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The 'truncated octahedron' is an Archimedean solid. It has 8 regular hexagonal faces, 6 regular square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron.
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Contents

Coordinates and permutations

Area and volume

Uniform colorings

Related polyhedra

Tessellations

References

External links

Coordinates and permutations

All permutations of (0, ±1, ±2) are Cartesian coordinates of the vertices of a truncated octahedron centered at the origin. The vertices are thus also the corners of 12 rectangles whose long edges are parallel to the coordinate axes.
The truncated octahedron can also be represented by even more symmetric coordinates in four dimensions: all permutations of (1,2,3,4) form the vertices of a truncated octahedron in the three-dimensional subspace ''x'' + ''y'' + ''z'' + ''w'' = 10. For this reason the truncated octahedron is also sometimes known as the permutohedron.

Area and volume

The area ''A'' and the volume ''V'' of a truncated octahedron of edge length ''a'' are:
:
:

Uniform colorings

There are two uniform colorings, with tetrahedral symmetry and octahedral symmetry:

122 coloring Oh symmetry Wythoff: 2 4 > 3

123 coloring Th symmetry Wythoff: 3 3 2 >

Related polyhedra

The truncated octahedron exists within the set of truncated forms between a cube and octahedron:

Cube

Truncated cube

cuboctahedron

Truncated octahedron

Octahedron

Tessellations

The truncated octahedron exists in three different convex uniform honeycombs (space-filling tessellations):

Bitruncated cubic

Cantitruncated cubic

Truncated alternated cubic

The cell-transitive bitruncated cubic honeycomb can also be seen as the Voronoi tessellation of the body-centred cubic lattice.

References

★ The Geometrical Foundation of Natural Structure: A Source Book of Design, , Robert, Williams, Dover Publications, Inc, 1979, ISBN 0-486-23729-X (Section 3-9)

★ Uniform space-filling using only truncated octahedra Freitas, Robert A., Jr

★ Adjacent vertices on a permutohedron, Gaiha, P., and Guha, S. K., , , SIAM Journal on Applied Mathematics, 1977

★ VRML model of truncated octahedron Hart, George W

★ The Uniform Polyhedra: Truncated Octahedron Mäder, Roman

★ Permutohedron Weisstein, Eric W

External links

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