Here's some PostScript for your colors. Hopefully the comments will guide how to modify for experimenting (unchanged it will just do monochrome): %!PS-Adobe-3.0 % Replace just the first occurrence of 2048 by the desired n, or, in interactive mode, % type "/n 100000 def" before running the file to produce 100000 seeds. % Scaling is done automatically to keep the entire graphic centered on a letter-sized page. % /default {1 index where {pop pop pop} {def} ifelse} bind readonly def /n 2048 default /s 5 n 2048 div sqrt div def /rgbcolors [[0 0 0]] default % list of rgb colors, defaults to just black % for 4 colors: %/rgbcolors [[1 0 0][1 .5 0] [0 1 0] [0 0 1]] def % list of rgb colors % for 2 colors: %/rgbcolors [[1 0 0] [0 1 0]] def /nc rgbcolors length def /360phinv -137.50776405003785464634873962837 def /360v2-1 149.11688245431421756860794071549 def /angle 360phinv default % angle defaults to golden ratio * 2pi % for 2 colors: %/angle 360v2-1 def currentpagedevice /PageSize get aload pop .5 mul exch .5 mul exch translate s s scale 1 setlinewidth 1 setlinecap /pt {dup nc mod rgbcolors exch get aload pop setrgbcolor sqrt 0} readonly bind def 1 1 n {angle rotate pt moveto closepath stroke} for showpage On 3/23/2012 4:10 PM, Warren Smith wrote:
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?
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