What is the most space-efficient way to pack tennis balls or oranges? Mathematicians have studied this 鈥渟phere-packing鈥� problem for centuries, but surprisingly little attention has been paid to replicating the results in the real world. Now, physical experiments involving microscopic plastic balls have confirmed what mathematicians had long suspected 鈥� with a small number of balls, it is best to stick them in a sausage.
Johannes Kepler was the first person to tackle sphere packing, suggesting in 1611 that a pyramid would be the best way to pack cannonballs for long voyages, but this answer was only fully proven by mathematicians in 2014.
This proof only considers the best way of arranging an infinite number of spheres, however. For finite sphere packings, simply placing the balls in a line, or sausage, is more efficient until there are around 56聽spheres. At this point, the balls experience what mathematicians call the 鈥渟ausage catastrophe鈥� and something closer to pyramid packing becomes more efficient.
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But what about back in the real world? Sphere-packing theories assume that the balls are perfectly hard and don鈥檛 attract or repel each other, but this is rarely true in聽real life 鈥� think of the squish of聽a聽tennis ball or an orange.
One exception is microscopic polystyrene balls, which are very聽hard and basically inert. at聽the聽University of Twente in聽the聽Netherlands and his team,聽who were unaware of mathematical sphere-packing theories, were experimenting with聽inserting these balls into empty cell membranes and were聽surprised to find them forming sausages.
鈥淥ne of my students observed a聽linear packing, but it was quite puzzling,鈥� says Vutukuri. 鈥淲e thought that there was some fluke,聽so he repeated it a couple of聽times and every time he observed similar results,鈥� says Vutukuri. 鈥淚 was wondering, 鈥榳hy聽is this happening?鈥� It鈥檚 a聽bit聽counterintuitive.鈥�
After reading up on sphere packing, Vutukuri and his team decided to investigate and carried out simulations for different numbers of polystyrene balls in a聽bag. They then compared their predictions with experiments using up to nine real polystyrene balls that had been squeezed into cell membranes immersed in a liquid solution. They could then聽shrink-wrap the balls by changing the concentration of聽the聽solution, causing the membranes to tighten, and see聽what formation the balls settled in聽using a microscope.
鈥淔or up to nine spheres, we showed, both experimentally and聽in simulations, that the sausage is the best packed,鈥� says聽team member at Utrecht University, the Netherlands. With more than nine balls, the membrane became deformed by the pressure of the balls. The team ran simulations for up to 150 balls and reproduced the sausage catastrophe, where it suddenly becomes more efficient to pack things in polyhedrons, with between 56鈥� and 70 balls.
The sausage formation for聽a聽small number of balls is unintuitive, says at聽Imperial College London, but makes sense because of the large surface area of the membrane with respect to the balls at low numbers. 鈥淲hen dimensions become really, really small, then the wall effects become very important,鈥� he says.
The findings could have applications in drug delivery, such as how to most efficiently fit hard antibiotic molecules, like gold, inside cell-like membranes, but the work doesn鈥檛 obviously translate at this point, says M眉ller.
Nature Communications
