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:
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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