{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Description\n\nThe *24-cell* is the convex polytope in $\\R^4$ whose vertices are the $24$ points obtained from $(\\pm 1, \\pm 1, 0, 0)$ by permuting the coordinates.\nIt is a convex [[regular polytope|F60PD912D5]] with [[Schläfli symbol|MAE6BF4DFK]] $\\set{3, 4, 3}$.\nIt has $24$ vertices, $96$ edges, $96$ triangular faces and $24$ cells, which are regular octahedra.\n\nCheck of the vertex and edge counts: there are $\\binom{4}{2} = 6$ ways to place the two nonzero coordinates and $4$ choices of signs, so $24$ vertices.\nAll vertices lie at distance $\\sqrt2$ from the origin.\nThe nearest neighbours of $(1,1,0,0)$ are the $8$ points $(1,0,\\pm1,0)$, $(1,0,0,\\pm1)$, $(0,1,\\pm1,0)$, $(0,1,0,\\pm1)$, at distance $\\sqrt2$.\nSo each vertex lies on $8$ edges, and there are $24 \\cdot 8 / 2 = 96$ edges.\nThe Euler characteristic check $24 - 96 + 96 - 24 = 0$ holds, as for every convex $4$-polytope.\n\n## Remarks\n\nThe vertex figure is a cube, in agreement with the symbol: the last two entries $\\set{4, 3}$ are the symbol of the cube.\nThe symbol reads the same backwards, and indeed the 24-cell is self-dual.\nApart from polygons and simplices, it is the only self-dual convex regular polytope, and it has no analogue in other dimensions.\nIts symmetry group has order $1152$.\nAnother model: the $24$ units $\\pm 1$, $\\pm i$, $\\pm j$, $\\pm k$ and $\\frac12(\\pm 1 \\pm i \\pm j \\pm k)$ of the [[Hurwitz quaternions|QHTAEPTKEB]] are the vertices of a 24-cell.\n\n## Explanations\n\n### L1 human\n\nThe 24-cell is a four-dimensional solid built from $24$ octahedra, glued so that each triangle is shared by two of them.\nIt is one of the six perfectly regular solids of four-dimensional space.\nUnlike the other five, it is its own dual and it has no cousin among the regular solids of three or five dimensions.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "F60PD912D5"
    ],
    "kind": "object",
    "msc": [
      "51M20",
      "52B15",
      "52B11"
    ],
    "name": "24-cell",
    "same_as": {
      "wikidata": "Q539008",
      "wikipedia": "24-cell"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/24-cell"
      }
    ],
    "tag": "SH9NBM8JRR",
    "uses": [
      "MAE6BF4DFK",
      "QHTAEPTKEB"
    ]
  },
  "kind": "object",
  "link": "https://tamnd.github.io/math2/node/SH9NBM8JRR/",
  "links": [
    {
      "field": "instance_of",
      "target": "F60PD912D5"
    },
    {
      "field": "uses",
      "target": "MAE6BF4DFK"
    },
    {
      "field": "uses",
      "target": "QHTAEPTKEB"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/SH9NBM8JRR.md",
  "msc": [
    "51M20",
    "52B15",
    "52B11"
  ],
  "name": "24-cell",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SH9/SH9NBM8JRR.md",
  "status": null,
  "tag": "SH9NBM8JRR",
  "uri": "m2://SH9NBM8JRR",
  "url": "https://tamnd.github.io/math2/wiki/24-cell/",
  "writing": {
    "tag": "SH9NBM8JRR",
    "kind": "object",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}