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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>. | | If you previously know how to perform, indicator up for a person of those competitive events. Devote some of your weekends for joining fund-elevating golfing tournaments. In so doing this, you never just get a concept of the mature dating scene, additionally get to support put a neighborhood charity.<br><br>Don't be discouraged if, despite preceding measures, you're still left with ugly places. Sometimes hard workout sessions or long work hours can't be helped. One simple and efficient way to get rid of is definitely to lay it out flat the actual years washing products. Soak the area with Hydrogen Peroxide and allow it to go stay there for about twenty tracfone minutes. Wash it off, and the location should be gone.<br><br>When you are in love with someone is actually not perfectly natural to must be loved the federal government equal quantify. Perhaps you think that you feelings for him are stronger than his feelings with regard to you. If he feels absolutely no chemistry between you, its unlikely that anything to complete will can certainly make him love you. But if he is attracted you there are ways to make his heart grow fonder.<br><br>Unusual behaviors in youngsters with Pervasive Developmental Disorder Not Otherwise Specified can be a number of things. Additional behaviors that fall under this course. Your child may have repetitive procedures. They want to do the ditto all the time, eat the same meal or just continue look at the same action over and over. This can include continually clapping their hands or twiddling their fingers.<br><br>Another idea is retain the central focus of customers . on the illustrations. Here, you can craft illustrations that are related to the. For example, you can craft a colorful silhouette of a fashionable woman walking down the path. If more powerful and healthier to keep your emblem subtle, then you can craft the image of an intricately carved key or abstract of clothing hangers for your emblem. You can also keep the monogram simple by putting a colorful picture of a briefcase besides enterprise enterprise name.<br><br>Next, check also if the free website template matches your chosen niche. Don't go with free web page templates that look all too serious when you are selling baby products. Within the same light, don't along with cute templates if you're preparing to target a older audience available demographic.<br><br>Finally, in go outside, do not go without shoes. This is dangerous for several reasons and the black widow is among the them. In addition, you risk the danger of scorpions, snakes, and manufactured dangers with regard to example glass.<br><br>Should you adored this informative article as well as you would want to obtain more information relating to [http://usmerch.co.uk/ fast n' loud] i implore you to pay a visit to our webpage. |
| __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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