Inquiry into the Packing of Spheres by Fabien Paillusson on Wednesday 8th October 2025
Imagine randomly pouring uniform spheres into a rectangular container. How densely do they pack together? What is the optimal packing formation? How can disordered packing be condensed? This seminar by Fabien Paillusson goes through and answers these questions.
Working with the packing fraction (defined by the product of number of spheres and the volume of the container divided by the volume of occupied space) will inform how densely the spheres are packed. Due to the random and chaotic nature of this experiment, the packing fraction will vary. Comparing this information to Malteasers, even having a small change, such as 0.035, can have a huge impact on the variation of the amount of Malteasers in that space.
Chaos in this system creates a lot of uncertainty when trying to predict the outcome. One small difference the initial state can results in very different results. To negate this problem, looking at regular packing formations for spheres will remove the uncertainty of chaos. In 2D and 3D formations, it has been proved that hexagonal tessellating formations have the optimal packing fractions. The former was proved by Lagrange in 18th century, having a packing fraction of estimably 0.90689 [1], and the latter being proved by exhaustion in 1998, having a packing fraction of estimably 0.74048 [2].
Even though randomness cannot be predicted, it can be improved upon. By gently tapping vertically on the container after a random poor, the packing fraction increases. Taping too forcefully can have the reverse effect and cause the packing to be less dense. However, if the tapping is light enough, continually over time causes the packing fraction to increase even more, asymptotically approaching 0.64. As said by Phillipe and Bideau: ‘the evolution of the mean volume fraction and of the mean potential energy of a granular packing under consecutive vertical taps presents a slow densification until a final steady state’ [3].
In 1960, Scott performed an experiment on the packing fractions of finite particle in a finite container and investigated the change packing fraction before and after tapping to find the upper bound for each case. By interpolating the data, Scott calculated that the upper bound for untouched and tapped results are 0.60 and 0.64 respectively [4].
Yet a better way to improve the packing fraction is to tap on the container horizontally. This way increases the upper limit of the packing fraction to 0.74, which more closely approaches the maximum for ordered hexagonal packing. Interestingly, adding a crushing force onto the spheres in the container whilst tapping creates the opposite effect, where increasing the force applied to the spheres decreases the packing fraction [5].
As it comes with all science, not everyone agrees. A main source of discourse is the definition of randomness. In this instance the term lacks a lot of clarity. Theoretically, a random poor could result in a perfectly ordered packing, which is not being investigated in this instance. Instead, it has been suggested to use the term disordered as it describes the ‘random’ element of not knowing where each sphere will be but also exclude the cases of ordered packing.
Overall, there is a lot of uncertainty when working with randomness. Removing it makes proofs simpler however only looking at simple cases restricts discovery. Investigating chaotic experiments, such as this one, may not have a final answer yet but doing the work still has benefits in, further research and applying it to current problems.
Bibliography
[1] L. Fukshansky, “Revisiting the hexagonal lattice: on optimal lattice circle packing,” Elemente der Mathematik, pp. 19, Jan. 2011, doi: https://doi.org/10.4171/em/163.
[2] T. HALES et al., “A FORMAL PROOF OF THE KEPLER CONJECTURE,” Forum of Mathematics, Pi, vol. 5, 2017, doi: https://doi.org/10.1017/fmp.2017.1.
[3] P. Philippe and D. Bideau, “Compaction dynamics of a granular medium under vertical tapping,” Europhysics Letters (EPL), vol. 60, no. 5, pp. 677683, Dec. 2002, doi: https://doi.org/10.1209/epl/i2002-00362-7.
[4] G. D. SCOTT, “Packing of Spheres: Packing of Equal Spheres,” Nature, vol. 188, no. 4754, pp. 908909, Dec. 1960, doi: https://doi.org/10.1038/188908a0.
[5] D. L. Blair, N. W. Mueggenburg, A. H. Marshall, H. M. Jaeger, and S. R. Nagel, “Force distributions in three-dimensional granular assemblies: Effects of packing order and interparticle friction,” Physical Review E, vol. 63, no. 4, Mar. 2001, doi: https://doi.org/10.1103/physreve.63.041304.
62557
Blog overall presentation is very clear with no grammatical mistakes, the title of the talk is correct and so is the name of the speaker and the date. (3/3)
I personally believe you have taken the message from the speaker correctly! (3/3)
This has an accurate contextualization for research and for society. For society it could be explained a little better with less technical terms perhaps. For research this is very well written and the point comes across very well. (2/3)
This has additional external sources. For example it has a quote from Phillipe and Bideau (3/3)
Majority of the writing style is appropriate for a lay of audience, however some parts could be explained a little better for people with no scientific background. An example of this is when you talk about ‘chaos’ most non-scientific people would not know what this could mean. (2/3)
Overall score - (13/15)
Well done :D
62451
1, Blogs overall presentation: The blogs overall presentaion is very clear it has good use of grammar making it flow nicely and makes it easy to read. The blog also contains the relevant informamotion on speaker, title and date. 3/3
2, Accuracy of reporting the seminars take home message: This blog explains and talks well about what the speaker was trying to get accross and is well inturpreted. 3/3
3, Accuracy of contextualisation for the research: I feel that the writer has managed to get the research contextualised well but it focuses more on the research side than a socitial one. its still good but if you could make make it more about how its used in society then that would take it a cut above. 2/3
4, Additional research and use of external sources: The writer has used a wide range of diffenentn sources and include a direct quote from an additional source showing that the writer has taken the time to research the topic in more detail. 3/3
5, Writing style and level for audiance: The writing style of this blog is slightly to advanced for people who have no prior knowledge to the topic, but it is well written and detailed. If you could make it a bit more friendly for newer people that would help. 2/3
13/15 almost there a few modifications and its perfect.
46509
Overall Blog Presentation: Good presentation, includes the title, speaker’s name, and the date. 3/3
Accurate Reporting: You have understood the subject well. 3/3
Accurate Contexualisation: Good contexualisation, but I would like to see more of how it has a societal impact. This is for the lay person to read, why should they care? 2/3
External Source: You’ve referenced several external sources with solid connections to the subjects. 3/3
Writing Style and Technical Level: Good technical understanding, but you have assumed that the lay man knows certain mathematical terms. 2/3
Overall: 13/15 Really good work, with a few tiny changes, it will be perfect. Good job!