Wow -- what an interesting paper! From the 2010 Hyderabad ICM. Also very easy to read (I'm through about half of it so far). At < http://arxiv.org/abs/1003.3053 >. --Dan Veit wrote: << Henry Cohn made an interesting parallel with equiangular line arrangements in complex space: "These two problems are finely balanced between order and disorder. Any Hadamard matrix or equiangular line configuration must have considerable structure, but in practice they frequently seem to have just enough structure to be tantalizing, without enough to guarantee a clear construction. This contrasts with many of the most symmetrical mathematical objects, which are characterized by their symmetry groups: once you know the full group and the stabilizer of a point, it is often not hard to deduce the structure of the complete object. That seems not to be possible in either of these two problems, and it stands as a challenge to find techniques that can circumvent this difficulty." from "Order and disorder in energy minimization"

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On Oct 11, 2012, at 4:48 PM, James Propp <jamespropp@gmail.com> wrote:

I liked this paper too.

Question: Does there exist a finite set S of radii such that it is plausible (or, better yet, provable) that the densest way to pack the plane with disks whose radii all belong to S is a particular aperiodic arrangement?

To reverse-engineer such a set S, one might start with the Penrose tiling, and then try to associate with it a dense disk-packing (using disks of two or three different sizes, whose placement is determined by the local combinatorics of the tiling).

You would have to make the disk packing encode the aperiodic matching rules, since periodic tilings, using the same tiles, will allow you to increase the density by either rounding up or down the ratio of tile frequencies (to a rational number) depending on which tile has the greater local density. It's been tried before, unsuccessfully to the best of my knowledge. -Veit

Jim Propp

On Mon, Oct 8, 2012 at 5:06 PM, Dan Asimov <dasimov@earthlink.net> wrote:

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Wow -- what an interesting paper! From the 2010 Hyderabad ICM.

Also very easy to read (I'm through about half of it so far).

At < http://arxiv.org/abs/1003.3053 >.

--Dan

Veit wrote:

<< Henry Cohn made an interesting parallel with equiangular line arrangements in complex space:

"These two problems are finely balanced between order and disorder. Any Hadamard matrix or equiangular line configuration must have considerable structure, but in practice they frequently seem to have just enough structure to be tantalizing, without enough to guarantee a clear construction. This contrasts with many of the most symmetrical mathematical objects, which are characterized by their symmetry groups: once you know the full group and the stabilizer of a point, it is often not hard to deduce the structure of the complete object. That seems not to be possible in either of these two problems, and it stands as a challenge to find techniques that can circumvent this difficulty."

from "Order and disorder in energy minimization"

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I suppose the potential energy e(C) for any configuration of n points C on S^2 is defined as the sum of reciprocal distances among all pairs of points in C. I guess distance d(x,y) could be reasonably defined by using either the arclength along S^2, or else Euclidean distance in R^3. Then E(n) is the minimum of e(C) over all configurations C of size n. We could call it E_a(n) if arclength is used, and E_e(n) if Euclidean distance is used. (It's not clear to me whether these would have distinct energy-minimizing configurations, but E_e is more physically realistic.) I would be very surprised if E(n) is algebraic. Reason being the great irregularity of the minimizing configurations in the classic problem P(n) of finding the maximum over size-n configurations C on S^2 of the minimum distance over all pairs of points in C. Very possibly, whenever sufficiently large n is of the form 6(a^2 - ab + b^2) for a, b in Z+ (if my hasty calculation is right), the optimal configuration will have isometry group = the icosahedral group I, #(I) = 60 (or occasionally the full isometry group of the icosahedron = I x Z/2Z, #(I x Z/2Z) = 120). (E.g., there is a continuum of equally minimum configurations for n = 5, and the maximum symmetry (of a cube) is broken for n = 8.) The connection is that using the inverse-square force law to define potential energy -- and then minimizing that P.E. over configurations C -- is like using the L^2 norm on reciprocal distances: the same configuration(s) will give minimum P.E. But if instead the L^p norm is used (on the 1/d(x,y)'s), then the classic problem above is what you get when you let p -> oo. --Dan P.S. I believe Neil has done extensive simulations to find minimum P.E. in a variety of situations (using Euclidean distance rather than arclength). On 2012-10-14, at 8:36 PM, James Propp wrote:

For n a positive integer, let E(n) be the minimal potential energy for any configuration of n points on the unit sphere subject to inverse-square-law repulsion. Must E(n) be algebraic? (That's gotta be true, but I don't see a rigorous argument.). Are there algorithms for computing E(n) (in the sense of finding an algebraic equation that it satisfies, and indicating which real root of the equation is intended)? How fast are these algorithms, and relatedly, what bounds can be given for the degree of E(n)?

Presumably this question would be part of a subject called "real algebraic programming" or something like that, but I don't know of any such subject (though I'm aware of Tarski's decision procedure for determining when a system of algebraic equations and inequalities admits a real solution).