Generalized polygon

Summary

A generalized $n$-gon is an incidence structure of points and lines whose incidence graph has diameter $n$ and girth $2n$. Ordinary $n$-gons are the trivial examples, projective planes are the generalized $3$-gons, and the generalized quadrangles are the generalized $4$-gons. Jacques Tits introduced them as the buildings of rank $2$, and the Feit–Higman theorem limits the finite thick ones to $n \in \set{2, 3, 4, 6, 8}$.