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  "body": "\n## Definition\n\nLet $\\Pi = (P, L, I)$ be a [[projective plane|DANFNN8C4Y]], seen as an [[incidence structure|FGKNAFNHA3]].\nIts *dual plane* is $\\Pi^{*} = (L, P, I^{\\top})$, whose points are the lines of $\\Pi$, whose lines are the points of $\\Pi$, and where $(\\ell, p) \\in I^{\\top}$ exactly when $(p, \\ell) \\in I$.\nThe *dual* of a statement about points, lines and incidence is the statement obtained by exchanging \"point\" with \"line\" and \"lies on\" with \"passes through\".\nA *duality* (or *correlation*) of $\\Pi$ is an isomorphism of incidence structures from $\\Pi$ to $\\Pi^{*}$.\n\n## Remarks\n\nThe axioms of a projective plane are self-dual, so $\\Pi^{*}$ is again a projective plane.\nHence the *principle of duality*: if a statement holds in every projective plane, so does its dual.\n\nExample: $\\mathbb{P}^2(k)$ over a field $k$ has a duality sending the point $[a : b : c]$ to the line $a x_0 + b x_1 + c x_2 = 0$ and the line $a x_0 + b x_1 + c x_2 = 0$ to the point $[a : b : c]$.\nCheck: the point $[x_0 : x_1 : x_2]$ lies on the line with coefficients $(a, b, c)$ exactly when $a x_0 + b x_1 + c x_2 = 0$, a condition symmetric in the two triples, so incidence is preserved in both directions.\n\nNon-example: the dual of an [[affine plane|EC7XT2ZM5S]] is not an affine plane.\nTwo parallel lines become two points with no line through both, which fails the first axiom of an affine plane.\n\nLet $\\mathbb{P}(V)$ be a projective space of finite dimension $n$.\nThe map $W \\mapsto W^{0}$, sending a subspace to its annihilator in the [[dual space|07JBSZE0V4]], reverses inclusion and exchanges projective dimensions $d$ and $n - 1 - d$.\n",
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