Densest lattice packing of a tetrahedron

The tetrahedron has 4 vertices, 6 edges and 4 triangular facets. It is one the five Platonic solids (it represents the element fire in Plato's Timaios) and it is selfdual.

In 1972 Hoylman calculated the lattice packing density of a tetrahedron which is equal to 18/49=0.3673... The pictures show a tetrahdron and its difference body, the cubeoctahedron. Both polytopes have the same optimal packing lattices, and the 14 points in the picture show the lattice points of a critical lattice of a cubeoctahedron lying in the boundary.

Model produced with: JavaView v2.00.a11

Keywords
| lattice packings; polytopes; packings; critical lattice; tetrahedron | |

MSC-2000 Classification
| 52C17 (11H31) | |

Zentralblatt No.
| 01683002 |

- Ulrich Betke and Martin Henk:
*Densest lattice packings of 3-polytopes*, Comp. Geom.**16**, 3 (2000), 157 - 186. - D.J. Hoylman:
*The densest lattice packing of tetrahedra*, Bull. Amer. Math. Soc**76**(1970), 135 - 137.

- Master File: tetrahedron_Master.jvx
- Applet File: tetrahedron_Master.jvx
- Applet File: cubeoctahedron_Applet.jvx
- Preview: tetrahedron_Preview.gif
- Preview: cubeoctahedron_Preview.gif

Gif-file was produced by Povray 3.02

Submitted: Thu Feb 1 16:41:52 CET 2001.

Accepted: Fri Apr 27 14:11:54 CET 2001.

University of Magdeburg

Department of Mathematics

Universitätsplatz 2

D-39106 Magdeburg

henk@mail.math.uni-magdeburg.de

http://www.math.uni-magdeburg.de/~henk