text: For hard particles, Pauling's rules are useful in understanding the structure of ionic compounds in the early days, and the later entropy maximization principle shows favor of dense packing in the system. Therefore, finding the densest packing for a given shape is a starting point for predicting the structure of hard nanoparticle superlattices. For spherical particles, the densest packings are face-centered cubic and hexagonal close-packed from the Kepler–Hales theorem.
  Different particle shapes / polyhedra create diverse complex packing structures in order to minimize the entropy of the system. By computer simulations, four structure categories are classified for faceted polyhedra nanoparticles according to their long-range order and short-range order, which are liquid crystals, plastic crystals, crystals, and disordered structures.