AdamG>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 http://mathworld.wolfram.com/KeplerConjecture.html says that Hales's proof will require 20 man-years to check. I rewrote http://gosper.org/kepvor.png to picture three of the aleph-null hexagonal packings (tp, tp1, and tp2). Apparently, only one is congruent to the octahedral packing. I expected two. With the staggered (tp2) packing, the Voronoi dodecahedron has six faces trapezoidal, but presumably tessellates 3space. (Twist a rhombic dodecahedron pi/3 about a hexagonal equator.) --rwg