Inquiry into the Packing Property of Spheres
Ever wondered if companies like Mars are really squeezing in as many Maltesers as they possibly can into each container? Well, thanks to a fascinating seminar on 8 October 2025, called Inquiry into the Packing Property of Spheres by Dr Fabien Palillusson, we can now begin to explore the answer.
In his seminar, Dr Fabien Palillusson explored how uniform spheres behave when poured into a container, specifically, how they interact and how much space they actually occupy. He introduced a concept from mathematics known as the “packing fraction”, which helps quantify this. Essentially, it’s a ratio: you take the number of spheres, multiply that by the volume of a single sphere, and then divide by the total volume those spheres now fill. It’s a good way to measure how efficiently space is being used.
Dr Palillusson then took things further by explaining how researchers have carried out a range of experiments to figure out the best way to pack spheres as tightly as possible into a container. The main goal was to reduce the empty space between the spheres, and one simple but effective method was tapping the container to help the spheres settle more compactly. These experiments did succeed in fitting more spheres into the space, but they didn’t really explain how the spheres were arranging themselves. That intriguing question was the next topic of discussion.
Dr Fabien Palillusson discussed how spheres can be packed in both two-dimensional (2D) and three-dimensional (3D) spaces. He focused on three main types of arrangements: square packing, where spheres touch to form a grid pattern; hexagonal packing, where each sphere is surrounded by six others in a honeycomb-like structure; and random packing, where spheres are distributed without a specific pattern.
In 2D, it was shown that hexagonal packing is the most efficient. This was first proven by the mathematician Lagrange in the 18th century, who demonstrated that hexagonal packing has the highest packing fractionmeaning it fills the most area with the least amount of empty space.
For 3D packing, the proof took much longer. The idea was originally proposed by Kepler in 1611, and later partially proven by Gauss in the 19th century for regular arrangements. In 1998, Hales and Ferguson developed a proof by exhaustion, and in 2003, Hales began a formal proof that was verified using automated proof-checking over the course of 12 years. This final proof was officially published in 2017. These findings confirmed that hexagonal close packing is the most efficient way to fill space with spheres in three dimensions.
The third arrangement: random packing, where spheres are distributed without a specific pattern, is harder to study. One of Dr Palillusson’s key points was the difficulty in defining “randomness”. Different scientific communities often interpret randomness in different ways, which makes it challenging to model consistently using mathematical tools.
To explore the issue of randomness in sphere packing, Dr Palillusson referred to a 1960 study by Scott Nature. This study found that the packing fraction, the proportion of space filled by spheres, tends to reach a maximum of 0.64 in random close packing and around 0.6 in random loose packing. To make this easier to understand, these decimal values can be thought of as percentages: random close packing fills about 64% of the container, while random loose packing fills roughly 60%.
However, these values have been debated over time. For example, a later study by Karl Michael Martin Jr reported an average packing fraction of just 0.52 [1], showing that results can vary significantly depending on the experimental setup. One factor that complicates things is the tapping method. After pouring the spheres into a container, researchers often tap it to help the spheres settle more tightly. In some experiments, this led to packing fractions exceeding 0.64, which raised questions about whether the arrangement could still be considered truly random.
In conclusion, Dr Fabien Palillusson’s seminar clearly highlighted the complexities and key principles behind sphere packing, especially when randomness is involved. If we return to the original question, whether a box of Maltesers is packed as efficiently as possible, then in theory, a perfectly packed box would leave no room for the chocolates to move. So, if you shake the box and it rattles, it’s a sign that it’s not fully packed. On the bright side, that satisfying rattle might just be designed to increase the anticipation of what is within.
[1] Martini, K., Michael, Amherst, R. and High School (n.d.). An Experimental Study of Random Loose Packing of Uniform Spheres An Experimental Study of Random Loose Packing of Uniform Spheres. [online] Available at: https://people.umass.edu/nmenon/martini.pdf [Accessed 22 Oct. 2025].
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Well done for including the date, title, and name; just be careful with the spelling, it’s Paillusson.
You’ve provided excellent coverage of the history of proofs for both 2D and 3D ordered packing. It might be worth expanding a little to discuss different protocols or to consider randomness in a statistical sense.
Your discussion of the contention surrounding random packing results and what is meant by “randomness” in research is really strong, and the real-life example with Maltesers is a nice inclusion. No notes there.
The post is very engaging; I really liked the jokes at the beginning and end. You’ve kept the tone and depth perfectly suited to your audience, and explaining the packing fraction as a percentage rather than just as a decimal was an excellent choice. Again, no notes.
Finally, your external reference looks good and supports the content well.