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  "body": "\n## Definition\n\nLet $\\mathcal{S} = (P, L, I)$ be an [[incidence structure|FGKNAFNHA3]] with point set $P$, line set $L$ and incidence relation $I \\subseteq P \\times L$.\nIts *incidence graph* is the [[bipartite graph|0ZVQKTRKVQ]] with vertex set $P \\cup L$, with an edge between $p \\in P$ and $\\ell \\in L$ when $(p, \\ell) \\in I$.\nLet $n \\ge 2$ be an integer.\nThe structure $\\mathcal{S}$ is a *generalized $n$-gon* if its incidence graph is connected, has diameter $n$, and has [[girth|D13K9GJT0B]] $2n$.\nHere the diameter is the largest distance between two vertices of the [[graph|00C0BR7D7H]].\nA generalized $n$-gon is *thick* if every point lies on at least three lines and every line has at least three points.\n\n## Remarks\n\nExample: the [[Fano plane|QQ8GW1ASZM]] is a generalized $3$-gon.\nTwo points lie on a common line, and two lines meet in a point, so in both cases the distance is $2$.\nA point and a line not through it are at distance $3$, so the diameter is $3$.\nA cycle of length $4$ would give two points on two common lines, which is impossible.\nA triangle of the plane gives a cycle of length $6$, so the girth is $6$.\nIt is thick, since each line has $3$ points and each point is on $3$ lines.\n\nNon-example: the affine plane of order $2$, with $4$ points and the $6$ lines of $2$ points each.\nIts girth is $6$, but two parallel lines have no common point, so they are at distance $4$; the diameter is $4 \\ne 3$.\n\nFor $n \\ge 3$, any two points lie on at most one line, so a generalized $n$-gon is a partial linear space.\nThe generalized $3$-gons with at least three points on each line and three lines on each point are exactly the [[projective planes|DANFNN8C4Y]], and the generalized $4$-gons are the generalized quadrangles.\nA finite thick generalized $n$-gon has an order $(s, t)$: every line has $s + 1$ points and every point is on $t + 1$ lines.\n",
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