Dungeons and Dragons dice pack densely
May 12, 2010 5 comments
From the apples at your local grocery store to the pills in your medicine cabinet, packing products in an efficient manner is an important consideration in many industries. In new research, a group of physicists in the US has investigated the packing properties of a less familiar object, though it may be recognizable to players of the game Dungeons and Dragons – the tetrahedral die. They find that these shapes pack incredibly densely, despite taking on a highly disordered configuration.
Tetrahedra are regular convex shapes possessing four triangular faces. To date very little research has been carried out on how these shapes pack together. But a better understanding of this process could be of interest to geological industries such as oil companies when choosing where to drill their wells. This is because granular matter is more similar to tetrahedra than spheres, which is how it is depicted in basic geological models.
In the past year or so the applied mathematics community has taken up the challenge to investigate tetrahedra, and it has become clear that these shapes could pack much more densely than spheres, at least in theory. In the extensive research on spheres over the years, they have never filled more than 64% of a container, despite a conjecture by Kepler that they could pack to a fundamental limit of 74.05%. In contrast, some recent numerical models have shown that tetrahedra can pack to fractions of more than 85%.
With this latest research, Alexander Jaoshvili at New York University in the US, working with colleagues, has taken a closer look at how tetrahedra pack together in the real world. In a fairly straightforward experiment, the researchers assembled a large number of identical tetrahedral-shaped dice and began adding these to different-shaped containers, shaking and adding more dice until no more could be added. Packing fractions were then determined by injecting a well known filling fluid until the containers were full and subtracting these volumes of fluid from the total volumes of the containers. For one of the large radius containers, a packing density of 0.76 was recorded, which compared with 0.64 for spheres added to the same container.
To probe a little deeper and examine the packing structure, Jaoshvili's team then placed the packed containers in an MRI scanner. This enabled the researchers to locate the centres of particles and to resolve the kinds of configurations that the dice were taking on. What they saw is that, despite their ability to pack so tightly, the dice are in fact highly disordered within the containers. This finding adds weight to recent theoretical work that suggests that the tetrahedra are aligning themselves into a form of quasicrystal structure upon compression.
All shook up
Jaoshvili and his team were slightly surprised by the disorder. "One would expect that if particles are highly packed they would be highly ordered as well, but with tetrahedrons we find that they are packed with high density and are highly disordered," says Jaoshvili.
This surprise is shared by Daan Frenkel, a theoretical chemist at the University of Cambridge, who believes that, at the moment, the result can only be explained qualitatively, by comparing tetrahedra with other shapes. "With cubes, the gap-less packing can be continued indefinitely – they can pack 100% of the space. Tetrahedra cannot "tile" space – but they are better at it than spheres."
Since Jaoshvili submitted his paper there has been a flurry of activity regarding tetrahedral packing and he expects further light to be shed on the quasicrystal structure of the packing in the near future.
This research is published in Physical Review Letters.
About the author
James Dacey is a reporter for physicsworld.com