m2://A67HV2KCHR",
  "url": "https://tamnd.github.io/math2/wiki/birkhoffs-axioms-definition/",
  "writing": {
    "tag": "A67HV2KCHR",
    "kind": "definition",
    "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
      }
    ]
  }
}