Let $S^2$ be the 2 dimensional sphere embedded in $\mathbb{R}^3$. It is well known, using the Euler characteristic, that it is not possible to tessellate $S^2$ with all congruent regular hexagons.

My question is, would be it possible to partition $S^2$ with congruent regular hexagons, and only a finite number of regular $k$-gons, with $k\neq 6$? That is, does it exist a constant $C$ such that, for any $N$, would it be possible to partition $S^2$ using at least $N-C$ congruent regular hexagons, with the remaining ones being regular $k$-gons, $k\neq 6$? ($C$ has to be independent of $N$)

I have tried using Goldberg polyhedra (icosahedral ones), which has only 12 regular pentagons and the rest being regular hexagons, and then projecting, using the gnomonic projection, the resulting partition on the sphere. However, equal sized hexagons on the icosahedron are projected to non equal sized ones on the sphere, due to the distorsions of the gnomonic projection...

Thanks!

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