March 2012 - mathfun@mailman.xmission.com

Re: [math-fun] 3D sunflowers are impossible?
by Dan Asimov 24 Mar '12

24 Mar '12

Actually, it's true that combings are possible for all odd-dimensional spheres. (Or in the language used below, for a sphere in any even dimension: S^(2n-1) := {x in R^(2n) | ||x|| = 1}. That can be readily seen because the above definition is equivalent to S^(2n-1) := {z in C^n | ||z|| = 1} and so there is a free SO(2) action on S^(2n-1) via e^it z := (e^it * z_1, e^it * z_2,...,e^it * z_n), which means that S^(2n-1) is fibred by consistently oriented circles -- hence there is a "combing". --Dan << One idea is to use the famous topology theorem that "it is impossible to comb the hairs on a sphere." (For a 2D surface of a sphere in 3D. Combings are possible, and known, for a sphere in 2, 4, or 8 dimensions.) >> ________________________________________________________________________________________ It goes without saying that .

1 0

[math-fun] 3D sunflowers are impossible?
by Warren Smith 24 Mar '12

24 Mar '12

In real physical crystals, it seems one always(?) gets "grain boundaries" if one tries to vary the local crystal orientation with position. [There are, however, "quasicrystals" which confuse the issue.] But as we saw from my sunflower lattice approximation theorem, the "sunflower" can do that while avoiding grain boundaries, thus "violating physical intuition." Why this discrepancy? Perhaps the reason is, physics is 3 dimensional, sunflowers 2 dimensional. Perhaps "3-dim'l sunflowers are impossible." Can one formulate & prove some theorem making such an impossibility claim? I think "yes" one can, though I'm not satisfied with my attempts... Anyway, here are two attempts, at least one of which somewhat succeeds, albeit I still feel confused about them both. I. One idea is to use the famous topology theorem that "it is impossible to comb the hairs on a sphere." (For a 2D surface of a sphere in 3D. Combings are possible, and known, for a sphere in 2, 4, or 8 dimensions.) 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." (or inward.) Red cannot be radially outward everywhere since then the nonparallel green directions would violate the hairy ball theorem. So red points radially outward in some places on the big sphere but not all, and by genericity only on a subset of measure=0. It seems to me for anything nice enough to be worthy of the name "sunflower" (a concept we get to define...) this subset indeed must have finite cardinality. But it seems to me if you do that, then the sunflower is going to have spherically unsymmetric statistical behavior. I mean, the 2D sunflower sort of has "the same behavior everywhere" at large radius, but this couldn't. So if we can figure out what that means exactly and define it to be impossible by choosing the right definition of what is allowed to be a "sunflower," voila. http://en.wikipedia.org/wiki/Hairy_ball_theorem II. Another fact is J.Liouville's theorem that in 3 and higher dimensions, "conformal maps" stop existing (except for a small known family) ... but when I tried to use this I ran into self-contradictions, I'm clearly misunderstanding something in that approach. -- Warren D. Smith http://RangeVoting.org

1 0

[math-fun] re; big sunflowers
by Stuart Anderson 24 Mar '12

24 Mar '12

The following small postscript program provides a similar graphic; %!PS 297 421 translate 0 .1 400{dup dup sqrt 4 div 0 360 arc fill 137.50775 rotate}for showpage 137.50775.. is the 'golden angle' If you gradually change the rotation angle and animate , you get this; http://www.youtube.com/watch?v=1GK1JfivJSU No editing was done, just a gradual change of angle. what interested me was the appearance and metamorphosis of different cycles. Stuart

5 5

[math-fun] sunflower lattice local approximation theorem
by Warren Smith 24 Mar '12

24 Mar '12

A "sunflower" means the set of points where Nth point has R=sqrt(N) and theta=g*2*pi*N in polar coordinates for some irrational number g. I CLAIM if g is a quadratic irrational and real P with 0<P<1/10 are fixed, then in the limit N-->infinity with D=N^P, that the set of sunflower points within distance D of the Nth sunflower point, all lie within epsilon of a set of points generated by translations and rotations of a fixed (g-dependent) finite set of 2-dimensional point lattices, where epsilon-->0 in the limit. FOURIER TRANSFORMS For a discrete point set, place unit-mass Dirac delta functions at each point, and then take the Fourier transform of the sum of those delta functions. Call that, for short, the "Fourier transform of the point set." So for the first 2048 points in a sunflower, the fourier transform "power spectrum" as a function of x and y is F(x,y) = sum(N=0..2047) [ cos(x*cos(N*2*pi*g)*sqrt(N))^2 + sin(y*sin(N*2*pi*g)*sqrt(N))^2 ] It seems to me that the lattice approximation theorem implies that the fourier transform of a an N-point sunflower will exhibit some kind of discrete structure in the N-->infinity limit, albeit it is not clear to me just what will happen. A greyscale picture of F(x,y), anyone?

