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  "body": "\n## Summary\n\nLet $P$ be a [[convex polytope|300K6CB1KH]] with the origin in its interior.\nIts polar polytope is the set $P^\\circ$ of vectors $y$ with $x \\cdot y \\le 1$ for all $x \\in P$.\nIt is again a convex polytope, and taking the polar twice gives back $P$.\nPolarity reverses the order of faces: vertices of $P$ correspond to [[facets|DPDXN2FJC4]] of $P^\\circ$ and facets of $P$ to vertices of $P^\\circ$.\nIt turns [[simple polytopes|7E6FXTGH2Y]] into [[simplicial polytopes|YVJ15PGKDM]] and back, and it is the geometric form of the dual polyhedron.\n\n## Explanations\n\n### L1 human\n\nPolarity swaps corners and walls.\nThe polar of a cube centred at the origin is an octahedron.\nThe $6$ faces of the cube become the $6$ corners of the octahedron, and the $8$ corners of the cube become its $8$ faces.\nThe polar of a regular dodecahedron is an icosahedron.\nA large polytope has a small polar, and the other way round.\n",
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  "name": "Polar polytope",
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