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  "body": "\n## Description\n\nThe Fano plane has the $7$ points $1, 2, \\dots, 7$ and the $7$ lines\n$$\\set{1,2,3},\\ \\set{1,4,5},\\ \\set{1,6,7},\\ \\set{2,4,6},\\ \\set{2,5,7},\\ \\set{3,4,7},\\ \\set{3,5,6}.$$\nCheck: each line has $3$ points, so the lines cover $7 \\cdot 3 = 21$ pairs of points.\nThese $21$ pairs are distinct, and there are $\\binom{7}{2} = 21$ pairs in all, so every two points lie on exactly one line.\nEvery two lines meet in exactly one point, as one sees from the list.\nSo the Fano plane is a [[finite projective plane|2HWB7P1DZN]] of order $2$ and a [[Steiner system|7BPSC4T4M3]] $S(2, 3, 7)$.\n\n## Remarks\n\nWrite each point $i$ in binary as a nonzero vector of $\\F_2^3$.\nThen the lines are the sets $\\set{a, b, a + b}$ with $a \\ne b$, for example $1 + 6 = 7$ in binary addition.\nSo the Fano plane is $PG(2, 2)$, the projective plane over the field with two elements.\nIts automorphism group is $GL(3, 2)$, a simple group of order $168$.\nIt is the smallest projective plane.\n\n## Explanations\n\n### L1 human\n\nThe Fano plane is a tiny geometry with $7$ points and $7$ lines, each line with $3$ points.\nIt is usually drawn as a triangle with its three medians and its inscribed circle; the circle counts as a line.\nAny two points share one line and any two lines share one point.\n",
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