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  "body": "\n## Definition\n\nLet $s \\ge 1$ and $t \\ge 1$ be integers.\nA *generalized quadrangle of order $(s, t)$* is an [[incidence structure|FGKNAFNHA3]] $(P, L, I)$ of points and lines such that:\n\n1. each point lies on exactly $t + 1$ lines, and two distinct points lie on at most one common line;\n2. each line contains exactly $s + 1$ points, and two distinct lines meet in at most one common point;\n3. if a point $p$ is not on a line $\\ell$, exactly one line through $p$ meets $\\ell$.\n\n## Remarks\n\nCounting gives $(s + 1)(st + 1)$ points and $(t + 1)(st + 1)$ lines.\nA generalized quadrangle is the same as a [[generalized polygon|PCT2HY9AAZ]] with $n = 4$.\nThe finite [[polar spaces|Z7T0V2VAWJ]] of rank $2$ are generalized quadrangles.\n\nExample: the $(s + 1) \\times (s + 1)$ grid, with the rows and columns as lines, has order $(s, 1)$.\nClauses 1 and 2 are clear.\nFor clause 3, let $p$ be off the row $\\ell$.\nThe column through $p$ meets $\\ell$ in one point $q$, and the row through $p$ is disjoint from $\\ell$, so exactly one line through $p$ meets $\\ell$.\nThe same check works with rows and columns exchanged.\n\nNon-example: in a projective plane of order $n \\ge 2$, every line through a point $p$ off $\\ell$ meets $\\ell$, so $n + 1 \u003e 1$ lines through $p$ meet $\\ell$ and clause 3 fails.\n\nThe symplectic quadrangle $W(q)$, formed by the points of $\\mathrm{PG}(3, q)$ and the totally isotropic lines of a symplectic form, has order $(q, q)$.\n",
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