Inquiring into the packing properties of spheres
It may seem like a simple question asking “How many spheres can you fit in this container?” but from attending a seminar by Fabien Paillusson titled ‘Inquiring into the packing properties of spheres’ on 8th October 2025, I can tell you it is not so simple.
Fabien discussed ways in which spheres can be optimally packed to ensure maximum volume can be filled. The three ways discussed were randomly, square and hexagonally in both 2D and 3D. It was determined that the arrangement which results in the greatest volume being covered is when they are packed hexagonally which results in roughly 0.74 of the container being occupied and can be backed up by an article from Nature Reviews Physics titled ‘Packing finite numbers of spheres efficiently’ stating “hexagonal close packing and face-centred cubic packing both achieve the maximum density of approximately 0.74048”[1].
When it comes to random assortment, the seminar discussed that it may be possible to make it more densely packed together. Even if it only increases density slightly, tapping on the container can cause movement of the spheres so that any bridges of spheres which have naturally formed can be broken. This will not, however, lead to a greater density than hexagonal packing as testing has shown that the proportion of the container’s volume that the spheres take up plateaus at around 0.64. Some may suggest that using pressure may be helpful to improve density, but the result is actually the opposite! Adding an additional load does not allow any room for movement for motion of the spheres and arches can’t be broken. So, if you ever do need to fit more spheres in a container make sure you tap instead of compressing.
This can be contextualised into real life with spherical sweet manufacturers wanting to fit as many of their products in a container as possible and how best to do this so that the space is optimised. This will benefit the company by reducing packaging and transport costs as you can fit more in the given container, so less containers are required. In addition, this will have a great environmental impact and increase sustainability, creating a very good brand image while also doing their part to not harm the world around us.
There has been lots of work and research around sphere packing over the past hundreds of years in 2D and 3D. The optimal arrangement for 2D being hexagonal was proven by LaGrange in the 18th century, however 3D was a bit more complicated. While it was first conjectured by Kepler in 1611, it was not until 406 years later in 2017 that there was a final publishing. During this time, Gaus proved it for all regular packings in the 19th century with Hales and Ferguson then proving by exhaustion in 1998 and then later proposing a formal solution. Research does continue to investigate sphere packing in higher dimensions. When reading a Bulletin of The American Mathematical Society it states that “In 2016 Maryna Viazovska solved the sphere packing problem in R 8 with an innovative use of modular forms, which was soon extended to R 24” [2] showing further research is still being done to this day. It is interesting to see how much research can be done of a topic that at surface-level may seem so simple.
[1] Budrikis, Z., 2024. Packing finite numbers of spheres efficiently. Nature Reviews Physics, 6, p.82.
[2] BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY, Volume 61, Number 1, January 2024, p.322
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Report has great presentation, details of the seminar have all been included, topics are well laid out, few obvious grammar or spelling mistakes. Good summary of most of the results discussed in the seminar, and Motivation for research in a wider context has been clearly discussed, leaving no doubt on the importance of the problems discussed.
Good use and reference of sources outside of the seminar, source 2 is well used to show further current research in the field. Source 1 seems only to have been used to restate results from the seminar, would be more interesting to reference some of the newer research discussed in the article.
Technical language is suitable for unfamiliar readers, but definitions and results could be stated more clearly at times. For example, “When it comes to random assortment, the seminar discussed that it may be possible to make it more densely packed together.” Could make clearer what it is that is being made more densely packed (purely random assortments, vs random with tapping).
Overall very engaging and well written report!
15100
Engaging introduction and good use of references. A simpler way of saying “spherical sweet manufacturers” could be used to make the section flow slightly better. Overall, engaging and clear.