In experiments with M&Ms and D&D dice, our intrepid researchers push the barriers of densely-packed packages. And fail. But at least they discover the ideal shape for packing the maximum number of objects into a stocking.
Every Christmas, parents are not only driven batty trying to hunt down intentionally rare toys and figure out which edition of an action figure their kid wants most, but they have to come up with 'stocking stuffers' as well. Stockings are long, narrow, lumpy pieces of fabric. They squish whatever is in the bottom of them. Their threads catch on whatever is put in them. And if a parent has to stuff more than one, it's guaranteed that they won't pack evenly - spawning family dramas that will end with one of their grown children hurling a glass of brandy at the fireplace and running off to Fiji to sell beaded necklaces.
It turns out that the best way to avoid this family drama is to fill the stockings with only spherical gifts. Repeated, increasingly frustrated testing shows that spheres are the ideal shape for filling random spaces - especially when hapharzardly thrown in. (Obviously, meticulously packed cubes will fill a cubic container best, but the holidays don't give us the luxury of regular shape.) Carelessly thrown spheres will fill 64 percent of a space. Spheres that are packed carefully, stacked up the way that cannonballs are, fill 74 percent of a given area. Thorough tests have been done, including ones in which scientists 'carefully modified' M&M candies to get the desired 74 percent density.
Recently scientists have tried packing other polygons into packages using various techniques, hoping to beat the packing barrier. They used 12-sided dodecahedrons, 4-sided pyramids, 8-sided polyhedrons and 20-sided isocohedrons. D&D players will have recognized these shapes as they read them, and come up with a question about the experiment. The answer is yes, they scientists used 'commercially available' dice. No word on if they rolled a natural twenty.
The scientists tried packing in many ways, from throwing the dice in randomly, shaking the container to let the dice setlle, using a machine to shake the dice in case human hands weren't precise enough, and even letting the dice settle in the containers through liquid and then draining off the liquid. The best case, cubes and tetrahedrons shaken by a machine, packed to a fullness of 64 percent - not any tighter a squeeze than with the spheres. Spheres randomly thrown into the containers packed 59 percent of the space, nine percent better than the average for other solids.
And so the stately sphere is still the shape to beat. Fill your loved one's stockings with oranges, marbles, and those little foil-covered chocolate balls to let them know that you love them to maximum efficiency.
Read the full scientific paper via Clark University.