Tetrahedron

Author

Description

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

References

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

Files

• 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

Submission information

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

Author's Address

University of Magdeburg
Department of Mathematics
Universitätsplatz 2
D-39106 Magdeburg
henk@mail.math.uni-magdeburg.de
http://www.math.uni-magdeburg.de/~henk