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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $E(n)$ be the [[Euclidean group|0H3YY8CPKH]] of all [[isometries|0J1N3WTKM4]] of $\\R^n$.\nEach element of $E(n)$ is a map $x \\mapsto Ax + b$ with $A$ in the [[orthogonal group|0TKYAEJ8N7]] $O(n)$ and $b \\in \\R^n$, and $E(n)$ carries the topology of $O(n) \\times \\R^n$.\nA subgroup $\\Gamma$ of $E(n)$ is a *crystallographic group* of dimension $n$ if:\n\n1. $\\Gamma$ is a discrete subgroup of $E(n)$.\n2. The quotient space $\\R^n / \\Gamma$ of orbits is [[compact|0TVJZYB6SH]].\n\n## Remarks\n\nExample: the group $\\Z^n$ of [[translations|087VJM5Q77]] $x \\mapsto x + b$ with $b \\in \\Z^n$.\nIt is discrete, since distinct elements differ by a translation of length at least $1$.\nThe quotient $\\R^n / \\Z^n$ is the image of the compact cube $[0, 1]^n$, so it is compact; it is the $n$-dimensional torus.\n\nNon-example: the translations $(x, y) \\mapsto (x + k, y)$ of $\\R^2$ with $k \\in \\Z$.\nThis group is discrete, but the quotient is an infinite cylinder, which is not compact, so clause 2 fails.\n\nEquivalently, $\\Gamma$ acts properly discontinuously on $\\R^n$ with a compact fundamental domain.\nCrystallographic groups of dimension $3$ are called space groups, and those of dimension $2$ are the [[wallpaper groups|MY3CBHARYQ]].\n",
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