Packing, tiling, and covering with tetrahedra
- 11 July 2006
- journal article
- Published by Proceedings of the National Academy of Sciences in Proceedings of the National Academy of Sciences
- Vol. 103 (28), 10612-10617
- https://doi.org/10.1073/pnas.0601389103
Abstract
It is well known that three-dimensional Euclidean space cannot be tiled by regular tetrahedra. But how well can we do? In this work, we give several constructions that may answer the various senses of this question. In so doing, we provide some solutions to packing, tiling, and covering problems of tetrahedra. Our results suggest that the regular tetrahedron may not be able to pack as densely as the sphere, which would contradict a conjecture of Ulam. The regular tetrahedron might even be the convex body having the smallest possible packing density.Keywords
This publication has 13 references indexed in Scilit:
- A proof of the Kepler conjectureAnnals of Mathematics, 2005
- Unusually Dense Crystal Packings of EllipsoidsPhysical Review Letters, 2004
- Densest lattice packings of 3-polytopesComputational Geometry, 2000
- A counter-example to Kelvin's conjecture on minimal surfacesPhilosophical Magazine Letters, 1994
- An ellipsoid packing in E 3 of unexpected high densityMathematika, 1991
- Polytetrahedral Order in Condensed MatterPublished by Elsevier BV ,1989
- The densest lattice packing of tetrahedraBulletin of the American Mathematical Society, 1970
- Complex alloy structures regarded as sphere packings. II. Analysis and classification of representative structuresActa Crystallographica, 1959
- Complex alloy structures regarded as sphere packings. I. Definitions and basic principlesActa Crystallographica, 1958
- LXIII. On the division of space with minimum partitional areaThe London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1887