What are we looking at here? According to a 2009 paper from the University of Michigan, the upper polyhedron is a pentagonal dipyramid “built from tetrahedral dice stuck together with modelling putty.” (Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra)
According to the same paper, the lower polyhedron made from red dice is a “quasicrystal with packing fraction φ = 0.8324.” (The lower photo is from The Ann Arbor Chronicle)
The big breakthrough in tetrahedral packing was Elizabeth Chen’s 2008 research uncovering substantially denser packing arrangements than previously thought possible.
Tetrahedral packing is interesting precisely because tetrahedrons cannot pack together perfectly. Although, Aristotle believed that they did…
In 1972 Stanislav Ulam, a Polish-American mathematician who worked on the Manhattan Project, conjectured that spheres were the worst-packing of all convex bodies. So from Ulam’s conjecture, it should follow that tetrahedra should pack denser than 0.74048. But in the mid-2000s, investigations of tetrahedron packing that used computer simulations, as well as experiments using physical tetrahedral dice, could not establish any configuration of tetrahedron packing that clearly surpassed the 0.74048 for spheres. Maybe tetrahedra were worse-packing than spheres?
Was Ulam wrong? No. We’ll get to that in a moment. Now’s a good chance to think about how very wrong Aristotle had been – wrong about tetrahedra and their ability to completely fill space. How did he manage to massively miss that one?
Part of the reason could have been that Aristotle had no ready source of tetrahedral dice and gummi putty to try pasting models of tetrahedra together – the way that Elizabeth Chen asked the audience of her thesis defense to do. Once you have them in your hands, it’s easy to paste together models and convince yourself that they will fill less than all of space – a pastes-great-less-filling experience.
Dave Askins, Packing Pyramids: UM and Ann Arbor
The Ann Arbor Chronicle, February 14, 2010
(See also NY Times article, Packing Tetrahedrons, and Closing In on a Perfect Fit)
What’s the “packaging” take away? If you’re not designing a package for tetrahedral dice, why should you even care about tetrahedral packing?
Well, as we’ve shown recently, there are some tetrahedron shaped packages out there. (See: Modissa, Абажурус, and Tetra-Pak) Denser, more efficient packing arrangements might lower the cubic space taken up by such packages in a shipping container.
And the remaining gaps between the tetrahedrons, might, in some way be exploited as a feature. (More about that in a future post.)