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Truncated octahedron
(Click here for rotating model)
Type	Archimedean solid
Elements	F=14, E=36, V=24 (χ=2)
Faces by sides	6{4}+8{6}
Schläfli symbol	t{3,4}
Wythoff symbol	2 4 | 3
Coxeter-Dynkin
Symmetry	Oh
References	U08, C20, W7
Properties	Semiregular convex zonohedron
Colored faces	4.6.6 (Vertex figure)
Tetrakis hexahedron (dual polyhedron)	Net

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 (or 180° rotational symmetry), the truncated octahedron is a zonohedron.

Contents 1 Coordinates and Permutations 2 Geometric relations 3 Related polyhedra 4 References 5 External links

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. The construction generalizes to any n, and forms an (n-1)-dimensional polytope the vertices of which represent the permutations of a set of n items; for instance, the six permutations of (1,2,3) form a regular hexagon in the plane x+y+z=6.

Truncated octahedra are able to tessellate 3-dimensional space, forming the bitruncated cubic honeycomb. This tessellation can also be seen as the Voronoi tessellation of the body-centred cubic lattice.

Compare:

Cube

Truncated cube

cuboctahedron

Truncated octahedron

Octahedron

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)
• Freitas, Robert A., Jr. Uniform space-filling using only truncated octahedra. Figure 5.5 of Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999. Retrieved on September 8, 2006.

• Gaiha, P., and Guha, S. K. (1977). "Adjacent vertices on a permutohedron". SIAM Journal on Applied Mathematics 32 (2): 323–327. 

• Hart, George W. VRML model of truncated octahedron. Virtual Polyhedra: The Encyclopedia of Polyhedra. Retrieved on September 8, 2006.

• Mäder, Roman. The Uniform Polyhedra: Truncated Octahedron. Retrieved on September 8, 2006.

• Weisstein, Eric W. Permutohedron. MathWorld–A Wolfram Web Resource. Retrieved on September 8, 2006.

• Eric W. Weisstein, Truncated octahedron at MathWorld.