Densest lattice packing of a dodecahedron

The dodecahedron has 20 vertices, 30 edges and 12 pentagonal facets. It is one the five Platonic solids (it represents the cosmos or ether in Plato's Timaios) and its dual is the icosahedron.

The density of a densest lattice packing was calculated with the algorithm of Betke and Henk. The density is equal to 0.9045..., and the 12 points in the picture show the lattice points of a critical lattice lying in the boundary. The critical lattice coincides with fcc-lattice, which is the critical lattice of a 3-dimensional sphere.

Model produced with: JavaView v2.00.a11

Keywords
| lattice packings; polytopes; packings; critical lattice; dodecahedron | |

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

Zentralblatt No.
| 01682994 |

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

- Master File: dodecahedron_Master.jvx
- Applet File: dodecahedron_Master.jvx
- Preview: dodecahedron_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