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  "body": "\n## Summary\n\nThe Gale diagram, or Gale transform, turns $n$ points spanning $\\R^d$ into $n$ vectors in $\\R^{n-d-1}$.\nThe new vectors encode the affine dependences of the old points.\nThe faces of the [[convex hull|0FQSDC2EQ2]] of the points can be read off from them.\nSuppose a [[convex polytope|300K6CB1KH]] has only a few more vertices than its dimension.\nThen its Gale diagram lives in a very low dimension and can be drawn.\nThis makes Gale diagrams a main tool for classifying such polytopes and for building polytopes with unexpected properties.\n\n## Explanations\n\n### L1 human\n\nA polytope in high dimension with few corners is hard to picture directly.\nThe Gale diagram trades the dimension of the space for the number of extra corners.\nFor example, a $d$-dimensional polytope with $d + 3$ corners becomes a set of vectors in the plane.\nQuestions about faces of the polytope then become questions about which vectors surround the origin.\n",
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