Dense periodic packings of tetrahedra with small repeating units

TL;DR

This paper introduces a family of dense, periodic packings of regular tetrahedra with small repeating units, achieving high packing fractions and inspired by numerical searches, with explicit analytic constructions and properties.

Contribution

It provides the first analytic constructions of dense tetrahedron packings with small repeating units, improving understanding of tetrahedral packing arrangements.

Findings

Packing fraction of approximately 0.8547 for four-tetrahedra units

Existence of a packing with two-tetrahedra units and a fraction of about 0.7194

Packings are transitive and analytically constructed

Abstract

We present a one-parameter family of periodic packings of regular tetrahedra, with the packing fraction $100/117\approx0.8547$, that are simple in the sense that they are transitive and their repeating units involve only four tetrahedra. The construction of the packings was inspired from results of a numerical search that yielded a similar packing. We present an analytic construction of the packings and a description of their properties. We also present a transitive packing with a repeating unit of two tetrahedra and a packing fraction $\frac{139+40\sqrt{10}}{369}\approx0.7194$.