Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra

TL;DR

This paper presents a comprehensive analytical framework for constructing dense periodic packings of tetrahedra, including the densest known packings with a four-particle basis, and discusses their potential optimality and bounds on packing density.

Contribution

It introduces the most general analytical formulation for dense tetrahedron packings with four particles per fundamental cell, unifying and extending previous findings.

Findings

Includes the densest known packings with density 0.856347.

Provides a six-parameter family of dense packings.

Suggests these packings may be the densest among all four-particle packings.

Abstract

The determination of the densest packings of regular tetrahedra (one of the five Platonic solids) is attracting great attention as evidenced by the rapid pace at which packing records are being broken and the fascinating packing structures that have emerged. Here we provide the most general analytical formulation to date to construct dense periodic packings of tetrahedra with four particles per fundamental cell. This analysis results in six-parameter family of dense tetrahedron packings that includes as special cases recently discovered "dimer" packings of tetrahedra, including the densest known packings with density $\phi= 4000/4671 = 0.856347...$. This study strongly suggests that the latter set of packings are the densest among all packings with a four-particle basis. Whether they are the densest packings of tetrahedra among all packings is an open question, but we offer remarks about this issue. Moreover, we describe a procedure that provides estimates of upper bounds on the maximal density of tetrahedron packings, which could aid in assessing the packing efficiency of candidate dense packings.