{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $r \u003e 0$.\nA *sphere packing* in the [[Euclidean space|0XPR1E85W0]] $\\R^n$ with radius $r$ is a set $C \\subset \\R^n$ of centres such that $\\abs{x - y} \\ge 2r$ for all distinct $x, y \\in C$.\nThe packing is the family of [[closed balls|0Q0H13G8ZB]] $B(x, r)$ with $x \\in C$; their interiors are pairwise disjoint.\nThe packing is a *lattice packing* if $C$ is a [[lattice|WN9V7WSJKF]], or a translate of one.\n\n## Remarks\n\nExample: in $\\R^2$ take $C = \\Z^2$ and $r = \\tfrac12$.\nTwo distinct integer points are at distance at least $1 = 2r$, so the discs of radius $\\tfrac12$ centred at them form a lattice packing, the square packing.\nNon-example: $C = \\Z^2$ with $r = 0.6$; the points $(0, 0)$ and $(1, 0)$ are at distance $1 \u003c 1.2$, so their discs overlap.\nFor a lattice $\\Lambda$ with shortest nonzero vector of length $m$, the largest radius is $r = m/2$.\nSome authors allow balls of different radii; the term \"sphere packing\" then refers to any family of balls with disjoint interiors.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "HF32CBN5MA",
    "kind": "definition",
    "msc": [
      "52C17",
      "11H31",
      "05B40"
    ],
    "name": "Sphere packing",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Sphere_packing"
      }
    ],
    "tag": "T4PGC6NH56",
    "uses": [
      "0XPR1E85W0",
      "0Q0H13G8ZB",
      "WN9V7WSJKF"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/T4PGC6NH56/",
  "links": [
    {
      "field": "defines",
      "target": "HF32CBN5MA"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    },
    {
      "field": "uses",
      "target": "0Q0H13G8ZB"
    },
    {
      "field": "uses",
      "target": "WN9V7WSJKF"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/T4PGC6NH56.md",
  "msc": [
    "52C17",
    "11H31",
    "05B40"
  ],
  "name": "Sphere packing",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/T4P/T4PGC6NH56.md",
  "status": null,
  "tag": "T4PGC6NH56",
  "uri": "m2://T4PGC6NH56",
  "url": "https://tamnd.github.io/math2/wiki/sphere-packing-definition/",
  "writing": {
    "tag": "T4PGC6NH56",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}