If physicists ran candy stores, gumball machines might be filled with pyramids instead of spheres. It seems that tetrahedra, with their four triangular faces, are the most efficient shape for filling a container randomly, as opposed to carefully stacking objects within it.
Graduate student Alexander Jaoshvili of New York University and his colleagues filled and shook containers of tetrahedral game dice. They found that the tetrahedra were packed tightly enough to occupy 76 per cent of their containers. In comparison, randomly packed spheres fill up to 64 per cent of space, while the figure for squashed spheres, or ellipsoids, can be as high as 74 per cent.
The work is sure to rock the confectionery world, but it has implications for developing stronger materials as well. That鈥檚 because MRI studies by the team show that a tetrahedral die can be locked into place by its immediate neighbours alone, making it harder to nudge out of place. Jumbled collections of spheres, by contrast, are less rigid because any sphere can be moved by objects as far away as six diameters.
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Tough dinnerware
The insight could lead to the creation of nearly unbreakable plates. 鈥淚f, for instance, you wanted to make a very dense, rigid, hard ceramic, you would probably be better off making the powder from tetrahedra,鈥� says Jaoshvili鈥檚 adviser, Paul Chaikin.
And tetrahedra may be able to pack together randomly even more efficiently, says of Princeton University.
In a recent simulation, Torquato and student Yang Jiao found a way to pack tetrahedra that took up more than 82 per cent of space. But this configuration may be more ordered than the one in Jaoshvili鈥檚 study.
Ordered or random
The question is important because it鈥檚 still not clear what kind of packing 鈥� random or ordered 鈥� is most efficient for tetrahedra.
Ordered, crystalline arrangements of tetrahedra can fill more than 85 per cent of available space, according to recent simulations, but randomly assembled objects might be able to pack more tightly. 鈥淣obody knows whether the densest packing is ordered or random,鈥� Chaikin told 快猫短视频.
鈥淧eople tend to think that the densest packings are always ordered, but there鈥檚 no fundamental reason why that has to be true,鈥� agrees Torquato. 鈥淲e can鈥檛 rule out the possibility that the densest packings of tetrahedra will be disordered.鈥�
Future simulations may help settle the matter. And shapes more complex than spheres, ellipsoids and tetrahedra are only beginning to be explored, so we may not yet have discovered the optimal candy shape.
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