Packings and Assemblies for Continuous Families of Polyhedra

Nanoparticles and colloids of various polyhedral shapes are synthesized and used as building blocks for self-assembly. It has been shown in simulations that a plethora of complex crystals (including quasicrystals) can form entropically solely due to the anisotropic shape of the particles. At the limit of maximum density, the densest packings, as have been studied in math for centuries, becomes the relevant quantity. However, a general understanding of how changes in shape affect packings and assemblies remains largely unexplored. Here, we study continuous families of polyhedra to demonstrate richness and complexity in behavior as a function of shape. We investigate connections between assemblies and densest packings, discuss the possibility of predicting one from the other and outline general guidelines for experiments.