Grocers stacking oranges demonstrate intuitive grasp of sphere-packing math

They may not know it, but grocers face some of the most difficult questions in mathematics when stacking produce each day.

Four centuries ago, the astronomer and mathematician Johannes Kepler guessed that the standard grocers’ method of piling oranges packs the most fruit into the least space. Confirming he was right had to wait until 1998, when mathematician Thomas Hales of the University of Pittsburgh, working with his student Samuel Ferguson, proved Kepler’s conjecture with the aid of 180,000 lines of computer code.

But even that achievement didn’t settle all the grocers’ dilemmas. Researchers have known that other sphere-stacking methods can be equally dense. So how does one recognize when a stack is the densest possible? A partial answer appears in a paper posted October 3 at arXiv.org.

Back in 1969, Hungarian mathematician László Fejes Tóth thought he’d figured out one simple method: If each sphere i