Is there some sense in which the sunflower is a "quasicrystal"? E.g. what is its fourier transform? It plainly has no symmetry at all unlike, e.g. Penrose tiling with 5-fold symmetry. http://www.cs.uml.edu/~jpropp/sunflowers.html Another question is, suppose plant has not just "seeds" (points in plane it wishes to keep separated) but K different-colored seeds, and it likes to keep them separated and also to keep same-colored seeds further-separated. For example with K=2 the checkerboard would be a good solution, albeit it is not a sunflower. The sunflower with K=1 colors was based on the fact that the golden ratio g=(1+-sqrt(5))/2 is the "worst approximable" irrational number, i.e. min |P*g-Q| minimized over integer P,Q with |Q|<N, is large. It seems to me to get good sunflowers with K=2 colors, you want g to be replaced by h such that h only has poor approximations by P/Q if P is demanded to be even, which suggests considering -1+-sqrt(2) whose best rational approximations 1, 3/2, 7/5, 17/12, etc have always-odd numerators. One reason K=2 is interesting is we can now construct an "fake elliptic" function with poles at the red and zeros at the blue locations. If this were a checkerboard then that'd be a genuine elliptic (doubly periodic meromorphic) function, but with a bicolored sunflower we get a fake-elliptic function which is "quasiperiodic" (with whatever definition of "quasiperiodic" is valid for sunflowers). Certain "Painleve transcendent" functions have got zero & pole patterns that look somewhat like this, and it'd be great if we could make a connection. Warren D. Smith http://rangevoting.org