{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nThe [[kissing number|B408FZYXF9]] in dimension $3$ is $\\tau(3) = 12$.\nThat is, at most $12$ unit balls in $\\R^3$ with pairwise disjoint interiors can touch a given unit ball, and $12$ is attained.\n\n## Remarks\n\nThe lower bound comes from the face-centred cubic packing: the $12$ vectors $(\\pm 1, \\pm 1, 0)$, $(\\pm 1, 0, \\pm 1)$, $(0, \\pm 1, \\pm 1)$, scaled to norm $2$, are pairwise at distance at least $2$.\nThe configuration of the $12$ vertices of a regular icosahedron also works, and in it no two outer balls touch, so they can move.\nThe question goes back to a discussion between Isaac Newton, who held that the answer is $12$, and David Gregory, who thought $13$ might fit.\nKurt Schütte and B. L. van der Waerden gave the first complete proof in 1953.\nLater proofs bound the number of points on the sphere with pairwise angular distance at least $60^\\circ$, for example with a refined form of the [[Delsarte linear programming bound|7RPEDHAT7R]].\n\n## Explanations\n\n### L1 human\n\nHold one ball and try to make as many equal balls as possible touch it.\nTwelve fit, with some space left over, so it seems a thirteenth might squeeze in after shuffling.\nThe theorem says it never does.\n\n### L2 human\n\nThe plain linear programming bound gives only $\\tau(3) \\le 13$, so the three-dimensional case needs extra geometric arguments.\nIn dimensions $8$ and $24$ the same linear programming method is exactly sharp and gives $240$ and $196560$.\nDimension $4$ was settled by Oleg Musin, who showed $\\tau(4) = 24$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "B408FZYXF9"
    ],
    "kind": "statement",
    "msc": [
      "52C17",
      "05B40"
    ],
    "name": "Kissing number problem in three dimensions",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kissing_number"
      }
    ],
    "tag": "Q1H6BCKTBE",
    "uses": [
      "B408FZYXF9"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/Q1H6BCKTBE/",
  "links": [
    {
      "field": "about",
      "target": "B408FZYXF9"
    },
    {
      "field": "uses",
      "target": "B408FZYXF9"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/Q1H6BCKTBE.md",
  "msc": [
    "52C17",
    "05B40"
  ],
  "name": "Kissing number problem in three dimensions",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Q1H/Q1H6BCKTBE.md",
  "status": null,
  "tag": "Q1H6BCKTBE",
  "uri": "m2://Q1H6BCKTBE",
  "url": "https://tamnd.github.io/math2/wiki/kissing-number-problem-in-three-dimensions/",
  "writing": {
    "tag": "Q1H6BCKTBE",
    "kind": "statement",
    "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
      }
    ]
  }
}