Truncated octahedron

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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.

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:

$A\; =\; (6+12sqrt\{3\})\; a^2\; approx\; 26.7846097a^2$

$V\; =\; 8sqrt\{2\}\; a^3\; approx\; 11.3137085a^3.$

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

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

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External links

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Last updated on Friday December 14, 2007 at 12:10:22 PST (GMT -0800)
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