Packing, Tiling, and Covering with Tetrahedra

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...

Full description

Saved in:
Bibliographic Details
Published inProceedings of the National Academy of Sciences - PNAS Vol. 103; no. 28; pp. 10612 - 10617
Main Authors Conway, J. H., Torquato, S.
Format Journal Article
LanguageEnglish
Published United States National Academy of Sciences 11.07.2006
National Acad Sciences
Subjects
Bubbles
Coordinate systems
Density
Euclidean space
Icosahedrons
Mathematical lattices
Octahedrons
Packing density
Physical Sciences
Studies
Symmetry
Tessellations
Tetrahedra
Tetrahedrons
Vertices
Online AccessGet full text

Cover

More Information
Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 23
Author contributions: J.H.C. and S.T. designed research, performed research, and wrote the paper.
Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved May 24, 2006
ISSN:0027-8424
1091-6490
DOI:10.1073/pnas.0601389103