Inquiry into the Packing Properties of Spheres
Are your balls packed as tightly as they could be? Dr Fabien Paillusson wants you to know this one trick! In his seminar, Inquiry into the Packing Properties of Spheres (8 October 2025), we explore the idea of the packing fraction. This quantifies how much of a given volume is actually filled by spheres, as a ratio of the total volume of the spheres and the volume of the container.
“Who cares?” I hear you asking. Well, let’s consider everyone’s favourite spherical treat: the humble Malteser. Small differences in packing fraction can have a surprisingly large effect on how many Maltesers fit in a given container. If we consider a container that holds on average 50,000 Maltesers, the variation in packing fraction for poured spheres (…according to Wikipedia) could lead to a difference of 2,869 Maltesers from one tub to another!
So, what’s the most efficient way to pack spheres as tightly as possible?
In two dimensions, the hexagonal arrangement (proved by Lagrange in the 18th century) gives the densest packing. In three dimensions, the analogous structure was conjectured by Johannes Kepler in 1611 and, after centuries of work, finally proven by Thomas Hales and Samuel Ferguson: a computer-verified proof completed over 12 years and formally published in 2017.
That’s the best possible ordered packing. But let’s be honest, no one wants to hand-stack 50,000 Maltesers into a perfectly neat pattern. So what about randomly packed spheres? How tightly can we pack them without imposing order?
Experiments show that vertical tapping can increase the packing fraction; the smaller the tap amplitude, the denser the final arrangement (as shown by Paillusson and Vale). Interestingly, pushing down on the spheres has the opposite effect: applying more force actually makes the packing looser, as demonstrated by Mueggenburg (2012).
But is there a universal limit for randomly packed spheres; a “new Kepler conjecture” for randomness? Not quite. There’s no single definition of randomness in this context. As we move away from perfect disorder, the packing fraction naturally increases.
Vertical tapping experiments suggest an upper limit, but horizontal tapping can achieve even higher densities, though it also introduces partial order into the system shown by Yu et al. (2006). Ultimately, the achievable packing fraction depends on how you handle the container, how much order you allow and the set-up.
Researchers continue to debate what “random” truly means here. Perhaps the best way forward is to define randomness not by spatial arrangement, but by the likelihood of certain configurations occurring. In a 2000 paper, Torquato et al. [1] argued that “The current picture of RCP [random close packing] cannot be made mathematically precise.” So, disagreements persist, and that ongoing discussion keeps the study of sphere packing alive and rolling.
[1] S. Torquato, T. M. Truskett, and P. G. Debenedetti, “Is random close packing of spheres well defined?,” Physical Review Letters, vol. 84, no. 10, pp. 2064–2067, Mar. 2000, doi: 10.1103/PhysRevLett.84.2064.
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