As Christmas day approaches, folks everywhere are stocking up on wrapping paper and ribbons. Geeks everywhere face an entirely different conundrum, though: what's the most efficient way to pack up and wrap a given set of objects? Ian Stewart at New Scientist has posted an interesting article about some of the theory behind efficient gift wrapping:
Suppose Santa's elves tie a ribbon around, say, six circular mince pies in a flat pack, so that the package fits into the smallest possible area. If they experimented, they would find that the best arrangement of the pies is in a line, so that the string round the outside forms a sausage shape. However, if the elves were tying a ribbon round seven identical mince pies, a sausage would no longer give the smallest area. They would do better arranging the mince pies in a hexagon, so that the one in the middle touches the other six (see diagram).
In a 3D space, Stewart goes on to say the sausage shape arrangement works best for bundles of 56 or fewer spheres. Add another one, though, and a more compact arrangement becomes optimal. Check the full article for even geekier revelations, including tips on how to wrap hyperspheres in a 4D space, and whether the sausage arrangement is always optimal when you're dealing with 41 dimensions.