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  "body": "\n## Definition\n\nA *projective plane* is an [[incidence structure|FGKNAFNHA3]] of points and lines such that:\n\n1. Any two distinct points lie on exactly one common line.\n2. Any two distinct lines have exactly one common point.\n3. There are four points, no three of which are [[collinear|0AFNNTRTJE]].\n\n## Remarks\n\nExample: for a field $k$, the projective plane $\\mathbb{P}^2(k)$ has as points the one-dimensional subspaces of $k^3$ and as lines the two-dimensional subspaces, with incidence given by inclusion.\nCheck: two distinct one-dimensional subspaces span exactly one plane.\nTwo distinct planes in $k^3$ meet in a line through $0$, by the dimension formula $2 + 2 - 3 = 1$.\nThe points $[1:0:0]$, $[0:1:0]$, $[0:0:1]$, $[1:1:1]$ have no three collinear.\nFor $k = \\F_2$ this is the Fano plane, with $7$ points and $7$ lines.\n\nNon-example: an [[affine plane|EC7XT2ZM5S]] satisfies clauses 1 and 3 but not clause 2, since it has parallel lines.\n\nClause 3 excludes degenerate cases, such as a single line with all points on it.\nThe axioms are self-dual: exchanging points and lines turns each clause into a statement that also holds, so the dual of a projective plane is a projective plane.\nIn a [[finite projective plane|2HWB7P1DZN]] every line has $n + 1$ points for some $n \\ge 2$, the order of the plane.\n",
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