Densest lattice packing of an octahedron
The octahedron has 6 vertices, 12 edges and 8 triangular facets. It is one of the five Platonic solids (it represents the element air in Plato's Timaios) and its dual is the cube.
The density of a densest lattice packing of an octahedron was already calculated by Minkowski in 1904. In 1948 Whitworth generalized Minkowski's result to a family of truncated cubes. The density of a densest lattice packing is equal to 18/19 = 0.9473..., and the 14 points in the picture show the lattice points of a critical lattice lying in the boundary of an octahedron.
Model produced with: JavaView v2.00.a11
|Keywords||lattice packings; polytopes; packings; critical lattice; octahedron|
|MSC-2000 Classification||52C17 (11H31)|
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