[jacobolus]

One thing you can do though is tile an octahedron by just hexagons. At each of the 6 corners you end up with 2 hexagons which border each-other along 2 edges (instead of the usual 1). If you blow this octahedron up into a sphere those hexagons appear to be pentagons, because two of their edges are colinear (i.e. the same great-circle arc).

This can be nicer in some cases: the edge case your hexagon-grid algorithms have to deal with is having a hexagon with one of the same neighbors twice, instead of needing to worry about pentagons per se.

[alanbernstein]

Would you care to explain this? When you mention tiling an octahedron with hexagons, the first thing I imagine is a truncated octahedron, which is composed of hexagons and squares.

[jacobolus]

You can start with 4 hexagons, each one covering an octant and a third of the neighboring octants.

Or another way to say this: if you start with 4 hexagons, with each glued together with each other along two adjacent edges, and you add the appropriate folds, you can make an octahedron.

Then you can subdivide each of those starting hexagons into n hexagons for any of these numbers, https://oeis.org/A003136 (power-of-4 sizes may be the most convenient among these, so that the overall grid has 2^n by 2^n size)