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image snub_cube_Preview.gif
image snub_cube_diff_Preview.gif
Electronic Geometry Model No. 2001.02.059


Martin Henk


Densest lattice packing of a snub cube

The snub cube has 24 vertices, 60 edges and 38 facets, 6 squares and 32 triangles. It is one of the thirteen Archimedean solids and its dual is called pentagonal icositetrahedron. Maybe the first presentation of this polytope can be found in the revised version of Albrecht Dürer's Underweysung der Messung (around 1538).

The density of a densest lattice packing was calculated with the algorithm of Betke and Henk. Since the snub cube is a non 0-symmetric polytope one first has to compute the difference body of a snub cube, which is a 0-symmetric polytope with 72 vertices, 144 edges and 74 facets, 6 octagons, 8 hexagons, 12 squares and 48 triangles. It is shown in the second picture.

The optimal packing lattices of a convex body and its difference body coincide. The lattice packing density of a snub cube is equal to 0.7876..., and the 12 points in the picture show the lattice points of a critical lattice of the difference body lying in the boundary.

Model produced with: JavaView v2.00.a11

Keywords lattice packings; polytopes; packings; critical lattice; snub cube
MSC-2000 Classification 52C17 (11H31)
Zentralblatt No. 01683000


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


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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