Truncated Tetrahedron EG-Models Home

image truncated_tetrahedron_Preview.gif
image truncated_tetrahedron_diff_Preview.gif
Electronic Geometry Model No. 2001.02.068


Martin Henk


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


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


Gif-file was produced by Povray 3.02

Submission information

Submitted: Thu Feb 1 16:41:52 CET 2001.
Accepted: Fri Apr 27 14:11:54 CET 2001.

Author's Address

Martin Henk
University of Magdeburg
Department of Mathematics
Universitätsplatz 2
D-39106 Magdeburg