Sphere Packings
Have you ever poured grains into a container and wondered if there is an effective way to fit more in? Perhaps by arranging into a pattern? Or tapping into place? Or how about applying force?
Many mathematicians have thought about these ideas. These ideas and calculations were gathered and discussed by Fabien Paillusson in a seminar entitled “Sphere Packings” on 08/10/25.
This seminar focused on comparing the packing value of different methods of packing spheres (Number of spheres * volume of container / volume occupied = packing value).
It was found that the most effective way to pack incompressible spheres is in a hexagonal pattern. This provides a maximum packing value of 0.74.
Kepler determined this effective packing formation to store cannonballs on a ship in 1611. Years later in the 19th century, Gauss mathematically proved the hexagonal formation was the most effective. Then, in 1998, Hales and Ferguson performed a proof by exhaustion, which means they calculated every possible packing formation. After that, a computer was given the task of checking the proof by exhaustion, and this task took 12 years to complete. The result was announced in 2014 [1].
At this point, it was proven that the hexagonal formation was effective. But what about the random formations? There were no calculations to see if the random is more effective. It was also considered that tapping with a set amplitude might affect the packing density. It was discovered that when an amplitude is applied, the packing value cannot increase above 0.64. This value is still less than the maximum value for the hexagonal of 0.74.
Perhaps adding force can be more effective? No. It was found that increasing the force on the spheres provides them with less space to move. This means an increase in force correlates to a decrease in packing value.
Is there no packing value in-between the hexagonal 0.74 and the random 0.64? The tests performed on the random formations were when they were only tapped vertically. Research conducted on tapping horizontally found that the maximum value for this method was 0.74.
Overall, this research is useful for manufacturing, as it provides estimations of the most effective storage methods. We understand a lot about hexagonal pattern packing and vertical tapping. However, there are still many things to be discovered about random packing.
[1] “Kepler’s Sphere-Packing Conjecture Is Finally Proved | Encyclopedia.com”. https://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/keplers-sphere-packing-conjecture-finally-proved
31102
I believe the equation for packing value you want is:
Number of spheres * Volume of single sphere / Volume of container
It might also be good to include the full title from the title slide “Inquiry into the packing of spheres” though I don’t imagine the title needs to be exact.
Overall though it flows well and the language is pitched at the right level for someone outside the mathematics / physics sphere. (ha)