Analytical Construction of A Dense Packing of Truncated Tetrahedra

TL;DR

This paper analytically constructs a nearly maximally dense packing of truncated tetrahedra, revealing new space tiling structures and phase behaviors, advancing understanding of dense polyhedron packings.

Contribution

It presents the first analytical construction of the densest known packing of truncated tetrahedra with near-unity density, based on a generalized organizing principle for non-centrally symmetric polyhedra.

Findings

Density of the packing is approximately 0.9952, close to the maximum possible.

Discovered a new space tiling involving regular tetrahedra and truncated tetrahedra.

Identified two stable crystal phases and a first-order liquid-crystal transition.

Abstract

Dense polyhedron packings are useful models of a variety of condensed matter and biological systems and have intrigued scientists mathematicians for centuries. Recently, organizing principles for the types of structures associated with the densest polyhedron packings have been put forth. However, finding the maximally dense packings for specific shapes remains nontrivial and is a subject of intensive research. Here, we analytically construct the densest known packing of truncated tetrahedra with $\phi=207/208=0.995~192...$, which is amazingly close to unity and strongly implies that this packing is maximally dense. This construction is based on a generalized organizing principle for polyhedra that lack central symmetry. Moreover, we find that the holes in this putative optimal packing are small regular tetrahedra, leading to a new tiling of space by regular tetrahedra and truncated tetrahedra. We also numerically study the equilibrium melting properties of what apparently is the densest packing of truncated tetrahedra as the system undergoes decompression. Our simulations reveal two different stable crystal phases, one at high densities and the other at intermediate densities, as well as a first-order liquid-crystal phase transition.