Re Asimov's conjecture, the fact that g=(sqrt(5)-1)/2 is the (essentially unique) "worst approximable" (by rationals) number is known and is rigorous and basically comes from fact its continued fraction is [1,1,1,1,1,...] Indeed there is a known theorem by A.Hurwitz and/or E.Borel somewhere in 1890-1905 that among any 3 successive CF convergents p/q, at least one must be as close or closer than 1/(sqrt(5)*q^2)... and sqrt(5) is the best possible constant for rational approximations because the golden ratio g meets the bound... This means the Weyl sequence 0, g, 2g, 3g, 4g... mod 1 has minimum gap shrinking asymptotically as slowly as possible for any Weyl sequence. But Asimov wanted the max gap to shrink as quickly as possible. For any Weyl sequence n*g mod 1, n=0,1,2,3... in the case where g is any quadratic irrational (periodic CF) the max and min gaps must shrink at the same rate which is gaps proportional to 1/#points. Plainly the two gaps to the point "0" are the only ones that matter by translation invariance (which is enough-valid) which hint should be good enough for you to prove. So I claim for any Weyl sequence based on any quadratic irrational, the max and min gaps both shrink like constants times 1/#points, and those plainly are best possible up to the values of the constants. Also this paper may be of interest (have not seen): N.B.Slater: Gaps and steps for the sequence nX mod 1, Proc. Cambridge Philos. Soc. 63 (1967) 1115-1123. About Propp's question, what are those hyperbolas etc, the sunflower pattern is basically a point lattice, except you've distorted things by use of polar coordinates. You would actually have had a genuine lattice if you'd drawn it on the surface of a cylinder spacing points along a geodesic, i.e. helix, at constant spacing. Point lattices contain lines. Your sunflowers therefore contain distorted lines, i.e. curves. If you view cylinder as a cone of angle 0 and flat paper as a cone of angle 180, and cones as angles a with 0<a<180 then the map consists of increasing the cone angle from 0 to 180 thus mapping the cylinder-lattice-pattern onto the plane-sunflower-pattern. Obviously, conic curves seem likely to be important... This is not yet a full explanation, but I'm confident is tied to the correct explanation. Now to return to my problem about K-colored generalization of sunflowers, how about some colored postscript code for them using multiple colors, the Jth color for points that are J mod K?