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image cubeoctahedron_Preview.gif
Electronic Geometry Model No. 2001.02.052


Martin Henk


Densest lattice packing of a cubeoctahedron

The cubeoctahedron has 12 vertices, 24 edges and 14 facets, 6 squares and 8 triangles. It is one of the thirteen Archimedean solids and it is the difference body of a tetrahedron. Its dual is called rhombic dodecahedron. It was already mentioned by Plato and rediscovered during the 15th century by the outstanding artist Piero della Francesca.

In 1972 Hoylman calculated the lattice packing density of a cubeoctahedron, which is equal to 45/49=0.9183... The 14 points in the picture show the lattice points of a critical lattice lying in the boundary.

Model produced with: JavaView v2.00.a11

Keywords lattice packings; polytopes; packings; critical lattice; cubeoctahedron
MSC-2000 Classification 52C17 (11H31)
Zentralblatt No. 01682993


  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.


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