Inquiry into the packing properties of spheres
The seminar given by Fabien Paillusson on the 8th of October 2025, called ‘Inquiry into the packing properties of spheres’ concerned the mathematics behind (in its simplest form) packing a multitude of identical spheres into a container as densely as possible. Paillusson explained that the density of the spheres is expressed through a mathematical equation known as the packing fraction. The larger the packing fraction, the more densely the spheres are packed into their container. The packing fraction (eta) is determined through the equation eta equals fraction numerator N asterisk times v over denominator V subscript packing end subscript end fraction, where N is the number of spheres, v is the volume of an individual sphere, and V subscript packing end subscript is the total volume of the container occupied by the spheres. Through this equation, mathematicians were able to prove that the densest way to pack spheres is hexagonally, in both 2d and 3d. Whilst the proof for two-dimensional packing was proven by Lagrange in the 18th century, proving the same for three-dimensions was much harder. Kepler conjectured this in 1611, but a formal proof was only published by Hales in 2017, 19 years after their proof of exhaustion published with Ferguson in 1998. However, the process of packing spheres hexagonally takes far longer than simply pouring those spheres into a container. Therefore, mathematicians have focused their efforts onto discovering the maximum packing fraction of randomly packed spheres in a closed space, or RCP.
Paillusson explored some of the methods that mathematicians have tested to try and improve the density of randomly packed spheres. Focusing on both tapping the system and applying a load. These experiments showed that applying a load by pushing down on the system, reduces the system’s maximum density potential. However, tapping the system is highly effective. These results showed that lighter tapping is more effective in increasing system density, and that combining vertical and horizontal taps yields the best results. In the end, it was discovered that RCP seems to find its maximum value at around 0.64. Which is a far cry from the hexagonally packed, packing fraction of approximately 0.74. This limit is further proved by Scott and Kilgour1, whose investigation involved the use of different types of spheres with different coefficients of friction, that also affects the packing of the spheres. The experiments, extrapolated with a ‘95%’ confidence limit’ resulted in the density being .
But why does this matter? Because it allows us to make estimates on the number of spheres within a space, helping to better calculate the variance. For companies shipping large amounts of spherical objects, the density of those objects is important to know because it will affect the quantity of how many can fit into the given space. If the variance between the expected amount of spheres and actual is large enough, said products could be delivered at a deficit, causing problems for the company, or at a surplus, resulting in a loss. This is just one example of the importance of sphere packing but suffice to say that it is an important sector of mathematics, and one which significantly affects everything around us.
1 G D Scott and D M Kilgour 1969 J. Phys. D: Appl. Phys. 2 863
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blog overall presentation: 2/3, it contains speaker name, date and tile of seminar. some small errors, for example last sentence of the 2nd to last paragraph is cut off.
Accurate reporting of the seminar’s take-home-message 3/3: the key points of the seminar are spoken about
Accurate contextualisation of the research topic 3/3. you talk about relevant topics for the seminar
external source 1/3. an external source is given but no direct quote is, a direct quote is required.
writing style and technical level 3/3: technical level is appropriate