Truncated octahedron

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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_0,1{3,4} t_0,1,2{3,3}
Wythoff symbol	2 4 \| 3 3 3 2 \|
Coxeter-Dynkin
Symmetry	O_h and T_h
References	U₀₈, C₂₀, W₇
Properties	Semiregular convex zonohedron permutohedron
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 the truncated octahedron is a zonohedron.

1 Construction
2 Coordinates and permutations
3 Area and volume
4 Uniform colourings
5 Related polyhedra
6 Tessellations
7 References
8 External links

[edit] Construction

A truncated octahedron is constructed from a regular octahedron with side length 3a by the removal of six right square pyramids, one from each point. These pyramids have both base side length, a, and lateral side length, e, of a. The base area is then a². Note that this shape is exactly similar to half an octahedron or Johnson solid J₁.

From the properties of square pyramids, we can now find the slant height, s, and the height, h of the pyramid:

$h = \sqrt{e^2-\frac{1}{2}a^2}=\frac{\sqrt{2}}{2}a$

$s = \sqrt{h^2 + \frac{1}{4}a^2} = \sqrt{\frac{1}{2}a^2 + \frac{1}{4}a^2}=\frac{\sqrt{3}}{2}a$

The volume, V, of the pyramid is given by:

$V = \frac{1}{3}a^2h = \frac{\sqrt{2}}{6}a^3$

Because six pyramids are removed by truncation, there is a total lost volume of √2 a³.

[edit] 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.

[edit] Area and volume

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

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

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

[edit] Uniform colourings

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

122 colouring
O_h symmetry
Wythoff: 2 4 | 3

123 colouring
T_h symmetry
Wythoff: 3 3 2 |

[edit] Related polyhedra

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

Cube

Truncated cube

cuboctahedron

Truncated octahedron

Octahedron

[edit] 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.

[edit] 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)
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 2006-09-08.

Gaiha, P., and Guha, S. K. (1977). "Adjacent vertices on a permutohedron". SIAM Journal on Applied Mathematics 32 (2): 323–327. doi:10.1137/0132025.