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  "body": "\n## Summary\n\nA polarity of a projective space is a correlation of order $2$.\nIt sends points to hyperplanes and hyperplanes to points, it reverses inclusion, and applying it twice gives the identity.\nPolarities are the geometric form of non-degenerate reflexive forms, such as symmetric, alternating and Hermitian forms.\nThey underlie [[duality in projective geometry|DCMP1A0138]], the theory of pole and polar for conics, and [[polar spaces|Z7T0V2VAWJ]].\n\n## Explanations\n\n### L1 human\n\nFor a circle in the plane, each point outside it has a polar line, the line through the two points where the tangents from it touch the circle.\nEach line has a pole, and going from point to line and back returns the starting point.\nThis exchange of points and lines that preserves incidence is a polarity.\nIn linear algebra, taking the orthogonal complement with respect to a dot product is the same kind of operation.\n",
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