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  "body": "\n## Summary\n\nA polar space is a geometry of points and lines with one rule.\nEach point is collinear with exactly one point or with all points of any line not through it.\nThe main examples are the points and lines lying on a non-degenerate [[quadric|Q3Q4FF23VG]].\nOthers come from the [[polarities|727TZQJ4ZA]] of symplectic and Hermitian forms, using the absolute points and totally isotropic lines.\nPolar spaces are the geometries of the classical groups other than the linear groups, and the non-degenerate ones of rank $2$ are the [[generalized quadrangles|GQJCPFSYVK]].\n\n## Explanations\n\n### L1 human\n\nTake the quadric surface $x_0 x_1 + x_2 x_3 = 0$ in projective space, which is covered by two families of straight lines.\nIts points and the lines lying on it form a polar space.\nPick a line on the surface and a point of the surface off that line.\nThen the point is joined by a line of the surface to exactly one point of the line.\nIn other polar spaces a point can also be joined to all points of a line, and this one-or-all rule turns out to characterize such geometries.\n",
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