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) | |

Zentralblatt No.
| 01682997 |

- Ulrich Betke and Martin Henk:
*Densest lattice packings of 3-polytopes*, Comp. Geom.**16**, 3 (2000), 157 - 186. - Hermann Minkowski:
*Dichteste gitterförmige Lagerung kongruenter Körper*, Nachr. K. Ges. Wiss. Göttingen, Math.-Phys. KL (1904) (1904), 311 - 355 (see also Gesammelte Abhandlungen vol. II, 3 - 42, Leipzig 1911). - J.V. Whitworth:
*On the densest packing of sections of a cube*, Annali Mat. Pura Appl.**27**(1948), 29 - 37.

- Master File: octahedron_Master.jvx
- Applet File: octahedron_Master.jvx
- Preview: octahedron_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