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  "body": "\n## Summary\n\nAn oriented matroid is a [[matroid|PX31ZNYS3Z]] in which each minimal dependence also carries signs, recording which elements lie on which side.\nIt abstracts the combinatorics of real vector configurations, of point configurations, of [[hyperplane arrangements|N602W0WAVT]] and of directed graphs.\nFaces of [[convex polytopes|300K6CB1KH]], regions of real arrangements and the results of linear programming can all be described in terms of oriented matroids.\nNot every oriented matroid comes from vectors; by the topological representation theorem, oriented matroids correspond to arrangements of pseudospheres.\n\n## Explanations\n\n### L1 human\n\nTake some vectors in the plane or in space.\nA dependence like $v_1 + v_2 - v_3 = 0$ tells you that $v_1$ and $v_2$ lie on one side and $v_3$ on the other.\nAn oriented matroid keeps only these signs, not the numbers.\nRemarkably, a lot of geometry, such as which points form the corners of their convex hull, can be recovered from the signs alone.\n",
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