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  "body": "\n## Summary\n\nA lattice in $\\R^n$ is the set of all integer combinations of $n$ linearly independent vectors.\nEquivalently, it is a discrete subgroup of $\\R^n$ that spans $\\R^n$.\nAs a group it is [[free abelian|0WJX8MGG37]] of rank $n$, and its covolume, the volume of a fundamental parallelepiped, measures how dense it is.\nLattices are the basic objects of the geometry of numbers, through [[Minkowski's theorem|0K387S2PKQ]], and they appear in number fields, in crystallography and in cryptography.\n\n## Explanations\n\n### L1 human\n\nPicture the points with integer coordinates in the plane: a regular grid.\nAny grid obtained by taking two independent arrows and all their integer combinations is a lattice.\nSquares, slanted parallelograms and the hexagonal pattern of a honeycomb all come from lattices.\nFor example, the vectors $(1, 0)$ and $(1/2, \\sqrt{3}/2)$ generate the hexagonal lattice.\nDifferent pairs of arrows can generate the same lattice, but the area of the parallelogram they span is always the same.\n",
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