The densest packing of congruent regular tetrahedra
Let $T \subset \mathbb{R}^3$ be a regular tetrahedron of edge length one. A packing is a locally finite family $\mathcal{P}$ of images of $T$ under Euclidean isometries with pairwise disjoint interiors; orientations are arbitrary and the packing need not be periodic. Let $$\delta(\mathcal{P}) = \limsup_{R \to \infty} \frac{|B_R(0) \cap \bigcup_{P \in \mathcal{P}} P|}{|B_R(0)|}, \qquad \delta_T = \sup_{\mathcal{P}} \delta(\mathcal{P}).$$ Hilbert asked how congruent copies of a given solid, for example regular tetrahedra, can be arranged in space as densely as possible. Chen, Engel and Glotzer constructed a periodic packing of density $4000/4671 \approx 0.8563$ (2010). Gravel, Elser and Kallus proved that every packing leaves a fraction at least $2.6 \times 10^{-25}$ of space empty, so $\delta_T \le 1 - 2.6 \times 10^{-25}$ (2011).
Problem. Determine $\delta_T$ exactly, by giving a matching rigorous upper bound and packing construction.
Source: Hilbert (1900), the eighteenth problem, which names regular tetrahedra; see Lagarias and Zong (2012). Stated by AIM — Open Applied Problems as the exact determination of the packing density.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- CO
- Keywords
- AIM Open Applied Problems, packing densities, regular tetrahedra
References
@article{hilbert1902,
author = {Hilbert, David},
title = {Mathematical problems},
journal = {Bull. Amer. Math. Soc.},
volume = {8},
number = {10},
year = {1902},
pages = {437--479},
doi = {10.1090/S0002-9904-1902-00923-3},
note = {Translated by Mary Winston Newson from Nachr. Ges. Wiss. G{\"o}ttingen (1900), 253--297},
}
@article{lagarias2012,
author = {Lagarias, Jeffrey C. and Zong, Chuanming},
title = {Mysteries in packing regular tetrahedra},
journal = {Notices Amer. Math. Soc.},
volume = {59},
number = {11},
year = {2012},
pages = {1540--1549},
doi = {10.1090/noti918},
}
@article{chen2010,
author = {Chen, Elizabeth R. and Engel, Michael and Glotzer, Sharon C.},
title = {Dense crystalline dimer packings of regular tetrahedra},
journal = {Discrete Comput. Geom.},
volume = {44},
number = {2},
year = {2010},
pages = {253--280},
doi = {10.1007/s00454-010-9273-0},
eprint = {1001.0586},
archivePrefix = {arXiv},
}
@article{gravel2011,
author = {Gravel, Simon and Elser, Veit and Kallus, Yoav},
title = {Upper bound on the packing density of regular tetrahedra and octahedra},
journal = {Discrete Comput. Geom.},
volume = {46},
number = {4},
year = {2011},
pages = {799--818},
doi = {10.1007/s00454-010-9304-x},
eprint = {1008.2830},
archivePrefix = {arXiv},
}
@misc{aim-regular-tetrahedron-packing,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The densest packing of congruent regular tetrahedra}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/238-regular-tetrahedron-packing.md},
note = {Entry 238 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00635, "The densest packing of congruent regular tetrahedra", clopen.solutions.