1 0

Re: [math-fun] The sequence that gets densest in the circle the fastest
by Dan Asimov 24 Mar '12

24 Mar '12

Re my questions about sequences in the circle, this freely downloadable .pdf file is fairly relevant. (Unfortunately it's scanned, so not searchable.): < http://www.google.com/url?q=http://www.fq.math.ca/Scanned/27-1/vanravenstei… >. (I've long known about the continued fraction of tau = (sqrt(5)-1)/2 and its being the worst number approximable by rationals. But the consequences of these re the properties of n*tau (mod Z) are not to my mind immediate.) --Dan ________________________________________________________________________________________ It goes without saying that .

2 1

Re: [math-fun] big sunflowers
by Stuart Anderson 24 Mar '12

24 Mar '12

> Neat animation! Why did you make the "seeds" get bigger with radius? > By the way, my calculator says the angle is 137.507764... > It's not my work, I picked it up from Bjorn Samuelsson's webpages a couple of years ago when I was learning postscript and collecting graphics. http://home.thep.lu.se/~bjorn/ http://home.thep.lu.se/~bjorn/postscript having "seeds' bigger with radius allowed me to do a zoom at the start of the animation with real sunflowers evidently one needs to understand the dynamics of plant buds; 'primordia' which has been the subject of much recent research also some Maple code for sunflowers; http://homepages.math.uic.edu/~howard/version5/spirals6v5.html Stuart

1 1

Re: [math-fun] big sunflowers
by Henry Baker 24 Mar '12

24 Mar '12

Thanks for posting this! Very interesting! Sadly, the pdf files don't look very good on the screen of my computer. I'll now have to download the ps files & try again. At 12:07 PM 3/23/2012, Thane Plambeck wrote: >This looks good > >http://diwww.epfl.ch/w3lsp/books/moire/prefaceKluwer.html > >Previewable on Amazon and if you dig, all the figures seem to be online > >On Fri, Mar 23, 2012 at 12:00 PM, James Propp <jamespropp(a)gmail.com> wrote: >> Most responses to my post have focussed on the frequency-domain, but what >> about the specific patterns (circles, hyperbolas, etc.) that we see in the >> spatial-domain? At some point these need to be explained. >> >> In fact, even leaving aside sunflowers, I'd like to see a nice mathematical >> explanation of Moire patterns of the conventional kind. Has such a thing >> ever been written? You'd think so, but I wasn't able to find one anywhere >> on the web last time I checked! >> >> Jim Propp > >-- >Thane Plambeck >tplambeck(a)gmail.com >http://counterwave.com/

1 0

[math-fun] sunflowers, asimov, propp, weyl
by Warren Smith 23 Mar '12

23 Mar '12

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?

3 2

Re: [math-fun] big sunflowers
by Dan Asimov 23 Mar '12

23 Mar '12

That's not exactly what I meant. Rather, one will see approximate short white line segments (in the complement of the dots) almost everywhere, since the dots are lined up parallel to two local axes. That's all I meant. The *details* of how these axes might line up, and why in some places they fit together to make apparently circular curves, would take a careful analysis of the specifics of the equation defining the locations of the dots, (r,theta) = (sqrt(n), n*(sqrt(5)-1)), n = 1,2,3,.... --Dan Allan wrote: << I think we are all agreed that most of the visible aliasing we see in the 50,000-dot image is due to pixel quantization. But Dan Asimov argues that even if we could get rid of pixel quantization, we would still see some moire-like effects. If I understand his argument, he is saying that in each area of the phi-based sunflower, the array of dots approximates the vertices of a lattice of parallelograms; the axis vectors of this lattice distort slowly as you move to nearby areas, and occasionally snap to a different set of axes. He anticipates "phase transitions" between domains governed by different axis vectors, and expects that these transitions will appear as visible discontinuities. I agree (again, hedging that I might not be following Dan's thoughts correctly) that different regions have different natural coordinate systems, but I disagree that the transitions will be abrupt or visible. Instead, I expect them to shade into each other imperceptibly; along the borders between these domains there will be regions that appear ambiguous, where one will be able to choose semiconsciouly (as in the Necker illusion) which lattice one sees. >> ________________________________________________________________________________________ It goes without saying that .

4 4

Re: [math-fun] big sunflowers
by Dan Asimov 23 Mar '12

23 Mar '12

I was also about to mention the same book. In fact, the author, Isaac Amidror, has written a second volume, "The theory of the Moiré phenomenon. Vol. II" as well as a 2nd edition of the first volume, now called "The theory of the Moiré phenomenon. Vol. I". According to Math Reviews, these are considered the definitive reference on the Moiré phenomenon. Also, the same guy has coauthored a recent paper: Isaac Amidror & Roger D. Hersch (2010): Mathematical moiré models and their limitations, Journal of Modern Optics, 57:1, 23-36, which I have downloaded and which looks quite readable. --Dan Thane wrote: << This looks good http://diwww.epfl.ch/w3lsp/books/moire/prefaceKluwer.html Previewable on Amazon and if you dig, all the figures seem to be online >> ________________________________________________________________________________________ It goes without saying that .

1 0