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  "body": "\n## Description\n\nLet $F$ be a field.\nThe *Klein quadric* is the [[quadric|Q3Q4FF23VG]] in $\\mathbb{P}^5(F)$, with coordinates $[p_{01} : p_{02} : p_{03} : p_{12} : p_{13} : p_{23}]$, given by\n$$p_{01} p_{23} - p_{02} p_{13} + p_{03} p_{12} = 0.$$\nSending a line of $\\mathbb{P}^3(F)$ to its [[Plücker coordinates|B8PAMSJ81T]] is a bijection from the set of lines of $\\mathbb{P}^3(F)$ onto the Klein quadric.\nThis bijection is the Klein correspondence.\n\nTwo lines of $\\mathbb{P}^3(F)$ meet if and only if the line of $\\mathbb{P}^5(F)$ joining their images lies entirely on the Klein quadric.\nCheck on an example: the lines $L_1$ through $e_0, e_1$ and $L_2$ through $e_0, e_2$ meet at $[e_0]$.\nTheir images are $[1 : 0 : 0 : 0 : 0 : 0]$ and $[0 : 1 : 0 : 0 : 0 : 0]$.\nEvery point $[s : t : 0 : 0 : 0 : 0]$ of the joining line satisfies the equation, since each product in it has a zero factor.\n\n## Remarks\n\nThe quadratic form $p_{01} p_{23} - p_{02} p_{13} + p_{03} p_{12}$ is a sum of three hyperbolic planes, so the Klein quadric is a non-degenerate quadric of maximal Witt index in $\\mathbb{P}^5(F)$.\nIts largest linear subspaces are planes, and they come in two families.\nThe planes of one family correspond to the sets of lines through a fixed point of $\\mathbb{P}^3(F)$.\nThe planes of the other family correspond to the sets of lines in a fixed plane of $\\mathbb{P}^3(F)$.\nOver a finite field $\\F_q$ the Klein quadric has $(q^2 + 1)(q^2 + q + 1)$ points, the number of lines of $PG(3, q)$.\n\n## Explanations\n\n### L1 human\n\nThe lines of three-dimensional projective space form a four-dimensional family.\nThe Klein quadric packages this family as a single curved hypersurface in a five-dimensional projective space.\nEach line becomes one point, and questions about lines, such as whether two lines meet, become questions about points on the quadric.\nThe coordinates are named after Julius Plücker and the quadric after Felix Klein.\n",
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