{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $P \\subseteq \\R^d$ be a [[convex polytope|300K6CB1KH]] of dimension $d$.\nA *flag* of $P$ is a sequence $(F_0, F_1, \\dots, F_{d-1})$ of [[faces|KY7SB854VH]] of $P$ with $F_0 \\subset F_1 \\subset \\cdots \\subset F_{d-1}$ and $\\dim F_i = i$ for each $i$.\nThe [[symmetry group|0ZG3Z5JZ85]] of $P$ is the group of isometries of $\\R^d$ that map $P$ onto itself.\nIt acts on the set of flags of $P$.\nThe polytope $P$ is *regular* if this action is [[transitive|02GK6HZYBT]].\nThat is, for any two flags $\\Phi$ and $\\Psi$ of $P$ there is a symmetry $g$ of $P$ with $g(\\Phi) = \\Psi$.\n\n## Remarks\n\nExample: the cube $[0,1]^3$ has $8$ vertices, each vertex lies on $3$ edges, and each edge lies on $2$ square faces, so it has $8 \\cdot 3 \\cdot 2 = 48$ flags.\nIts symmetry group has order $48$.\nIt can move any vertex to any vertex and then permute the three edges at a vertex freely, so it acts transitively on flags.\nSo the cube is regular.\n\nNon-example: the rectangle $[0,2] \\times [0,1]$ has $8$ flags but only $4$ symmetries.\nNo symmetry maps a flag whose edge has length $2$ to a flag whose edge has length $1$, so the action is not transitive.\n\nFor a convex polytope, a symmetry that fixes one flag is the identity, so the action of the symmetry group on the flags of a regular polytope is simply transitive.\nIn dimension $2$ the regular polytopes are the [[regular polygons|01KJJ9A400]], and in dimension $3$ they are the [[Platonic solids|017E5PRS1Q]].\nThe same flag condition defines regular star polytopes, which are not convex.\nIt also defines regular abstract polytopes, where the symmetry group is replaced by the automorphism group of a partially ordered set.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "F60PD912D5",
    "kind": "definition",
    "msc": [
      "51M20",
      "52B15"
    ],
    "name": "Regular polytope",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Regular_polytope"
      }
    ],
    "tag": "NHVQ6G3Z85",
    "uses": [
      "300K6CB1KH",
      "KY7SB854VH",
      "0ZG3Z5JZ85",
      "02GK6HZYBT"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/NHVQ6G3Z85/",
  "links": [
    {
      "field": "defines",
      "target": "F60PD912D5"
    },
    {
      "field": "uses",
      "target": "300K6CB1KH"
    },
    {
      "field": "uses",
      "target": "KY7SB854VH"
    },
    {
      "field": "uses",
      "target": "0ZG3Z5JZ85"
    },
    {
      "field": "uses",
      "target": "02GK6HZYBT"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/NHVQ6G3Z85.md",
  "msc": [
    "51M20",
    "52B15"
  ],
  "name": "Regular polytope",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NHV/NHVQ6G3Z85.md",
  "status": null,
  "tag": "NHVQ6G3Z85",
  "uri": "m2://NHVQ6G3Z85",
  "url": "https://tamnd.github.io/math2/wiki/regular-polytope-definition/",
  "writing": {
    "tag": "NHVQ6G3Z85",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}