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 Asimov
--My impossibility argument (such as it was -- as I said it is not as satisfactory as one would want) focused on the spherical boundary of the expanding large ball. Each point on said boundary feels locally (where "locally" means within a distance large enough compared to 1 but small enough compared to radius of ball) that it is a member of a periodic "finite set of lattices" point arrangement, to within epsilon [note this is NOT true of the whole sunflower ball, only little pieces of it near boundary], and all boundary points "feel the same" in some statistical sense. All these feelings, for all boundary points at once, are mutually incompatible with each other, I tried to argue. I think we should also demand that an N-point sunflower in D dimensions have radius of order N^(1/D), min point separation of order 1, and a simple formula for the point locations. Your (Asimov's) 3D proposal seems to fail the radius desideratum. The existence of a "spiral" in 2D which "covers the plane at distance=1," is something that seems pretty unique to 2D. One can in 3D make curves or surfaces that do it, but they unlike the 2D spiral can make no claim to being "nearly rotationally symmetric." What is this vague "near rotationally symmetric" notion, you ask. Well, one possible answer could perhaps be that if a 2D spiral is scaled and rotated, you get the spiral back again (or something very closely approximating it) if you look only within a narrow-width annulus. The scaling factor has to be chosen as a continuous function of the rotation angle which is 1 for small angles. So a spiral is not continuum-rotationally symmetric, but it can be regarded as such to a high degree of approximation if "rescaling is used to correct matters" and if you only look within a narrow width annulus. But I do not think this is true of any 1D curve or 2D surface that tries to cover the whole of 3D. So I'm putting forward here a vague idea that the spiral as so-characterized, is a uniquely 2D notion. It would be better to formalize all this non-vaguely... -- Warren D. Smith http://RangeVoting.org