The 2D sunflower constitutes an expanding *ball* (disk) in R^2, not a circle. So the 3D analogy would be an expanding ball in 3D. There's no reason that a sunflower-like arrangement can't be improvised on an expanding ball in 3D. For example, the spherical component could be defined by saying that any 3 consecutive points s_n, s_(n+1), s_(n+2) of the 3D sunflower sequence satisfy that their unit vectors s_n/||s_n||, s_(n+1)/||s_(n+1), s_(n+2)/s_(n+2) are congruent as a triple on the unit sphere, for all n. While at the same time their lengths ||s_n|| slowly grow larger exponentially, say ||s_n|| := e^(n*(1+eps)), for very small eps > 0. --Dan << 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."
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