# Dense Crystalline Dimer Packings of Regular Tetrahedra

@article{Chen2010DenseCD, title={Dense Crystalline Dimer Packings of Regular Tetrahedra}, author={Elizabeth R. Chen and Michael Engel and Sharon C. Glotzer}, journal={Discrete \& Computational Geometry}, year={2010}, volume={44}, pages={253-280} }

We present the densest known packing of regular tetrahedra with density $\phi =\frac{4000}{4671}=0.856347\ldots\,$. Like the recently discovered packings of Kallus et al. and Torquato–Jiao, our packing is crystalline with a unit cell of four tetrahedra forming two triangular dipyramids (dimer clusters). We show that our packing has maximal density within a three-parameter family of dimer packings. Numerical compressions starting from random configurations suggest that the packing may be optimal…

## 84 Citations

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

- MathematicsThe Journal of chemical physics
- 2011

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.

Golden, Quasicrystalline, Chiral Packings of Tetrahedra

- Materials Science
- 2019

Since antiquity, the packing of convex shapes has been of great interest to many scientists and mathematicians. Recently, particular interest has been given to packings of three-dimensional…

Quasi-random packing of tetrahedra

- Computer Science
- 2013

A new order metric for tetrahedral particle packing is presented, which is observed to have a strong linear correlation with the packing density, and the nematic order of clusters can be used to classify the ordered and disordered packing of tetrahedra.

Phase diagram of hard tetrahedra.

- Materials ScienceThe 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.

Crystalline assemblies and densest packings of a family of truncated tetrahedra and the role of directional entropic forces.

- Materials ScienceACS nano
- 2012

Polyhedra and their arrangements have intrigued humankind since the ancient Greeks and are today important motifs in condensed matter, with application to many classes of liquids and solids. Yet,…

Dense regular packings of irregular nonconvex particles.

- MathematicsPhysical review letters
- 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.

Classifying Crystals of Rounded Tetrahedra and Determining Their Order Parameters Using Dimensionality Reduction

- Materials ScienceACS nano
- 2020

It is argued that these characteristic combinations of q̅l are also useful as reliable order parameters in nucleation studies, enhanced sampling techniques, or inverse-design methods involving odd-shaped particles in general.

Self-assembly of a space-tessellating structure in the binary system of hard tetrahedra and octahedra.

- Materials ScienceSoft matter
- 2016

Both known one-component phases - the dodecagonal quasicrystal of tetrahedra and the densest-packing of octahedra in the Minkowski lattice - are found to coexist with the binary phase.

## References

SHOWING 1-10 OF 35 REFERENCES

Dense Periodic Packings of Tetrahedra with Small Repeating Units

- MathematicsDiscret. Comput. Geom.
- 2010

We present a one-parameter family of periodic packings of regular tetrahedra, with the packing fraction 100/117≈0.8547, that are simple in the sense that they are transitive and their repeating units…

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

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2010

This study provides the most general analytical formulation to date to construct dense periodic packings of tetrahedra with four particles per fundamental cell and suggests that the latter set of packings are the densest among all packings with a four-particle basis.

Analytical Constructions of a Family of Dense Tetrahedron Packings and the Role of Symmetry

- Mathematics
- 2009

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…

Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra

- Materials ScienceNature
- 2009

The first example of a quasicrystal formed from hard or non-spherical particles, which can be compressed to a packing fraction of φ = 0.8324, is reported, demonstrating that particle shape and entropy can produce highly complex, ordered structures.

Experiments on the random packing of tetrahedral dice.

- PhysicsPhysical review letters
- 2010

The authors' experiments on tetrahedral dice indicate the densest (volume fraction phi=0.76+/-.02, compared with phi(sphere)=0.64), most disordered, experimental, random packing of any set of congruent convex objects to date.

A Picturebook of Tetrahedral Packings.

- Mathematics
- 2010

A Picturebook of Tetrahedral Packings by Elizabeth R. Chen Chair: Jeffrey C. Lagarias We explore many different packings of regular tetrahedra, with various clusters & lattices & symmetry groups. We…

THE DENSEST LATTICE PACKING OF TETRAHEDRA

- Mathematics
- 1970

The problem of finding the densest packing of tetrahedra was first suggested by Hubert [3, p. 319]. Minkowski [4] attempted to find the densest lattice packing of tetrahedra, but his result is…

Packing, tiling, and covering with tetrahedra.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 2006

The results suggest that the regular tetrahedron may not be able to pack as densely as the sphere, which would contradict a conjecture of Ulam.

Dense packings of polyhedra: Platonic and Archimedean solids.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2009

The conjecture that the densest packings of Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings is made, which can be regarded to be the analog of Kepler's sphere conjecture for these solids.