# The number of faces in a Farey diagram/graph

A Farey line of order $(m, n)$ is a line given by an equation $ux + vy + w = 0$ with $u, v, w ∈ \mathbb{Z}$, $-m \leq u \leq m$, $-n \leq v \leq n$ and which intersects the border of $[0, 1]^2$ in at least two points.

A Farey diagram of order $(m, n)$ is a planar graph whose vertices are intersection points of Farey lines of order $(m, n)$ in $[0, 1]^2$ and the edges are the segments of the Farey lines connecting two consecutive vertices.

This paper gives an upper bound on the number of vertices and faces in a Farey diagram.

I am looking for any results/ideas which would help to estimate asymptotically the number of faces of a Farey diagram. Also, it seems that all faces of a Farey diagram are triangles or quadrangles, and it would be helpful to have an explanation of this observation.

The Farey diagram of (3,3)-order