Degenerate Quasicrystal of Hard Triangular Bipyramids Stabilized by Entropic Forces

The assembly of hard polyhedra into novel ordered structures has recently received much attention. Here we focus on triangular bipyramids (TBPs)- i.e.~dimers of hard tetrahedra- which pack densely in a simple triclinic crystal with two particles per unit cell [1]. This packing is referred to as the TBP crystal. We show that hard TBPs do not form this densest packing in simulation. Instead, they assemble into a different, far more complicated structure, a dodecagonal quasicrystal, which, in the level of monomers, is identical to the quasicrystal recently discovered in the hard tetrahedron system [2], but the way that tetrahedra pair into TBPs in the nearest neighbor network is random, making it the first degenerate quasicrystal reported in the literature [3]. This notion of degeneracy is in the level of decorating individual tiles and is different from the degeneracy of a quasiperiodic random tiling arising from phason flips [4]. The $(3.4.3^2.4)$ approximant of the quasicrystal is shown to be more stable than the TBP crystal at densities below $79.7\%$.\\[4pt] [1] Chen ER, Engel M, Sharon SC, Disc. Comp. Geom. 44:253 (2010).\\[0pt] [2] Haji-Akbari A, Engel M, et al.~Nature 462:773 (2009).\\[0pt] [3] Haji-Akbari A, Engel M, Glotzer SC, arXiv:1106.5561 [PRL, in press].\\[0pt] [4] Elser V, PRL 54: 1730 (1985)