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| {{Semireg polyhedra db|Semireg polyhedron stat table|tO}}
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| In [[geometry]], the '''truncated octahedron''' is an [[Archimedean solid]]. It has 14 faces (8 regular [[hexagon]]al and 6 [[Square (geometry)|square]]), 36 edges, and 24 vertices. Since each of its faces has [[point symmetry]] the truncated octahedron is a [[zonohedron]]. It is also the [[Goldberg polyhedron]] G<sub>IV</sub>(1,1), containing square and hexagonal faces.
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| If the original truncated octahedron has unit edge length, its dual [[tetrakis cube]] has edge lengths <math>\tfrac{9}{8}\scriptstyle {\sqrt{2}}</math> and <math>\tfrac{3}{2}\scriptstyle{\sqrt{2}}</math>.
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| __TOC__
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| ==Construction==
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| {| align=center
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| |[[image:Truncated_Octahedron_with_Construction.svg]]
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| |width=50px|
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| |[[image:Square Pyramid.svg]]
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| |}
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| A truncated octahedron is constructed from a regular [[octahedron]] with side length 3''a'' by the removal of six right [[square pyramid]]s, one from each point. These pyramids have both base side length (''a'') and lateral side length (''e'') of ''a'', to form [[equilateral triangle]]s. The base area is then ''a''<sup>2</sup>. Note that this shape is exactly similar to half an octahedron or [[Johnson solid]] J<sub>1</sub>.
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| From the properties of square pyramids, we can now find the slant height, ''s'', and the height, ''h'', of the pyramid:
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| ::<math>h = \sqrt{e^2-\frac{1}{2}a^2}=\frac{\sqrt{2}}{2}a</math>
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| ::<math>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</math>
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| The volume, ''V'', of the pyramid is given by:
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| ::<math>V = \frac{1}{3}a^2h = \frac{\sqrt{2}}{6}a^3</math>
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| Because six pyramids are removed by truncation, there is a total lost volume of <math>\scriptstyle {\sqrt{2}a^3}</math>.
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| ==Orthogonal projections==
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| The ''truncated octahedron'' has five special [[orthogonal projection]]s, centered, on a vertex, on two types of edges, and two types of faces: Hexagon, and square. The last two correspond to the B<sub>2</sub> and A<sub>2</sub> [[Coxeter plane]]s.
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| {|class=wikitable width=640
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| |+ Orthogonal projections
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| |-
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| !Centered by
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| !Vertex
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| !Edge<br>4-6
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| !Edge<br>6-6
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| !Face<br>Square
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| !Face<br>Hexagon
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| |-
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| !Truncated<BR>octahedron
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| |[[File:Cube t12 v.png|100px]]
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| |[[File:Cube t12 e46.png|100px]]
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| |[[File:Cube t12 e66.png|100px]]
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| |[[File:3-cube t12_B2.svg|100px]]
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| |[[File:3-cube t12.svg|100px]]
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| |-
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| !Hexakis<BR>hexahedron
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| |[[File:Dual cube t12 v.png|100px]]
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| |[[File:Dual cube t12 e46.png|100px]]
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| |[[File:Dual cube t12 e66.png|100px]]
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| |[[File:Dual cube t12_B2.png|100px]]
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| |[[File:Dual cube t12.png|100px]]
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| |- align=center
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| !Projective<BR>symmetry
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| |[2]
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| |[2]
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| |[2]
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| |[4]
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| |[6]
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| |}
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| ==Coordinates ==
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| {| class=wikitable width=300 align=right style="margin-left:1em"
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| |[[File:Truncated_octahedron_in_unit_cube.png|150px]]
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| |[[File:Triangulated truncated octahedron.png|150px]]
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| |- valign=top
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| |[[Orthogonal projection]] in [[bounding box]]<BR>(±2,±2,±2)
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| |Truncated octahedron with hexagons replaced by 6 coplanar triangles. There are 6 new vertices at: (±1,±1,±1).
