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  "body": "\n## Summary\n\nA geometric lattice is a finite lattice that is [[atomistic|99DVVV7BH2]] and [[semimodular|C1NET23GG0]].\nThese lattices are exactly the lattices of flats of finite [[matroids|PX31ZNYS3Z]].\nSo they give an order-theoretic description of matroids, in which the atoms are the points and the rank function measures dimension.\nLattices of subspaces of a finite vector space, Boolean lattices of subsets of a finite set and lattices of partitions are geometric.\n\n## Explanations\n\n### L1 human\n\nTake finitely many points in space and consider the \"flats\" they span: the points, the lines through two of them, the planes through three, and so on.\nOrdered by inclusion, these flats form a geometric lattice.\nEvery flat is the join of the points it contains, and joining one more point raises the dimension by at most one.\nFor three points in general position in the plane, the lattice has the empty flat, three points, three lines and the whole plane.\n",
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