# Tetrahedron packing

In geometry, tetrahedron packing is the problem of arranging identical regular tetrahedra throughout three-dimensional space so as to fill the maximum possible fraction of space.

The currently densest known packing structure for regular tetrahedra is a double lattice of triangular bipyramids and fills 85.63% of space

Currently, the best lower bound achieved on the optimal packing fraction of regular tetrahedra is 85.63%.[1] Tetrahedra do not tile space,[2] and an upper bound below 100% (namely, 1 − (2.6...)·10−25) has been reported.[3]

## Historical results

Tetrahedral packaging

Aristotle claimed that tetrahedra could fill space completely.[4] [5]

In 2006, Conway and Torquato showed that a packing fraction about 72% can be obtained by constructing a non-Bravais lattice packing of tetrahedra (with multiple particles with generally different orientations per repeating unit), and thus they showed that the best tetrahedron packing cannot be a lattice packing (with one particle per repeating unit such that each particle has a common orientation).[6] These packing constructions almost doubled the optimal Bravais-lattice-packing fraction 36.73% obtained by Hoylman.[7] In 2007 and 2010, Chaikin and coworkers experimentally showed that tetrahedron-like dice can randomly pack in a finite container up to a packing fraction between 75% and 76%.[8] In 2008, Chen was the first to propose a packing of hard, regular tetrahedra that packed more densely than spheres, demonstrating numerically a packing fraction of 77.86%.[9][10] A further improvement was made in 2009 by Torquato and Jiao, who compressed Chen's structure using a computer algorithm to a packing fraction of 78.2021%.[11]

In mid-2009 Haji-Akbari et al. showed, using MC simulations of initially random systems that at packing densities >50% an equilibrium fluid of hard tetrahedra spontaneously transforms to a dodecagonal quasicrystal, which can be compressed to 83.24%. They also reported a glassy, disordered packing at densities exceeding 78%. For a periodic approximant to a quasicrystal with an 82-tetrahedron unit cell, they obtained a packing density as high as 85.03%.[12]

In late 2009, a new, much simpler family of packings with a packing fraction of 85.47% was discovered by Kallus, Elser, and Gravel.[13] These packings were also the basis of a slightly improved packing obtained by Torquato and Jiao at the end of 2009 with a packing fraction of 85.55%,[14] and by Chen, Engel, and Glotzer in early 2010 with a packing fraction of 85.63%.[1] The Chen, Engel and Glotzer result currently stands as the densest known packing of hard, regular tetrahedra. Surprisingly, the quasicrystal approximant[12] packs denser than this double lattice of triangular bipyramids when tetrahedra are slightly rounded (the Minkowski sum of a tetrahedron and a sphere), making the 82-tetrahedron quasicrystal approximant the largest unit cell for a densest packing of identical particles to date.[15]

## Relationship to other packing problems

Because the earliest lower bound known for packings of tetrahedra was less than that of spheres, it was suggested that the regular tetrahedra might be a counterexample to Ulam's conjecture that the optimal density for packing congruent spheres is smaller than that for any other convex body. However, the more recent results have shown that this is not the case.

• Packing problem
• Disphenoid tetrahedral honeycomb - an isohedral packing of irregular tetrahedra in 3-space.
• The triakis truncated tetrahedral honeycomb is cell-transitive and based on a regular tetrahedron.

## References

1. Chen, Elizabeth R.; Engel, Michael; Glotzer, Sharon C. (2010). "Dense crystalline dimer packings of regular tetrahedra". Discrete & Computational Geometry 44 (2): 253–280. doi:10.1007/s00454-010-9273-0.
2. Struik, D. J. (1925). "Het probleem 'De Impletione Loci'". Nieuw Archief voor Wiskunde. 2nd ser 15: 121–134.
3. Simon Gravel; Veit Elser; Yoav Kallus (2010). "Upper bound on the packing density of regular tetrahedra and octahedra". Discrete & Computational Geometry 46 (4): 799–818. doi:10.1007/s00454-010-9304-x.
4. Jeffrey Lagarias and Chuanming Zong (2012-12-04). "Mysteries in Packing Regular Tetrahedra".
5. Conway, J. H. (2006). "Packing, tiling, and covering with tetrahedra". Proceedings of the National Academy of Sciences 103 (28): 10612–10617. doi:10.1073/pnas.0601389103. PMID 16818891. Bibcode2006PNAS..10310612C.
6. Hoylman, Douglas J. (1970). "The densest lattice packing of tetrahedra". Bulletin of the American Mathematical Society 76: 135–138. doi:10.1090/S0002-9904-1970-12400-4.
7. Jaoshvili, Alexander; Esakia, Andria; Porrati, Massimo; Chaikin, Paul M. (2010). "Experiments on the Random Packing of Tetrahedral Dice". Physical Review Letters 104 (18): 185501. doi:10.1103/PhysRevLett.104.185501. PMID 20482187. Bibcode2010PhRvL.104r5501J.
8. Chen, Elizabeth R. (2008). "A Dense Packing of Regular Tetrahedra". Discrete & Computational Geometry 40 (2): 214–240. doi:10.1007/s00454-008-9101-y.
9. Cohn, Henry (2009). "Mathematical physics: A tight squeeze". Nature 460 (7257): 801–802. doi:10.1038/460801a. PMID 19675632. Bibcode2009Natur.460..801C.
10. Torquato, S.; Jiao, Y. (2009). "Dense packings of the Platonic and Archimedean solids". Nature 460 (7257): 876–879. doi:10.1038/nature08239. PMID 19675649. Bibcode2009Natur.460..876T.
11. Haji-Akbari, Amir; Engel, Michael; Keys, Aaron S.; Zheng, Xiaoyu; Petschek, Rolfe G.; Palffy-Muhoray, Peter; Glotzer, Sharon C. (2009). "Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra". Nature 462 (7274): 773–777. doi:10.1038/nature08641. PMID 20010683. Bibcode2009Natur.462..773H.
12. Kallus, Yoav; Elser, Veit; Gravel, Simon (2010). "Dense Periodic Packings of Tetrahedra with Small Repeating Units". Discrete & Computational Geometry 44 (2): 245–252. doi:10.1007/s00454-010-9254-3.
13. Torquato, S.; Jiao, Y. (2009). "Analytical Constructions of a Family of Dense Tetrahedron Packings and the Role of Symmetry". arXiv:0912.4210 [cond-mat.stat-mech].
14. Jin, Weiwei; Lu, Peng; Li, Shuixiang (December 2015). "Evolution of the dense packings of spherotetrahedral particles: from ideal tetrahedra to spheres". Scientific Reports 5 (1): 15640. doi:10.1038/srep15640. PMID 26490670. Bibcode2015NatSR...515640J.