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Packings of hard polyhedra have been studied for centuries due to theirmathematical aesthetic and more recently for their applications in fields suchas nanoscience, granular and colloidal matter, and biology. In all thesefields, particle shape is important for structure and properties, especiallyupon crowding. Here, we explore packing as a function of shape. By combiningsimulations and analytic calculations, we study three 2-parameter families ofhard polyhedra and report an extensive and systematic analysis of the densestpackings of more than 55,000 convex shapes. The three families have thesymmetries of triangle groups (icosahedral, octahedral, tetrahedral) andinterpolate between various symmetric solids (Platonic, Archimedean, Catalan).We find that optimal (maximum) packing density surfaces that reveal unexpectedrichness and complexity, containing as many as 130 different structures withina single family. Our results demonstrate the utility of thinking of shape notas a static property of an object in the context of packings, but rather as butone point in a higher dimensional shape space whose neighbors in that space mayhave identical or markedly different packings. Finally, we present andinterpret our packing results in a consistent and generally applicable way byproposing a method to distinguish regions of packings and classify types oftransitions between them.

Complexity in surfaces of densest packings for families of polyhedra