Densest lattice packing of a truncated tetrahedron

The truncated tetrahedron has 12 vertices, 18 edges and 8 facets, 4 hexagons and 4 triangles. It is one of the thirteen Archimedean solids and its dual is called trikis tetrahedron. It was rediscovered during th 15th century by the outstanding artist Piero della Francesca.

The density of a densest lattice packing was calculated with the algorithm of Betke and Henk. Since the truncated tetrahedron is a non 0-symmetric polytope one first has to compute the difference body of a truncated tetrahedron, which is a 0-symmetric polytope with 24 vertices, 36 edges and 14 facets, 8 hexagons and 6 squares. It is shown in the second picture.

The optimal packing lattices of a convex body and its difference body coincide. The density of a truncated tetrahedron is equal to 0.6809..., and the 14 points in the picture show the lattice points of a critical lattice of the difference body lying in the boundary.

Model produced with: JavaView v2.00.a11

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

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

Zentralblatt No.
| 01683009 |

- Ulrich Betke and Martin Henk:
*Densest lattice packings of 3-polytopes*, Journal of Comp. Geom.**16**, 3 (2000), 157 - 186.

- Master File: truncated_tetrahedron_Master.jvx
- Applet File: truncated_tetrahedron_Master.jvx
- Applet File: truncated_tetrahedron_diff_Applet.jvx
- Preview: truncated_tetrahedron_Preview.gif
- Preview: truncated_tetrahedron_diff_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