# Exact constructions of a family of dense periodic packings of tetrahedra.

@article{Torquato2010ExactCO, title={Exact constructions of a family of dense periodic packings of tetrahedra.}, author={Salvatore Torquato and Yang Jiao}, journal={Physical review. E, Statistical, nonlinear, and soft matter physics}, year={2010}, volume={81 4 Pt 1}, pages={ 041310 } }

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…

## 45 Citations

Upper Bound on the Packing Density of Regular Tetrahedra and Octahedra

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- 2011

The existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedral edges, so that each sphere is not fully covered by the packing, is shown.

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

- Physics, MedicineThe Journal of chemical physics
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This work 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.

Phase diagram of hard tetrahedra.

- Chemistry, PhysicsThe Journal of chemical physics
- 2011

It is shown that the quasicrystal approximant is more stable than the dimer crystal for packing densities below 84% using Monte Carlo computer simulations and free energy calculations.

Dense Crystalline Dimer Packings of Regular Tetrahedra

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The packing is crystalline with a unit cell of four tetrahedra forming two triangular dipyramids (dimer clusters) and it is shown that the packing has maximal density within a three-parameter family of dimer packings.

Self-assembly of uniform polyhedral silver nanocrystals into densest packings and exotic superlattices.

- Materials Science, MedicineNature materials
- 2011

It is shown with experiment and computer simulation that a range of nanoscale Ag polyhedra can self-assemble into their conjectured densest packings, and that octahedra form an exotic superstructure with complex helical motifs rather than the densest Minkowski lattice.

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- 2011

A new numerical scheme to study systems of nonconvex, irregular, and punctured particles in an efficient manner is presented and it is proved that the densest packing is obtained for both rhombiuboctahedra and rhombic enneacontrahedra.

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- 2012

This work presents the structures of the densest-known packings and demonstrates that a broad array of different densest mechanically stable structures consisting of only two types of components can form without any consideration of attractive or anisotropic interactions.

Disordered strictly jammed binary sphere packings attain an anomalously large range of densities.

- Physics, MedicinePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2013

It is reported that disordered strictly jammed binary packings can be produced with an anomalously large range of average packing fractions 0.634≤φ≤0.829 for small to large sphere radius ratios α restricted to α≥0.100, and an unusual feature of the packing fraction of jammed backbones (packings with rattlers excluded).

Perspective: Basic understanding of condensed phases of matter via packing models.

- Physics, MedicineThe Journal of chemical physics
- 2018

This perspective reviews pertinent theoretical and computational literature concerning the equilibrium, metastable, and nonequilibrium packings of hard-particle packings in various Euclidean space dimensions and emphasizes the "geometric-structure" approach, which provides a powerful and unified means to quantitatively characterize individual packings via jamming categories and "order" maps.

Equilibrium phase behavior and maximally random jammed state of truncated tetrahedra.

- Materials Science, PhysicsThe journal of physical chemistry. B
- 2014

A Monte Carlo implementation of the adaptive-shrinking-cell (ASC) numerical scheme and free-energy calculations are employed to ascertain with high precision the equilibrium phase behavior of systems of congruent Archimedean truncated tetrahedra over the entire range of possible densities up to the maximal nearly space-filling density.

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