Generalized polygon

Remarks

Example: the Fano plane is a generalized $3$-gon. Two points lie on a common line, and two lines meet in a point, so in both cases the distance is $2$. A point and a line not through it are at distance $3$, so the diameter is $3$. A cycle of length $4$ would give two points on two common lines, which is impossible. A triangle of the plane gives a cycle of length $6$, so the girth is $6$. It is thick, since each line has $3$ points and each point is on $3$ lines.

Non-example: the affine plane of order $2$, with $4$ points and the $6$ lines of $2$ points each. Its girth is $6$, but two parallel lines have no common point, so they are at distance $4$; the diameter is $4 \ne 3$.

For $n \ge 3$, any two points lie on at most one line, so a generalized $n$-gon is a partial linear space. The generalized $3$-gons with at least three points on each line and three lines on each point are exactly the projective planes, and the generalized $4$-gons are the generalized quadrangles. A finite thick generalized $n$-gon has an order $(s, t)$: every line has $s + 1$ points and every point is on $t + 1$ lines.