Consider a tiling of 2D space with polygons (tiles do not have to be the same shape). If every vertex must be connected to three lines, can we make a statement about the ratio of vertices to faces? In the hexagonal and truncated hexagonal tilings this ratio is 2:1. But how can this be proven for all tilings, if it is true?
We can use the Euler characteristic chi for that. It is defined as
chi = v - e + f
v ... number of vertices
e ... number of edges
f ... number of faces
A finite plane has Euler characteristic chi = 1, an infinite plane (which is similar to a torus) has Euler characteristic chi = 0.
Given the constraint that each vertex is connected to three edges (and each edge is connected to two vertices), we have
2e = 3v
Plugging that into the definition of the Euler characteristic, we get:
chi = v - 3/2 v + f
= f - 1/2 v
In the case of an infinite plane (chi = 0), we then get
0 = f - 1/2 v
1/2 v = f
v = 2 f
And this is the ratio that you mentioned. Hence, it is not only true for hexagonal tilings but all tilings where each vertex is connected to three edges, regardless of what polygons are used.