TITLE: Is it possible to have a spherical object with only hexagonal faces?
QUESTION [45 upvotes]: If so, what would be the most efficient algorithm for generating spheres with different number of hexagonal faces at whatever interval required to make them fit uniformly or how might you calculate how many hexagonal faces are required for each subdivision?

REPLY [10 votes]: If a compromise is acceptable, the $2p, 4p, 8p, ... $ subdivisions of  each side of an icosahedron in Buckminster Fuller domes  leave behind 12 pentagons at each of its 12 vertices. 
Else, possible by means of a stereographic projection of a flat regular hexagonal net (a node of which touches south pole and other nodes/junctions connect to north pole), the curvilinear hexagon boundary cells shrink to zero towards north pole according to standard stereographic scaling. They can be seen on POV Ray image provided by user PM 2Ring below. A part of spiral has been traced connecting opposite vertices of some hexagons.

It is a collection of log spirals centered at south pole that  isogonally ( i.e., conformally) project to rhumb-lines (loxodromes constant inclination to meridians at $\pm \pi/6$) with corresponding latitude circles drawn.  I could make an image later if you wish to see. Since these loxodromes are not overtly seen on the above image,  some segments are indicated across some hexagon diameters.