Re: [math-fun] how to make a rhombic dodecahedon
Both the hexagonal and square lattices of spheres, when laminated, yield the FCC lattice, which is the unique optimal lattice packing of spheres. I think that it was recently proved that it is optimal amongst all packings, although there are 2^aleph-null optimal packings. (Proof: In the hexagonal lattice, there are two sets of `deep holes' in which we can rest another layer. Hence, a countably infinite string over the alphabet {1,2} can be encoded in an optimal lattice packing; there are 2^aleph-null such possibilities. Consequently, it is a lower bound on the number of optimal packings. Also, it is an upper bound, since it is the total number of arrangements of countably many spheres.) Sincerely, Adam P. Goucher http://cp4space.wordpress.com
----- Original Message ----- From: Bill Gosper Sent: 06/21/13 11:43 AM To: math-fun@mailman.xmission.com Subject: [math-fun] how to make a rhombic dodecahedon
I asked Neil if Kepler's sphere-stacking conjecture was about tetrahedral or square pyramids of cannonballs. He fooled with Mathematica briefly and said it doesn't matter <http://gosper.org/kepvor.png>. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Just to add to this, Wikipedia reminds me that there are two distinct lattice packings that are special cases of the 2^aleph_0 packings that Adam refers to below: the cubic close-packing and the hexagonal close-packing: < http://en.wikipedia.org/wiki/Sphere_packing >. These are defined, respectively, by whether the alternation of the layers Adam mentions is in the pattern a b c a b c ... or a b a b a b .... Also, minor nitpicking point: the enumeration of the distinct not-necessarily-lattice ways the layers can be alternated should involve the equivalence of any shifted sequence of the 3 types of layers, and also possible equivalence by rotations between two arrangements. (Finally, the optimality of a sphere packing is determined by the limit of its density in a ball B(R) of radius R, as R -> oo, when this limit exists, which is independent of the center of the ball. This means that any modification of a packing in a bounded region will have no effect on the limiting density. --Dan On 2013-06-21, at 6:59 AM, Adam P. Goucher wrote:
Both the hexagonal and square lattices of spheres, when laminated, yield the FCC lattice, which is the unique optimal lattice packing of spheres. I think that it was recently proved that it is optimal amongst all packings, although there are 2^aleph-null optimal packings.
(Proof: In the hexagonal lattice, there are two sets of `deep holes' in which we can rest another layer. Hence, a countably infinite string over the alphabet {1,2} can be encoded in an optimal lattice packing; there are 2^aleph-null such possibilities. Consequently, it is a lower bound on the number of optimal packings. Also, it is an upper bound, since it is the total number of arrangements of countably many spheres.)
Bill said he asked me, but he didn't! If he had, I would have referred him to my paper with Conway, *What Are All the Best Sphere Packings in Low Dimensions?* [pdf<http://neilsloane.com/doc/BEST/Me194.pdf>], J. H. Conway and N. J. A. Sloane, *Discrete and Computational Geometry (Laszlo Fejes Toth Festschrift)*, 13 (1995), pp. 383-403. Available from my home page, http://NeilSloane.com/doc/BEST/Me194.pdf Neil On Sat, Jun 22, 2013 at 8:18 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Just to add to this, Wikipedia reminds me that there are two distinct lattice packings that are special cases of the 2^aleph_0 packings that Adam refers to below: the cubic close-packing and the hexagonal close-packing: < http://en.wikipedia.org/wiki/Sphere_packing >. These are defined, respectively, by whether the alternation of the layers Adam mentions is in the pattern a b c a b c ... or a b a b a b ....
Also, minor nitpicking point: the enumeration of the distinct not-necessarily-lattice ways the layers can be alternated should involve the equivalence of any shifted sequence of the 3 types of layers, and also possible equivalence by rotations between two arrangements.
(Finally, the optimality of a sphere packing is determined by the limit of its density in a ball B(R) of radius R, as R -> oo, when this limit exists, which is independent of the center of the ball. This means that any modification of a packing in a bounded region will have no effect on the limiting density.
--Dan
On 2013-06-21, at 6:59 AM, Adam P. Goucher wrote:
Both the hexagonal and square lattices of spheres, when laminated, yield the FCC lattice, which is the unique optimal lattice packing of spheres. I think that it was recently proved that it is optimal amongst all packings, although there are 2^aleph-null optimal packings.
(Proof: In the hexagonal lattice, there are two sets of `deep holes' in which we can rest another layer. Hence, a countably infinite string over the alphabet {1,2} can be encoded in an optimal lattice packing; there are 2^aleph-null such possibilities. Consequently, it is a lower bound on the number of optimal packings. Also, it is an upper bound, since it is the total number of arrangements of countably many spheres.)
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-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
participants (3)
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Adam P. Goucher -
Dan Asimov -
Neil Sloane