Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra

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 $\varphi =4000/4671=\mathrm{0.856347...}$. 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.

It contains 25 pages, 5 figures.

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• Maximum and Minimum Stable Random Packings of Platonic Solids
• Dense crystalline dimer packings of regular tetrahedra
• Phase Diagram of Hard Tetrahedra
• Densest binary sphere packings
• Jammed frictional tetrahedra are hyperstatic
• Equilibrium Phase Behavior and Maximally Random Jammed State of Truncated Tetrahedra
• Organizing Principles for Dense Packings of Nonspherical Hard Particles: Not All Shapes Are Created Equal
• Further details on the phase diagram of hard ellipsoids of revolution
• Athermal jamming of soft frictionless Platonic solids
• Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximally random jammed particle packings. II. Anisotropy in particle shape
• Isostaticity of Constraints in Jammed Systems of Soft Frictionless Platonic Solids
• Upper bound on the packing density of regular tetrahedra and octahedra
• Jamming transition in non-spherical particle systems: pentagons vs. disks
• Hard convex lens-shaped particles: Characterization of dense disordered packings
• A method for dense packing discovery
• Kinetic Frustration Effects on Dense Two-Dimensional Packings of Convex Particles and Their Structural Characteristics
• On the translative packing densities of tetrahedra and cubooctahedra