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  "body": "\n## Summary\n\nA generalized $n$-gon is an incidence structure of points and lines whose incidence graph has diameter $n$ and girth $2n$.\nOrdinary $n$-gons are the trivial examples, [[projective planes|DANFNN8C4Y]] are the generalized $3$-gons, and the [[generalized quadrangles|GQJCPFSYVK]] are the generalized $4$-gons.\nJacques Tits introduced them as the [[buildings|6BQJ50NEJ0]] of rank $2$, and the [[Feit–Higman theorem|P2V0XS4F63]] limits the finite thick ones to $n \\in \\set{2, 3, 4, 6, 8}$.\n\n## Explanations\n\n### L1 human\n\nDraw a triangle: three points, three sides, each side with two points and each point on two sides.\nMake a graph whose vertices are the points and the sides, with an edge when a point lies on a side.\nThis graph is a cycle of length $6$: any two vertices are at distance at most $3$, and the shortest cycle has length $6 = 2 \\cdot 3$.\nA generalized $3$-gon keeps these two numbers but allows many more points on each line and many more lines through each point.\nThat is exactly a projective plane: two points always share a line, and two lines always share a point.\n",
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