Symmetry, shape, and order

TL;DR

This paper analytically explores dense packings of truncated cones, revealing high-density arrangements with broken symmetry and identifying two regimes based on cone dimensions, contributing to understanding of packing and material structures.

Contribution

It introduces new analytical models for packing truncated cones, demonstrating high-density arrangements and symmetry-breaking phenomena depending on shape parameters.

Findings

Biaxial packing of solid cones with 0.7854 density.

Two distinct packing regimes for truncated cones separated at c*.

Analytical characterization of symmetry-breaking in cone packings.

Abstract

Packing problems have been of great interest in many diverse contexts for many centuries. The optimal packing of identical objects has been often invoked to understand the nature of low temperature phases of matter. In celebrated work, Kepler conjectured that the densest packing of spheres is realized by stacking variants of the face-centered cubic lattice and has a packing fraction of $\pi/(3\sqrt{2}) \sim 0.7405$. Much more recently, an unusually high density packing of approximately 0.770732 was achieved for congruent ellipsoids. Such studies are relevant for understanding the structure of crystals, glasses, the storage and jamming of granular materials, ceramics, and the assembly of viral capsid structures. Here we carry out analytical studies of the stacking of close-packed planar layers of systems made up of truncated cones possessing uniaxial symmetry. We present examples of high density packing whose order is characterized by a {\em broken symmetry} arising from the shape of the constituent objects. We find a biaxial arrangement of solid cones with a packing fraction of $\pi/4$. For truncated cones, there are two distinct regimes, characterized by different packing arrangements, depending on the ratio $c$ of the base radii of the truncated cones with a transition at $c^*=\sqrt{2}-1$.