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| |}
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| All [[permutation]]s of (0, ±1, ±2) are [[Cartesian coordinates]] of the [[vertex (geometry)|vertices]] of a [[Truncation (geometry)|truncated]] [[octahedron]] of edge length a = √ 2 centered at the origin. The vertices are thus also the corners of 12 rectangles whose long edges are parallel to the coordinate axes.
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| The edge vectors have Cartesian coordinates {{math|(0,± 1,±1)}} and permutations of these. The face normals (normalized cross products of edges that share a common vertex) of the 6 square faces are {{math|(0,0,±1)}}, {{math|(0,±1,0)}} and {{math|(±1,0,0)}}. The face normals of the 8 hexagonal faces are {{math|(± 1/√ 3, ± 1/√ 3, ± 1/√3)}}. The dot product between pairs of two face normals is the cosine of the dihedral angle between adjacent faces, either {{math|-1/3}} or {{math|-1/√3}}. The dihedral angle is approximately 1.910633 rad (109.471 ° {{OEIS2C|A156546}}) at edges shared by two hexagons or 2.186276 rad (125.263 ° {{OEIS2C|A195698}}) at edges shared by a hexagon and a square.
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| === Permutohedron===
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| 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 {{nowrap|''x'' + ''y'' + ''z'' + ''w'' {{=}} 10}}. Therefore, the truncated octahedron is the [[permutohedron]] of order 4.
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| [[Image:Permutohedron.svg|300px]]
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| ==Area and volume==
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| The area ''A'' and the [[volume]] ''V'' of a truncated octahedron of edge length ''a'' are:
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| :<math>A = \left(6+12\sqrt{3}\right) a^2 \approx 26.7846097a^2</math>
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| :<math>V = 8\sqrt{2} a^3 \approx 11.3137085a^3.</math>
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| ==Uniform colorings==
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| There are two [[uniform coloring]]s, with [[tetrahedral symmetry]] and [[octahedral symmetry]]:
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| {|class="wikitable"
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| |- valign=top
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| ![[Octahedral symmetry]]
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| ![[Tetrahedral symmetry]]<br>([[Omnitruncated tetrahedron]])
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| |-
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| |[[Image:Truncated Octahedron 122 Colouring.svg|160px]]<br>122 coloring<br>[[Wythoff symbol|Wythoff]]: 2 4 | 3
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| |[[Image:Truncated Octahedron 123 Colouring.svg|160px]]<br>123 coloring<br>Wythoff: 3 3 2 |
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| |}
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| ==Related polyhedra==
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| The truncated octahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.
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| {{Octahedral truncations}}
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| It also exists as the omnitruncate of the tetrahedron family:
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| {{Tetrahedron family}}
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| {{Omnitruncated table}}
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| This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and [[Coxeter-Dynkin diagram]] {{CDD|node_1|p|node_1|3|node_1}}. For ''p'' < 6, the members of the sequence are [[Omnitruncation (geometry)|omnitruncated]] polyhedra ([[zonohedron|zonohedra]]), shown below as spherical tilings. For ''p'' > 6, they are tilings of the hyperbolic plane, starting with the [[truncated triheptagonal tiling]].
