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  "body": "\n## Description\n\nA finite projective plane is a finite set of points with a family of subsets, called lines, with three properties.\nEvery two distinct points lie on exactly one line, and every two distinct lines meet in exactly one point.\nThere are four points with no three on a line.\nEach such plane has an order: an integer $n \\ge 2$ such that each line has $n + 1$ points and each point lies on $n + 1$ lines.\nA plane of order $n$ has $n^2 + n + 1$ points and $n^2 + n + 1$ lines.\n\nThe standard construction is $PG(2, q)$ for a prime power $q$.\nLet $\\F_q$ be the [[finite field|08RTDHX1SK]] with $q$ elements.\nThe points are the $1$-dimensional [[subspaces|0EPTJA5QV4]] of $\\F_q^3$ and the lines are its $2$-dimensional subspaces, with incidence given by inclusion.\nCheck: there are $(q^3 - 1)/(q - 1) = q^2 + q + 1$ points.\nTwo distinct $1$-dimensional subspaces span exactly one $2$-dimensional subspace.\nTwo distinct $2$-dimensional subspaces $U, W$ satisfy $\\dim(U \\cap W) = 2 + 2 - \\dim(U + W) = 1$, so they meet in exactly one point.\nThe four points spanned by $(1,0,0)$, $(0,1,0)$, $(0,0,1)$ and $(1,1,1)$ have no three on a line.\nSo $PG(2, q)$ is a projective plane of order $q$.\n\n## Remarks\n\nA projective plane of order $n$ is the same thing as a [[Steiner system|7BPSC4T4M3]] $S(2, n + 1, n^2 + n + 1)$, a symmetric [[block design|4JCSS2Z1BA]].\nThe plane of order $2$ is the [[Fano plane|QQ8GW1ASZM]].\nA projective plane of order $n$ exists if and only if there are $n - 1$ [[mutually orthogonal Latin squares|H6TN6BADVY]] of order $n$.\nEvery prime power is the order of a plane.\nThe [[Bruck–Ryser–Chowla theorem|R43K0GTTDS]] rules out the orders $6$, $14$, $21$ and many more.\nA long computer search by Lam and others, finished in 1989, showed that there is no plane of order $10$.\nNo plane of an order that is not a prime power is known.\n\n## Explanations\n\n### L1 human\n\nA finite projective plane is a small geometry with finitely many points in which any two points lie on one line.\nUnlike in the usual plane, any two lines meet.\nThere are no parallel lines.\nThe smallest one has $7$ points and $7$ lines, each line with $3$ points.\n",
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