I thought of this question after reading a bit about Goldberg polyhedrons.
Does it make sense to suggest that a complete regular polyhedron consisting of only hexagons could exist, given an infinite size?
I don't think so. At least, I suspect it wouldn't be complete, but I'm not sure how to better reason about the question.
Intuitively, it seems as though one would always need other polygons to complete the face. However, in my mind, I can imagine that at any given localized region of the face the tiling resembles a two dimensional plane, so there's no "need" for any other polygons to appear in the face. In such a case, would there be anywhere on the face that would need a pentagon or other polygon to complete the face? If not, wouldn't that mean we've defined a regular polyhedron consisting of only hexagons?
Thanks!