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  "body": "\n## Example\n\nA wallpaper group is a [[crystallographic group|7KTJKFQT2N]] of dimension $2$: a discrete group of isometries of the plane $\\R^2$ with compact quotient.\nA concrete one, called $p4$, is the group $\\Gamma$ of all maps\n$$v \\mapsto R^k v + t, \\qquad k \\in \\set{0, 1, 2, 3}, \\ t \\in \\Z^2,$$\nwhere $R(x, y) = (-y, x)$ is the rotation by a quarter turn about the origin.\nThe set $\\Gamma$ is a group, because $R$ maps $\\Z^2$ onto itself, so composites and inverses of such maps have the same form.\nIt is discrete, because its elements have only four linear parts and their translation parts lie in $\\Z^2$.\nThe quotient is compact, because every orbit meets the unit square $[0, 1]^2$, already under the translations alone.\nThe translation subgroup is $\\Z^2$, of index $4$, as part 1 of the [[Bieberbach theorems|YE87R5CDJY]] predicts.\n\n## Remarks\n\nUp to isomorphism, and also up to affine change of coordinates, there are exactly $17$ wallpaper groups.\n\n## Explanations\n\n### L1 human\n\nA wallpaper group is the symmetry group of a pattern that repeats in two directions across the whole plane, like tiles on a bathroom floor.\nThe pattern $p4$ is a grid of square tiles each marked with a pinwheel, so it can be turned by a quarter turn but not reflected.\nAlthough such patterns come in endless designs, only $17$ different symmetry groups occur.\n",
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