I tried to make the simplest case that is not the icosahedron. I intended it to have just 1 hexagon. But it turned out I made the hexagonal antiprism, which has 2 hexagons!
The thing is, I can't seem to make a case with only 1 hexagon. It would not violate the Euler characteristic being 2, that always works as long as there are 12 5-nodes. So there must be other restrictions.
