We say that a goldberg polyhedron $\operatorname{GP}(m,n)$ has a hexagon diameter if, in spherical form, there is a diameter of the sphere that only intersects hexagons, and goes through the center of any hexagon it intersects. For which $m$ and $n$ does $\operatorname{GP}(m,n)$ have a hexagon diameter?
Observations:
- For $k \in \mathbb Z$, it appears that $\operatorname{GP}(2k,0)$ is an example and $\operatorname{GP}(2k+1,0)$ is not, ($\operatorname{GP}(2k+1,0)$ has a diameter going through hexagons only, but it doesn't go through their centers).
- If $\operatorname{GP}(m,n)$ is an example, then so is $\operatorname{GP}(n,m)$.
(The reason I'm wondering is I'm trying to make a board game a projective sphere, and hemisphere where the diameter is all tiles is easier to use.)