Only in 3D are spheres the worst packers

IT'S official. Spheres waste the most space of any shape if you are packing identical objects into a box. Although most people suspected this, proof is now available - and that has revealed a surprise: spheres are only the worst in a three-dimensional universe.

Cubes are the best box-fillers, because they can fit together exactly with no wasted space. To simplify the problem of finding the worst, mathematicians add restrictions, such as only considering shapes that are symmetrical.

Spheres make intuitive sense as the worst packers, so Yoav Kallus, a condensed matter physicist at Princeton University, and Fedor Nazarov, a mathematician at Kent State University, Ohio, used them as their starting point - and tried to find something worse. They started by aligning neighbouring rows of spheres in the most efficient way possible. Then they showed that adding small dents to the surface of a sphere decreases the space they use up by providing new opportunities for spheres to nestle closer. This led to proof that no known shape packs less efficiently, the pair told the International Workshop on Packing Problems in Dublin, Ireland, at the start of September.

Kallus and Nazarov also applied the same logic to higher-dimensional space, where spheres still exist as mathematical objects, even though they are much more difficult to picture. The more dimensions, the more neighbours each sphere has. Surprisingly, adding dents now increased the amount of wasted space. The pair showed this was true for 4, 5, 6, 7, 8 and 24 dimensions.

The finding is all the more intriguing when you compare it with the result for two dimensions, where an octagon with curved corners, not a circle, is the least efficient way to pack. Three dimensions are a sort of a "happy middle" for spheres, says Kallus.

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