# Geodesic sphere using only Regular pentagons and hexagons

I know geodesic approximation to a construct a spherical dome shape needs 12 pentagons and these pentagons are regular pentagons.

However when I look closely hexagons are slightly different in their shapes and sizes. Is it mathematically possible to construct a geodesic sphere using 12 pentagons and REGULAR hexagons of the SAME SIZE? (for example like Truncated Icosahedron)

Would putting extra pentagons to force curvature between the regular hexagons solve this?

Using an $\ne 12$ pentagons (and otherwise only hexagons) will not give you a sphere because of Euler's polyhedron formula (unless you do not let three polygons meet at every vertex, but then your shape would be even more irregular).