H. K. Chan/Trinity College Dublin

Holiday math. Ho-Kei Chan's package of Christmas ornaments has a cylinder-to-sphere diameter ratio slightly greater than 2.7013, but all of the ornaments touch the cylinder's sides, showing that it cannot be the densest possible arrangement.

Christmas Ornaments, Packed Like Sardines

One day, physicist Ho-Kei Chan of Trinity College Dublin was playing with steel ball bearings, trying to pack them into a little cylindrical tube in the most efficient way possible. It's a tricky problem that can take even a powerful computer a week to calculate. But after thinking about it for a while, Chan has figured out a way to simplify the math. The advance could help engineers pack all sorts of spheres more efficiently, from nano-sized buckyballs to Christmas tree ornaments.

The challenge of packing as many spheres as possible into a cylinder comes up all the time. Microfluidics engineers grapple with it when they try to pack as many drug-delivery bubbles as they need into a tiny capillary tube, and manufacturers confront it when they try to cut shipping costs by packing as many bouncy balls into as small a package as possible. There are lots of possible packing patterns. The tennis ball-style single stack of balls in a tight-fitting cylinder is the simplest; pairs of balls stacked two by two in alternating directions is another. But as the cylinder gets wider, the possibilities spiral into mind-bogglingly complex helices and patterns. What's the best way to pack them?

Chan realized he could solve the problem by imagining his cylinder full of ball bearings as a stack of disks, with a single layer of ball bearings on each disk. He then wrote a computer simulation to model it that way. The computer would lay spheres in a disk, and when it ran out of room, it would move up until it found enough space to fit another sphere. It would then rotate around fitting spheres on that level until there was no more room, moving up again, and repeating. Not all the spheres on a certain disk needed to be level; if a sphere could nestle down a little into a space created by two spheres on the disk below to get the tightest fit, it would. If the spheres below pushed it up a little higher than the other spheres on its disk, that was okay, too.

Chan was pleased to see that the program, running just 15 minutes on a laptop, produced results very close to those of weeklong computer simulations run by his colleagues at Trinity and Aberystwyth University in the United Kingdom. And he also saw why he couldn't get the same results using steel ball bearings in a cylinder. The cylinder's flat bottom allowed the ball bearings to shift out of alignment and into a less dense pack. Chan found a similar problem when he went shopping for Christmas ornaments in the store (see picture). But that insight showed him that the first layer of spheres could serve as a template. As long as you had the right form at the bottom of your cylinder, getting the densest possible pack would be a cinch.

Chan's current method, which he will report in an upcoming paper in Physical Review E, works only for situations in which the ratio of the cylinder's diameter to the sphere's diameter is less than 2.7013. That's pretty common: A single column of tennis balls would have a cylinder-to-sphere diameter ratio close to 1. (Chan's Christmas ornaments were just a touch over the 2.7013 limit.) If the ratio is higher than that, the densest possible pack requires that some of the spheres pack in the middle without touching the cylinder's walls, and that is "more complicated," Chan says. He and colleagues at Trinity and Aberystwyth are working on a solution to that problem now.

Chan has "proposed a quite surprisingly new and simple way to obtain these packings," says Douglas Galvão, a physicist at the State University of Campinas in São Paulo, Brazil. "Considering the importance and broad existence of similar problems, I believe this work will have a high impact in the field, and many people will revisit this problem from this new and exciting perspective."

Although Chan's insight into sphere packing is primarily theoretical, he is excited about the practical applications. Microfluidics researchers packing drug-bearing bubbles into slim capillaries, materials scientists packing buckyballs to create multiwalled nanotubes, and manufacturers searching for more efficient shipping options for balls, globes, candy, or virtually anything at all spherical could all use templates for packing. Another potential application is liquid crystal displays such as those used in televisions and computer monitors, an area Chan worked in previously. If we could make liquid crystal molecules obey these rules, "we could potentially create a whole new class" of liquid crystals, he says.

Posted in Physics, Math