Same size Hexagon on spheroid

Here’s a go at this.
It relaxes a triangular mesh to get as close to regular as it can, then takes the incircular dual, then fits a perfectly regular (and flat) hexagon to each cell of this.
The base mesh before subdivision I used here is a hexagonal antiprism, which I believe gives a better fit for this shape than simply an icosahedron. It still has only 12 pentagons, but in a different arrangement. I think any other coarse mesh with more irregular faces is likely to result in worse gaps.
Realistically, when covering a curved surface with only a single size and shape of tile, there will always have to be significant gaps. I think it might be possible to improve the result here a little, by using a slightly different energy in the relaxation, but to get much closer fit, you’d need to vary the size/shape of the tiles. (60.5 KB)