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Discrete Comput Geom (2008) 40: 214240 DOI 10.1007/s00454-008-9101-y
A Dense Packing of Regular Tetrahedra
Elizabeth R. Chen
Received: 1 January 2007 / Revised: 9 July 2008 / Accepted: 9 July 2008 / Published online: 22 August 2008 Springer Science+Business Media, LLC 2008
Abstract We construct a dense packing of regular tetrahedra, with packing density D>.7786157.
Keywords Crystallography Packing Regular solid Hilbert problem
1 Introduction
How can one arrange most densely in space an innite number of equal solids of given form, e.g., spheres with given radii or regular tetrahedra with given edges (or in prescribed position), that is, how can one so t them together that the ratio of the lled to the unlled space may be as great as possible? (excerpt from David Hilberts 18th problem [8]).
This is a very old problem. Aristotle [1] believed that you could tile space with regular tetrahedra. Everyone believed him for the next 1800 years, until Johannes Mller (aka Regimontanus) proved everyone wrong (histories by Dirk Struik [13] and Marjorie Senechal [12]).
This is a very understandable mistake, because tetrahedra almost tile space locally, but not quite. The cluster E5 (consisting of 5 tetrahedra joined symmetrically about an edge) has total solid angle 5 2 cos1 13 12.309594173408 about the edge, and local density D .979566380077. The cluster V20 (consisting of 20 tetrahedra joined symmetrically about a vertex) has total solid angle 20 ( + 3 cos1 13 ) 11.025711968651 about the vertex, and local density D .877398280459.
For the sphere, the analogous problem was very challenging. A long time ago, Johannes Kepler conjectured that the densest packing is the hexagonal close-packing (HCP), with density D = /18 .740480489693. Carl Friedrich Gauss proved
This research was partially supported by the NSF-RTG grant DMS-0502170.
E.R. Chen ( )
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA e-mail: [email protected]
Discrete Comput Geom (2008) 40: 214240 215
that the HCP is the densest lattice packing. Only recently, Thomas Hales & Samuel Ferguson [6, 7] proved that the HCP is the densest packing in general.
Helmut Grmer [5] constructed a lattice packing of the single tetrahedron B1 with density D = 1849 .367346938775. Douglas Hoylman [9] proved that Grmers packing was the densest lattice packing.
Andrew Hurley [10]...