Complexity in surfaces of densest packings for families of polyhedra

TL;DR

This study systematically analyzes the densest packings of over 55,000 convex polyhedra across three shape families, revealing complex and rich packing density surfaces with numerous structural transitions.

Contribution

It provides the first extensive systematic analysis of densest packings across multiple polyhedral shape families, uncovering unexpected complexity and proposing a new shape space perspective.

Findings

Up to 130 different packing structures within a single shape family.

Identification of complex, multi-structure packing density surfaces.

Development of a method to classify packing regions and transitions.

Abstract

Packings of hard polyhedra have been studied for centuries due to their mathematical aesthetic and more recently for their applications in fields such as nanoscience, granular and colloidal matter, and biology. In all these fields, particle shape is important for structure and properties, especially upon crowding. Here, we explore packing as a function of shape. By combining simulations and analytic calculations, we study three 2-parameter families of hard polyhedra and report an extensive and systematic analysis of the densest packings of more than 55,000 convex shapes. The three families have the symmetries of triangle groups (icosahedral, octahedral, tetrahedral) and interpolate between various symmetric solids (Platonic, Archimedean, Catalan). We find that optimal (maximum) packing density surfaces that reveal unexpected richness and complexity, containing as many as 130 different structures within a single family. Our results demonstrate the utility of thinking of shape not as a static property of an object in the context of packings, but rather as but one point in a higher dimensional shape space whose neighbors in that space may have identical or markedly different packings. Finally, we present and interpret our packing results in a consistent and generally applicable way by proposing a method to distinguish regions of packings and classify types of transitions between them.