Communication: A packing of truncated tetrahedra that nearly fills all of space and its melting properties

Yang Jiao,Salvatore Torquato

doi:10.1063/1.3653938

Yang Jiao, Salvatore Torquato

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https://doi.org/10.1063/1.3653938

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Journal: The Journal of Chemical Physics	Publication Date: Oct 17, 2011
Citations: 29	License type: CC BY 3.0

Affiliation: Princeton University

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Dense polyhedron packings are useful models of a variety of condensed matter and biological systems and have intrigued scientists and mathematicians for centuries. Here, we analytically construct the densest known packing of truncated tetrahedra with a remarkably high packing fraction φ = 207/208 = 0.995192..., which is amazingly close to unity and strongly implies its optimality. This construction is based on a generalized organizing principle for polyhedra lacking central symmetry that we introduce here. The "holes" in the putative optimal packing are perfect tetrahedra, which leads to a new tessellation of space by truncated tetrahedra and tetrahedra. Its packing characteristics and equilibrium melting properties as the system undergoes decompression are discussed.

Highlights

The three-dimensional Platonic and Archimedean polyhedra possess beautiful symmetries and arise in many natural and synthetic structures
General organizing principles have been established for the densest packings of polyhedra in R3.10,11 For centrally symmetric Platonic and Archimedean solids, it has been conjectured that the densest packings can be achieved by arranging the polyhedra on an appropriate Bravais lattice with the same orientation.10 (A centrally symmetric solid has a center of inversion symmetry.) For non-centrally symmetric polyhedra, the optimal packings are generally not Bravais lattice packings
A tetrahedron lacks central symmetry, and it is known that its densest packing must exceed that of the densest Bravais lattice packing

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Introduction

The three-dimensional Platonic and Archimedean polyhedra possess beautiful symmetries and arise in many natural and synthetic structures. To be a non-Bravais lattice packing.9 An Archimedean truncated tetrahedron has four regular hexagonal faces and four regular triangular faces, as obtained by truncating the corners (small tetrahedra with edge length 1/3 of that of the original tetrahedra) of a tetrahedron (see Fig. 1(a)). In this paper, we find an exact construction for the densest known packing of truncated tetrahedra with a remarkably high packing fraction φ = 207/208 = 0.995192.

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