An overview of Dr Paillusson’s seminar: “Inquiry into the packing property of spheres”
By Fabien paillusson
08/10/25
Consider this: say you’re trying to fit marbles into a box and they don’t quite fit, does that mean they can’t ever fit? Well, this idea has been around for thousands of years and was first officially proposed by Kepler in 1611. Dr Paillusson’s lecture on the packing of spheres explains these key points and highlights some questions that still need answering.
When you think about a box that contains objects such as spheres, there’s going to be tiny gaps or spaces in between the spheres where they aren’t touching; this is what we call the packing fraction, and it refers to how “well packed” everything is. In other words, it’s the percentage of the box that has been filled by the material, so a packing fraction of 0.5 means that 50% of the total volume of the box has been filled with the contents. Well, what is the highest packing fraction you can get with spheres? It turns out that stacking spheres hexagonally gives the highest packing fraction – around 0.91 in 2D which was proved by Lagrange in the 18th century, and 0.74 in 3D which was proved by Hales & Ferguson in 2017. In practice it takes time to pack many spheres into a hexagonal structure and as a result there has been lots of research into optimising random packing, which is intrinsically less efficient than hexagonal.
Simulations agree on a maximum packing fraction of around 0.61 for random packing in 3D, that is if you are just randomly pouring the spheres into a container. But we all know that if you’re packing things into a container and it just doesn’t quite fit, then sometimes shaking it a little solves the problem. This is when Dr Paillusson introduced the idea of applying external forces to the container in the form of tapping and pushing. If you gently tap the container vertically, the packing fraction slowly increases up to a maximum value of 0.64, with an interesting discovery that the harder the tap, the lower the maximum packing fraction becomes. It matters how you tap the container, as a paper by Yu, A et al [1] shows that tapping the sides of the container results in a higher packing fraction than tapping the bottom.
Similarly, if you are tapping the container and have a force pushing down on the spheres, the packing fraction still increases, but the more force you push down with, the lower the final value becomes. In any case the results all show the same thing, it is possible to increase the random packing fraction while never reaching the maximum hexagonal packing fraction.
There is still lots we don’t know, such as the densest random packing, and the lack of definitions for words such as randomness causes disagreements. But we do know the densest structure, and further research may provide answers and allow more efficient packing in the future.
[1] Yu, A.B. and Hall, J.S. (1994). Packing of fine powders subjected to tapping. Powder Technology, 78(3), pp.247–256. doi:https://doi.org/10.1016/0032-5910(93)02790-h.
46551
This is really clear, engaging, and relatable, especially with the marble analogy which was great. To improve accessibility for non-scientists, maybe consider shortening some of the sentences and also try focusing more on intuitive ideas than specific studies or proofs.
Well done!