Densest Tetrahedral Packing
Obsolete news: The cover article for the 13 August 2009 issue of Nature published a packing method for tetrahedra with a packing density of 0.782021, a new record. For complex packings, space is divided into an orderly arrangement of identical cells. In this packing, each cell has 72 tetrahedra, shown here.[more]
The cells of a packing are usually parallelepipeds, but can be any space-filling polyhedron. The objects in a packing can extend outside of the cell, so long as they do not overlap objects in the neighboring cell. The packing density is the percentage of filled space within a particular cell.
Updated news: Aristotle mistakenly claimed that regular tetrahedra fill space. 1800 years later (in ~1470), Regiomontanus caught the error. In 1900, Hilbert asked for the densest tetrahedral packing as a part of his problem. In 2006, Conway and Torquato published a packing with density 0.7175. In 2008, Chen found a packing with density 0.7820, which was used to make the original version of this Demonstration. In 2009, Kallus, Elser, and Gravel found a quasicrystalline packing with density 0.8547. Torquato and Jiao increased the density to 0.8555.
The latest results, as of July 27, 2010, come from Chen, Engel, and Glotzer. There is a packing with density 4000/4671, or 0.856347, shown here. The unit cell has 16 tetrahedra.[less]
 S. Torquato and Y. Jiao, "Dense Packings of the Platonic and Archimedean Solids," Nature, 460(13), 2009 pp. 876–879.
 E. Chen, M. Engel, and S. Glotzer, "Dense Crystalline Dimer Packings of Regular Tetrahedra," Discrete and Computational Geometry, 44(2), 2010 pp. 253–280. http://arxiv.org/abs/1001.0586.