Inquiry into the Packing of Spheres
Stacking fruit and storing cereal may not seem complex, but the research that governs these was started centuries ago and there is still much up for debate to this day. The topic in question here is sphere packing and, although it was first discussed by Kepler in 1611, it has no lack of applications in a more modern world. This is the subject of Fabien Paillusson’s lecture ‘Inquiry into the packing of spheres’ on the 8th of October 2025 the basis of this blog.
The first question was finding the ideal way to store spheres. When this was proposed this was about transporting cannonballs as efficiently as possible (but feel free to sub in your favourite sphere). The correct solution was proposed when the problem was written in the early 17th century and is the likely the way you’d intuitively stack spherical objects if you were tasked with such a problem. However, this was only proven in 1998 by Thomas Hales by which point Kepler had, some would say unsurprisingly, passed away. A note of mild interest is that there are infinitely many different ways to arrive at this optimal packing in 3D. Starting by arranging the first layer in a hexagonal pattern, further layers can be formed by placing spheres in the depressions created between the others on the layer beneath. When doing this you can either choose to repeat the pattern every 2 or 3 layers, or you can mix and match giving rise to infinitely many unique arrangements. You can also start with a square grid on the first layer, which might seem less like a looser packing, but creating further layers in a similar way actually leads to the same arrangement as the 3-layer repeating hexagonal one albeit rotated [1]. Any of these will grant you an optimal ~74% of space filled by the spheres.
That’s well and good but, as nice as the that regular arrangement is, it’s not always practical to achieve it or possible at all. Noone is going to arrange their (approximately spherical) grains of flour or salt in a nice regular pattern to reduce the amount of space they take up in the cupboard (Maltesers were Fabien’s spherical food of choice during the lecture, but should you want to try this practically I suspect the volume they occupy would decrease on their own). If you don’t want to arrange the spheres manually, the best you can do is tap the container against some other surface. This has the effect of closing some of the empty space inside by breaking so-called bridges. This is also something you’d probably intuitively do, but how close does it get the the optimal 74%? For a while many papers based on both computer simulations and practical experiments seemed to suggest that the limit was around 64% [2][3], but doubt was cast on this as densities of 66% became possible if horizontal vibrations were introduced as well. It might not surprise you that ‘random’ is not necessarily an easy thing to define. Are these results a limit for a particular set-up rather than a geometric law? Does vibrating the spheres inherently reduce the randomness of them? These questions spawned a paper in 2000 suggesting that no one had really put together a formal explanation for what a random packing of spheres was.
All of that is to say that we still don’t have a firm answer on either the best way to compact a random arrangement of spheres, nor the maximum density of such an arrangement. However, for everyday purposes at least, somewhere in the mid 60s is plenty close enough to 74 for me. So, you’ll find me continuing to tap my containers of salt and spices against the table to try and get the last bit in. I might even throw in a sharp sideways shake in for fun (with the lid on of course).
[1] N. J. A. Sloane, “The Packing of Spheres,” vol. 250, no. 1, 1984, [Online]. Available: https://www.jstor.org/stable/24969283
[2] G. D. Scott and D. M. Kilgour, “The density of random close packing of spheres,” vol. 2, no. 6, June 1969, doi: 10.1088/0022-3727/2/6/311.
[3] W. S. Jodrey and E. M. Tory, “Computer simulation of close random packing of equal spheres,” vol. 32, no. 4, Oct. 1985.
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Presentation Date, title and name of speaker: Present in the introduction: “Fabien Paillusson’s lecture ‘Inquiry into the packing of spheres’ on the 8th of October 2025”. Grammar and English: The grammar is strong, and the language flows well. Minor informal touches (“feel free to sub in your favourite sphere”, “with the lid on of course”) add personality without compromising clarity. Only a couple of minor Grammatical errors and Typos Score: Good
Content Coverage of seminar content: The piece accurately explains the historical context (Kepler, 1611), the proof by Hales (1998), and practical implications (random packing, tapping containers). It includes technical details like packing percentages (74%, 64%, 66%) and mentions key concepts such as hexagonal and square arrangements. Accuracy: References are correctly cited, and the explanation aligns with known research. Score: Good.
Context Societal and research contexts: The blog connects the topic to everyday life (stacking fruit, storing cereal, tapping containers) and situates it within historical and modern research contexts. It even raises open questions about randomness and geometric laws, showing awareness of ongoing debate. Score: Excellent.
Style Suitable for a lay audience and engaging: The tone is conversational and accessible, using relatable examples (Maltesers, salt and spices) while explaining technical ideas clearly. Humour and informal phrasing make it engaging without losing substance. Score: Excellent.
External Source Reference and quoted opinion: References [1], [2], [3] are included and relevant. However, there is no direct quoted opinion from an external sourceonly citations. The mark scheme specifies “quoted opinion balancing/supporting the reporting”, which is missing. Score: Good (would be excellent if a quotation from one of the sources was integrated into the narrative).
Final Review The report is well presented, including the correct date, title, and name of the speaker, and demonstrates fine grammar and English with only minor typos.
It covers the seminar content accurately and explains the two main points effectively. However, it omits an important foundation introduced at the start of the seminar, the concept of packing fraction. The discussion focuses on random packing and regular arrangements in 3D space but does not address this key introductory element.
The report uses varied and relatable examples to provide societal context, such as storing fruit, cereal, salt, and spices, which makes the technical topic accessible. The language is clear and engaging, and technical details are introduced gradually, making it suitable for a lay audience.
Multiple references are included to support the discussion, which strengthens credibility. However, the report does not incorporate any direct quotations from these sources, as required by the marking scheme. Adding a quoted opinion from one of the cited papers would improve the balance and depth of the review.
Recommendation: Include a short, relevant quotation from one of the referenced sources to fully meet the “External source” criterion and enhance the overall quality.
Overall Grade: Good
42890
It is suitable for a lay audience and reads smoothly. You convey an understanding of the topic. The historical context also adds to the authenticity of the writing piece.