For instance there is a hexagonal tiling of the plane. There is also one using quadrilaterals. It seems intuitive that both of these tilings also apply on a torus. Is it the case that anything that maps 1-to-1 to a plane surface has a "complete" tiling with regular polygons? It seems like I should be able to tile hexes over a torus, and even over the sphere; although this seems like an edge case, since it's not 'on' the sphere except with a certain granularity and so on. I saw a Buckminster Fuller house the other days and have been wondering about this.
I will try to spell out the sense of the one-to-one mapping requirement a bit more clearly. Can I tile any surface homeomorphic to the plane? To what degree are constraints around gluing edges together the same as those for performing a tiling? It seems plausible to me that tiling is at least somewhat independent of homeomorphicity -- for instance I can imagine that there could be certain constructions permitting smooth tiling even over particularly awkward edges. (That is to say it seems possible that tiling properties don't need to have quite the same "algebra" as proximity relations; that possibly a slightly different homeomorphic relation is required?)
I note that on further reflection it seems like I'm conflating two tiling strategies in the first graph. The first being an 'internal' tiling, i.e., drawing on the actual surface. The second style being an 'approximation' tiling (like the way the Bucky house approximates a sphere with hexagonal tiles.) I hope this clarifies somewhat: that is, the question is whether the first kind of tiling is 'limited' to manifolds homeomorphic to the plane.
Some assumptions that might help clarify my intent:
- By tiling I mean the isometric "filling" of a surface by compact tiles; and in particular I am concerned with the case of simple polygonal tilings, e.g., quadrilaterals and hexagons
- By surface I think I mean a 2-d Riemannian manifold (at any rate: I am particularly curious about the cases of the torus, cylinder and 2-sphere)