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  "body": "\n## Summary\n\nA crystallographic group is a discrete group of isometries of [[Euclidean space|0XPR1E85W0]] $\\R^n$ whose quotient space is compact.\nThese groups describe the symmetries of periodic patterns, such as crystals in dimension $3$ and wallpapers in dimension $2$.\nBy the [[Bieberbach theorems|YE87R5CDJY]], each one contains a lattice of translations of finite index.\nIn each dimension there are only finitely many of them up to isomorphism.\nIn dimension $3$ they are the space groups of crystallography, and in dimension $2$ the [[wallpaper groups|MY3CBHARYQ]].\n\n## Explanations\n\n### L1 human\n\nImagine a pattern that fills all of space and repeats in every direction, such as the atoms of a salt crystal.\nThe rigid motions that carry the pattern onto itself form a crystallographic group.\nThe translations that shift the pattern by one period are always among them.\nFor the square grid of integer points in the plane, the group contains the shifts by integer vectors and the quarter turns about the grid points.\n",
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