{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a finite simple [[graph|00C0BR7D7H]].\nThen the following are equivalent.\n\n1. $G$ is isomorphic to the graph of vertices and edges of a $3$-dimensional [[convex polytope|300K6CB1KH]].\n2. $G$ is [[planar|0NC9R64TXP]] and [[3-vertex-connected|7FZB7YYBXA]].\n\n## Remarks\n\nOne direction is easier.\nThe graph of a $3$-polytope is planar: project its boundary onto one face from a point just outside that face.\nIt is $3$-connected by Balinski's theorem.\nFor the converse, one proof reduces every $3$-connected planar graph to the complete graph $K_4$.\nIt uses moves that replace a triangle by a three-legged star and back, and shows that each move can be carried out on polytopes.\nAnother proof uses the [[circle packing theorem|GR1DDT1B6B]] to realize the graph with all edges tangent to the unit sphere.\n\nExample: the cube graph, with $8$ vertices of degree $3$, is planar and $3$-connected, and the cube realizes it.\nWithout $3$-connectivity the conclusion fails: the cycle graph $C_4$ is planar but only $2$-connected.\nNo $3$-polytope has it as its graph, because every vertex of a $3$-polytope has degree at least $3$.\n\n## Explanations\n\n### L1 human\n\nWhich networks of dots and lines are the skeletons of convex solids in space?\nSteinitz's theorem gives a complete answer: exactly those that can be drawn in the plane without crossings and that stay connected after removing any two dots.\nSo the combinatorics of three-dimensional polytopes is pure graph theory.\n\n### L2 human\n\nThe theorem lets one count and list combinatorial types of $3$-polytopes by counting $3$-connected planar graphs.\nIt also shows that every combinatorial $3$-polytope can be realized with rational, even integer, coordinates, and with many extra properties, such as all edges tangent to a sphere.\nIn dimension $4$ and higher no such characterisation is known, and realizability of a given face lattice is a hard problem.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "300K6CB1KH",
      "0NC9R64TXP"
    ],
    "kind": "statement",
    "msc": [
      "52B10",
      "05C10"
    ],
    "name": "Steinitz's theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7606897",
      "wikipedia": "Steinitz's theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Steinitz%27s_theorem"
      }
    ],
    "tag": "2PYAZX5VD4",
    "uses": [
      "300K6CB1KH",
      "0NC9R64TXP",
      "7FZB7YYBXA",
      "00C0BR7D7H",
      "GR1DDT1B6B"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/2PYAZX5VD4/",
  "links": [
    {
      "field": "about",
      "target": "300K6CB1KH"
    },
    {
      "field": "about",
      "target": "0NC9R64TXP"
    },
    {
      "field": "uses",
      "target": "300K6CB1KH"
    },
    {
      "field": "uses",
      "target": "0NC9R64TXP"
    },
    {
      "field": "uses",
      "target": "7FZB7YYBXA"
    },
    {
      "field": "uses",
      "target": "00C0BR7D7H"
    },
    {
      "field": "uses",
      "target": "GR1DDT1B6B"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/2PYAZX5VD4.md",
  "msc": [
    "52B10",
    "05C10"
  ],
  "name": "Steinitz's theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2PY/2PYAZX5VD4.md",
  "status": null,
  "tag": "2PYAZX5VD4",
  "uri": "m2://2PYAZX5VD4",
  "url": "https://tamnd.github.io/math2/wiki/steinitzs-theorem/",
  "writing": {
    "tag": "2PYAZX5VD4",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}