 Truncated octahedron

Truncated octahedron
(Click here for rotating model)Type Archimedean solid
Uniform polyhedronElements 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 CoxeterDynkin
Symmetry O_{h}, [4,3], (*432)
T_{h}, [3,3] and (*332)Dihedral Angle References U_{08}, C_{20}, W_{7} Properties Semiregular convex zonohedron
permutohedron
Colored faces
4.6.6
(Vertex figure)
Tetrakis hexahedron
(dual polyhedron)
NetIn geometry, the truncated octahedron is an Archimedean solid. It has 14 faces (8 regular hexagonal and 6 square), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron.
If the original truncated octahedron has unit edge length, its dual tetrakis cube has edge lengths and .
Contents
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, to form equilateral triangles. The base area is then a². Note that this shape is exactly similar to half an octahedron or Johnson solid J_{1}.
From the properties of square pyramids, we can now find the slant height, s, and the height, h of the pyramid:
The volume, V, of the pyramid is given by:
Because six pyramids are removed by truncation, there is a total lost volume of √2 a³.
Images
Orthographic projections Coordinates and permutohedron
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 threedimensional subspace x + y + z + w = 10. Therefore, the truncated octahedron is the permutohedron of order 4.
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:
Octahedral symmetry Tetrahedral symmetry
(Omnitruncated tetrahedron)
122 coloring
Wythoff: 2 4  3
123 coloring
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
It also exists as the omnitruncate of the tetrahedron family:
Tetrahedron
Truncated
tetrahedron
Rectified
tetrahedron
Cantellated
tetrahedron
Omnitruncated
tetrahedron
Snub
tetrahedronTessellations
The truncated octahedron exists in three different convex uniform honeycombs (spacefilling tessellations):
Bitruncated cubic Cantitruncated cubic Truncated alternated cubic The celltransitive bitruncated cubic honeycomb can also be seen as the Voronoi tessellation of the bodycentred cubic lattice. The truncated octahedron is one of five threedimensional primary parallelohedron.
References
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X. (Section 39)
 Freitas, Robert A., Jr. "Uniform spacefilling using only truncated octahedra". Figure 5.5 of Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999. http://www.nanomedicine.com/NMI/Figures/5.5.jpg. Retrieved 20060908.
 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.
 Hart, George W. "VRML model of truncated octahedron". Virtual Polyhedra: The Encyclopedia of Polyhedra. http://www.georgehart.com/virtualpolyhedra/vrml/truncated_octahedron.wrl. Retrieved 20060908.
 Mäder, Roman. "The Uniform Polyhedra: Truncated Octahedron". http://www.mathconsult.ch/showroom/unipoly/08.html. Retrieved 20060908.
 Alexandrov, A. D. (1958). Convex polyhedra. Berlin : Springer, cop.. pp. 539. doi:3540231587.
External links
 Eric W. Weisstein, Truncated octahedron (Archimedean solid) at MathWorld.
 Weisstein, Eric W., "Permutohedron" from MathWorld.
 Richard Klitzing, 3D convex uniform polyhedra, x3x4o  toe
 Editable printable net of a truncated octahedron with interactive 3D view
Polyhedron navigator Platonic solids (regular) Archimedean solids
(Semiregular/Uniform)Catalan solids
(Dual semiregular)triakis tetrahedron · rhombic dodecahedron · triakis octahedron · tetrakis cube · deltoidal icositetrahedron · disdyakis dodecahedron · pentagonal icositetrahedron · rhombic triacontahedron · triakis icosahedron · pentakis dodecahedron · deltoidal hexecontahedron · disdyakis triacontahedron · pentagonal hexecontahedronDihedral regular Dihedral uniform Duals of dihedral uniform Dihedral others Degenerate polyhedra are in italics.Categories: Uniform polyhedra
 Archimedean solids
 Spacefilling polyhedra
 Zonohedra

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