Sphere Packings - Dr Fabien Paillusson
In our daily life, we can see many commodities that is in the shape of sphere such as orange and some chocolate. The sphere cannot fit into a container without leaving empty space. Hundreds of years ago, the concept of the sphere packing was first mathematically analyzed by Thomas Harriot, mathematical assistant to Sir Walter Raleigh, in around 1587. In this blog, I’m going to talk about the sphere packings base on the lecture of Dr Fabien Paillusson “Inquiry into the packing of spheres” on the 8th of October 2025.
In the seminar, Dr Fabien Paillusson has introduced what will happen when a numerous uniform spheres poured into a container. Then, he explained a mathematical concept which known as “packing fraction”, this can tell us the amount of sphere that we had in the container. For the formula, we that the number of spheres and the volume of the sphere to multiply together, and then divide that by the total volume occupied by the container. Because it is a ratio, the answer, which is the Aitor, is always smaller than or equal to one. This is a good method to calculate is the space used efficiently.
Also, Dr Fabien Paillusson had talked about several examples of finding the best way to pack sphere products into a container. He introduced three types of arrangements, “random”, “square” and “hexagonal” and proofing which arrangement can be packed the most efficiently in both two-dimensional (2D) and three dimensional (3D) containers. In 2D, it was shown that hexagonal packing is the densest way, and it was proved for all packings by Lagrange in the 18th century. In 3D, people also find that hexagonal packing is the most efficient way to pack sphere. This was conjectured by Kepler in 1611, proved for regular packings by Gauss in the 19th century and proof by exhaustion by Hales and Ferguson in 1988. The formal proof was started by Hales in 2003 and confirmed by automated proof checking over 12 years. For the random arrangement, where the spheres are distributed in the form of structural disordered, is hard to study because it depends on the protocol, order metric and system. Fabien Paillusson had said it was difficult to define “randomness”. The Scott Nature’s study in 1960 found that the packing fraction for the random close packing reach the maximum of 0.64 and tend to be around 0.6 for the random loose packing.
This study is first to make an improvement on the lower bound of sphere packing by more than a constant factor since 1947. It can first dates to the explorer Sir Walter Raleigh, who ask Thomas Harriott how her should pack cannonballs on the ships so he can carry as many as possible. Few years later, Thomas Harriott had passed this problem to Johannes Kepler, which had proposed that the most efficiently way to pack spheres is to arrange them in a “face-centered cubic” in 1611. “Face-centered cubic” is similarly putting the sphere into the square and triangle-based pyramids. This known as “Kepler’s conjecture” and using this method to pack 3D sphere would account for 74.05% of the total volume of the box. This claim was proven by Thomas Hales in 2017.[1]
In conclusion, Dr Fabien Paillusson had discussed the main theory of sphere packing clearly. But this also tell us that we still don’t have a final answer in high dimensions for the random arrangement of spheres. Like what Dr Jenssen said,“ We are still a long way off from definitively solving the sphere packing problem in high dimensions like Kepler and Hales did for 3D spheres. There has been some great progress in recent years, but there’s still plenty of work to be done!”
[1] Matthew Jenssen, “King’s Mathematician helps solve sphere packing problem centuries in the making” [online]. Available:
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Overall the presentation is okay however there are some grammatical errors that need addressing such as missing words and wrong tense. You covered the contents of the seminar well. The context is addressed with how we use sphere packing but could be expanded upon. There is proof of some additional reading but only one additional source. The style of writing works well for someone with no knowledge of the subject however could be improved if the equation was given instead of written out with words.
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You have covered this in good depth and it’s also easy to follow. It feels like anyone without a maths background would be able to understand the concepts you mention. Maybe some more explanations and examples could make it read slightly better, especially when talking about randomness, but overall it looks good.