In real physical crystals, it seems one always(?) gets "grain boundaries" if one tries to vary the local crystal orientation with position. [There are, however, "quasicrystals" which confuse the issue.] But as we saw from my sunflower lattice approximation theorem, the "sunflower" can do that while avoiding grain boundaries, thus "violating physical intuition." Why this discrepancy? Perhaps the reason is, physics is 3 dimensional, sunflowers 2 dimensional. Perhaps "3-dim'l sunflowers are impossible." Can one formulate & prove some theorem making such an impossibility claim? I think "yes" one can, though I'm not satisfied with my attempts... Anyway, here are two attempts, at least one of which somewhat succeeds, albeit I still feel confused about them both. I. One idea is to use the famous topology theorem that "it is impossible to comb the hairs on a sphere." (For a 2D surface of a sphere in 3D. Combings are possible, and known, for a sphere in 2, 4, or 8 dimensions.) For a lattice, there are 3 distinguished basis directions, call them the red, blue and green directions. Were these directions to vary smoothly as we moved around on the surface of a big sphere, then either we would violate the hairy ball theorem, or we would have to have some points on the big sphere at which the red lattice direction was "radially outward." (or inward.) Red cannot be radially outward everywhere since then the nonparallel green directions would violate the hairy ball theorem. So red points radially outward in some places on the big sphere but not all, and by genericity only on a subset of measure=0. It seems to me for anything nice enough to be worthy of the name "sunflower" (a concept we get to define...) this subset indeed must have finite cardinality. But it seems to me if you do that, then the sunflower is going to have spherically unsymmetric statistical behavior. I mean, the 2D sunflower sort of has "the same behavior everywhere" at large radius, but this couldn't. So if we can figure out what that means exactly and define it to be impossible by choosing the right definition of what is allowed to be a "sunflower," voila. http://en.wikipedia.org/wiki/Hairy_ball_theorem II. Another fact is J.Liouville's theorem that in 3 and higher dimensions, "conformal maps" stop existing (except for a small known family) ... but when I tried to use this I ran into self-contradictions, I'm clearly misunderstanding something in that approach. -- Warren D. Smith http://RangeVoting.org