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  "body": "\n## Summary\n\nThe geometry of numbers studies [[lattices|WN9V7WSJKF]] in $\\R^n$ together with convex bodies and other regions, and uses volume arguments to find lattice points in them.\nIt was founded by Hermann Minkowski, whose [[convex body theorem|0K387S2PKQ]] says that a large symmetric convex set contains a nonzero lattice point.\nIts main tools include [[Blichfeldt's theorem|A3FTHW4NES]], [[successive minima|YDMBKD8FP2]] and [[Minkowski's second theorem|5H4XZVF9B9]].\nIt gives results in algebraic number theory, Diophantine approximation and quadratic forms, and it is closely tied to sphere packing.\n\n## Explanations\n\n### L1 human\n\nMany questions about whole numbers can be turned into questions about points of a grid.\nThe geometry of numbers answers them by looking at shapes and their areas or volumes.\nIf a shape is large enough and symmetric about the origin, it must contain a grid point other than the origin.\nFor instance, this idea shows that every prime of the form $4k + 1$ is a sum of two squares.\n",
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