I think this, if it’s any worth, will eventually follow the same road. As incomplete a product as it looks now, it may become the de-facto way of taking pictures, just as the telephone became the standard means for communication and the lightbulb replaced burning lights.

http://arxiv.org/abs/cond-mat/0512563

Deriving Finite Sphere Packings
Arkus, N., Manoharan, V. N., & Brenner, M. P.
SIAM Journal on Discrete Mathematics, 25(4), 1860 (2011). d

Taking slices of that dodecahedron will give you caps for 5, 10, or 15 balls circumference tubes… you can make larger by interspersing loops of 6 balls in among the loops of 5 balls.

@Alemi: That’s the magic incantation I was looking for! (Most of what I’ve found so far deals with polar fluids and colloids, whereas I think the magnet balls might better be treated as jammed granular matter. But at least now I have a point of entry into the literature. Thanks.

@Jim Mahoney: Inspired by your comment and the linked photo, I’ve wasted a few more hours trying to create an fcc or hcp crystal out of magnet balls. Partial success: I get some lumpy blobs that appear to be locally close-packed in some regions, but with lots of defects. Fascinating. By the way, I find that a 2 x 2 x 2 cube is indeed stable–or metastable enough that I can make one and let it sit on the table without spontaneous rearrangement. I can’t make it by folding an octagon, but I can stack two squares.

@john Cowan: I splurged and bought the 6mm size.

But you can do 3D hex packing. Here’s a picture : http://goo.gl/JGfJ6 .

Starting small (like a tetrahedral clump of 4) doesn’t work; the edge effects make things unstable. That’s true of cubic packing too. A 6×6x6 cube is stable; a 2×2x2 is not.

Instead start with several large 6×6 or so hex pack planar sheets. With a bit of trial and error you’ll find that they can be aligned on top of each other in several different ways, with different sorts of symmetry.

See for example http://youtu.be/Rm-i1c7zr6Q for a nice video on 3D lattice sphere packing.

It is a well known statistical mechanical model of polarizable liquids, and in particular is sometimes used as the simplest model of water available.

Very interesting post. Keep up the path of discovery.

http://arxiv.org/pdf/1204.3924v1.pdf

“Indeed, we believe that our results do not solve
any problem, but pose important, new ones.”