We study the thermodynamics of hard regular tetrahedra using Monte Carlo simulations. It is a well-known fact that hard particles can assemble into ordered structures that are stabilized by entropy. It follows from simple thermodynamic reasoning that in the limit of infinite pressure, the arrangement of hard particles occupying the highest fraction of three-dimensional space is thermodynamically stable, and therefore there is an intimate relationship between thermodynamics of hard particle systems and the packing problem in geometry.

However the packing problem is far from solved for regular tetrahedra; Aristotle erroneously thought that regular tetrahedra fill Euclidean space and eighteen centuries elapsed before he was proven to be wrong. Within the last few years, several mathematicians have constructed dense arrangements of tetrahedra with packing fractions ranging from 72-78%. In our simulations, we observe much denser packings (83.38%) that are achieved by compressing a dodecagonal quasicrystal, which spontaneously forms from a fluid of disordered tetrahedra at densities as low as 50%. Quasicrystals are an exotic class of solids with long-range order but lack periodicity; hard tetrahedra are the first example of hard particles spontaneously assembling into a quasicrystal. A periodic structure closely related to the quasicrystal- known as the approximant- can even be compressed further to packing fractions as high as 85.03%, a record which was beaten just recently by a slightly denser crystalline arrangements of tetrahedra dimers with a packing fraction of (85.63%).

Amir Haji-Akbari, Michael Engel, Aaron S. Keys, Xiaoyu Zheng, Rolfe Petschek, Peter Palffy-Muhoray, Sharon C. Glotzer, Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra, Nature 462, 773-777 (2009).