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| The truncated octahedron is topologically related as a part of sequence of uniform polyhedra and tilings with [[vertex figures]] n.6.6, extending into the hyperbolic plane:
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| {{Truncated figure2 table}}
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| The truncated octahedron is topologically related as a part of sequence of uniform polyhedra and tilings with [[vertex figures]] 4.2n.2n, extending into the hyperbolic plane:
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| {{Truncated figure3 table}}
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| == Related polytopes ==
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| The ''[[Truncation (geometry)|truncated]] [[octahedron]]'' ([[bitruncation|bitruncated]] cube), is first in a sequence of bitruncated [[hypercube]]s:
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| {{Bitruncated hypercube polytopes}}
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| ==Tessellations==
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| The truncated octahedron exists in three different [[convex uniform honeycomb]]s ([[Honeycomb (geometry)|space-filling tessellations]]):
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| {|class="wikitable" style="text-align:center"
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| ![[Bitruncated cubic honeycomb|Bitruncated cubic]]
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| ![[Cantitruncated cubic honeycomb|Cantitruncated cubic]]
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| ![[Truncated alternated cubic honeycomb|Truncated alternated cubic]]
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| |-
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| |align="center"|[[Image:Bitruncated Cubic Honeycomb.svg|150px]]
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| |align="center"|[[Image:Cantitruncated Cubic Honeycomb.svg|150px]]
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| |align="center"|[[Image:Truncated Alternated Cubic Honeycomb.svg|150px]]
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| |}
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| The [[cell-transitive]] [[bitruncated cubic honeycomb]] can also be seen as the [[Voronoi tessellation]] of the [[Crystal structure|body-centered cubic lattice]]. The truncated octahedron is one of five three-dimensional primary [[Parallelohedron#Zonohedra that tile space|parallelohedra]].
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| ==References==
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| *{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
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| *{{cite web
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| |author=Freitas, Robert A., Jr
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| |title=Uniform space-filling using only truncated octahedra
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| |publisher=Figure 5.5 of [http://www.nanomedicine.com/NMI.htm Nanomedicine, Volume I: Basic Capabilities], Landes Bioscience, Georgetown, TX, 1999
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| |url=http://www.nanomedicine.com/NMI/Figures/5.5.jpg
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| |accessdate=2006-09-08}}
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| *{{cite journal
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| |author=Gaiha, P., and Guha, S.K.
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| |year=1977
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| |title=Adjacent vertices on a permutohedron
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| |journal=SIAM Journal on Applied Mathematics
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| |volume=32
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| |issue=2
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| |pages=323–327
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| |doi=10.1137/0132025}}
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| *{{cite web
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| |author=Hart, George W
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| |title=VRML model of truncated octahedron
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| |publisher=[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Polyhedra: The Encyclopedia of Polyhedra]
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| |url=http://www.georgehart.com/virtual-polyhedra/vrml/truncated_octahedron.wrl
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| |accessdate=2006-09-08}}
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| *{{cite web
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| |author=Mäder, Roman
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| |title=The Uniform Polyhedra: Truncated Octahedron
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| |url=http://www.mathconsult.ch/showroom/unipoly/08.html
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| |accessdate=2006-09-08}}
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| *{{cite book
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| |author=Alexandrov, A.D.
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| |year=1958
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| |title=Convex polyhedra
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| |location=Berlin
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| |publisher=Springer
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| |pages=539
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| |isbn=3-540-23158-7
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| }}
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| *{{cite book
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| |author=Cromwell, P.
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| |year=1997
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| |title=Polyhedra
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| |location=United Kingdom
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| |publisher=Cambridge
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| |pages=79-86 ''Archimedean solids''
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| |isbn=0-521-55432-2
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| }}
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| ==External links==
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| {{Commons category|Truncated octahedron}}
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| *{{mathworld2 |urlname=TruncatedOctahedron |title=Truncated octahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
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| *{{mathworld |urlname=Permutohedron |title=Permutohedron}}
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| *{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x4o - toe}}
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| *[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=3UtM7vifCnAn5PXmSbX99Eu3LJZAs0nAWn3JyT7et98rnPxTGYml4FXjuQ2tE4viYN0KMgAstBRd0otTWLThQWl9BNNC4uigRoZQUQOQibYqtCuLQw9Ui3OofXtQPEsqQ7#applet Editable printable net of a truncated octahedron with interactive 3D view]
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| {{Archimedean solids}}
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| {{Polyhedron navigator}}
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| [[Category:Uniform polyhedra]]
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| [[Category:Archimedean solids]]
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| [[Category:Space-filling polyhedra]]
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| [[Category:Zonohedra]]
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