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| {{More footnotes|date=September 2013}}
| | Hi there. Allow me start by introducing the writer, her name is Sophia. Alaska is exactly where he's always been residing. For years she's been operating as a travel agent. To perform lacross is the factor I love most of all.<br><br>Here is my website: online psychic ([http://www.stanfest.com/media/profile.php?u=ShLutz linked internet site]) |
| {{Semireg polyhedra db|Semireg polyhedron stat table|CO}}
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| In [[geometry]], a '''cuboctahedron''' is a [[polyhedron]] with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a [[quasiregular polyhedron]], i.e. an [[Archimedean solid]], being [[vertex-transitive]] and [[edge-transitive]].
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| Its [[dual polyhedron]] is the [[rhombic dodecahedron]].
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| ==Other names==
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| *''Heptaparallelohedron'' ([[Buckminster Fuller]])
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| **Fuller applied the name "[[Dymaxion]]" to this shape, used in an early version of the [[Dymaxion map]].
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| *With O<sub>h</sub> symmetry, it is a ''[[Rectification (geometry)|rectified]] [[cube]]'' or ''rectified octahedron'' ([[Norman Johnson (mathematician)|Norman Johnson]])
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| *With T<sub>d</sub> symmetry, it is a ''[[Cantellation (geometry)|cantellated]] [[tetrahedron]]''.
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| *With D<sub>3d</sub> symmetry, it is a ''triangular [[gyrobicupola]]''.
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| ==Area and volume==
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| The area ''A'' and the volume ''V'' of the cuboctahedron of edge length ''a'' are:
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| :<math>A = \left(6+2\sqrt{3}\right)a^2 \approx 9.4641016a^2</math>
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| :<math>V = \frac{5}{3} \sqrt{2}a^3 \approx 2.3570226a^3.</math>
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| ==Orthogonal projections==
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| The ''cuboctahedron'' has four special [[orthogonal projection]]s, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B<sub>2</sub> and A<sub>2</sub> [[Coxeter plane]]s. The skew projections show a square and hexagon passing through the center of the cuboctahedron.
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| {|class=wikitable
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| |+ Orthogonal projections
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| |-
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| !Centered by
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| !Square<br>Face
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| !Triangular<br>Face
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| |- align=center
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| !Face plane<BR>projections
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| |[[File:3-cube t1 B2.svg|100px]]
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| |[[File:3-cube t1.svg|100px]]
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| |- align=center
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| !Projective<BR>symmetry
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| |[4]
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| |[6]
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| |- align=center
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| !Skew<BR>projections
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| |[[File:Cuboctahedron B2 planes.png|90px]]
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| |[[File:Cuboctahedron 3 planes.png|100px]]
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| |- align=center
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| !Vertex/edge<BR>projections
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| ||[[File:Cube t1 v.png|100px]]
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| |[[File:Cube t1 e.png|100px]]
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| |}
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| ==Cartesian coordinates==
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| The [[Cartesian coordinates]] for the vertices of a cuboctahedron (of edge length √2) centered at the origin are:
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| :(±1,±1,0)
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| :(±1,0,±1)
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| :(0,±1,±1)
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| An alternate set of coordinates can be made in 4-space, as 12 permutations of:
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| :(0,1,1,2)
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| This construction exists as one of 16 [[orthant]] [[Facet (geometry)|facets]] of the [[cantellated 16-cell]].
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| ===Root vectors===
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| The cuboctahedron's 12 vertices can represent the root vectors of the [[simple Lie group]] A<sub>3</sub>. With the addition of 6 vertices of the [[octahedron]], these vertices represent the 18 root vectors of the [[simple Lie group]] B<sub>3</sub>.
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| ==Geometric relations==
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| A cuboctahedron can be obtained by taking an appropriate [[cross section (geometry)|cross section]] of a four-dimensional [[16-cell]].
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| A cuboctahedron has octahedral symmetry. Its first [[stellation]] is the [[polyhedral compound|compound]] of a [[cube (geometry)|cube]] and its dual [[octahedron]], with the vertices of the cuboctahedron located at the midpoints of the edges of either.
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| The cuboctahedron is a [[Rectification (geometry)|rectified]] [[cube]] and also a rectified [[octahedron]].
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| It is also a [[Cantellation (geometry)|cantellated]] [[tetrahedron]]. With this construction it is given the [[Wythoff symbol]]: {{nowrap|3 3 {{!}} 2}}. [[Image:Cantellated tetrahedron.png|50px]]
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| A skew cantellation of the tetrahedron produces a solid with faces parallel to those of the cuboctahedron, namely eight triangles of two sizes, and six rectangles. While its edges are unequal, this solid remains ''vertex-uniform'': the solid has the full tetrahedral [[symmetry group]] and its vertices are equivalent under that group.
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| The edges of a cuboctahedron form four regular [[hexagon]]s. If the cuboctahedron is cut in the plane of one of these hexagons, each half is a [[triangular cupola]], one of the [[Johnson solid]]s; the cuboctahedron itself thus can also be called a triangular [[bicupola (geometry)|gyrobicupola]], the simplest of a series (other than the [[gyrobifastigium]] or "digonal gyrobicupola"). If the halves are put back together with a twist, so that triangles meet triangles and squares meet squares, the result is another Johnson solid, the [[triangular orthobicupola]], also called an anticuboctahedron.
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| Both triangular bicupolae are important in [[sphere packing]]. The distance from the solid's center to its vertices is equal to its edge length. Each central [[sphere]] can have up to twelve neighbors, and in a face-centered cubic lattice these take the positions of a cuboctahedron's vertices. In a [[hexagon]]al close-packed lattice they correspond to the corners of the triangular orthobicupola. In both cases the central sphere takes the position of the solid's center.
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| Cuboctahedra appear as cells in three of the [[convex uniform honeycomb]]s and in nine of the convex [[uniform polychoron|uniform polychora]].
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| The volume of the cuboctahedron is 5/6 of that of the enclosing cube and 5/8 of that of the enclosing octahedron.
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| === Vertex arrangement ===
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| The cuboctahedron shares its edges and vertex arrangement with two [[nonconvex uniform polyhedron|nonconvex uniform polyhedra]]: the [[cubohemioctahedron]] (having the square faces in common) and the [[octahemioctahedron]] (having the triangular faces in common). It also serves as a cantellated [[tetrahedron]], as being a rectified [[tetratetrahedron]].
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| {| class="wikitable" width="400" style="vertical-align:top;text-align:center"
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| |[[Image:cuboctahedron.png|100px]]<br>Cuboctahedron
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| |[[Image:cubohemioctahedron.png|100px]]<br>[[Cubohemioctahedron]]
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| |[[Image:octahemioctahedron.png|100px]]<br>[[Octahemioctahedron]]
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| |}
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| The cuboctahedron [[covering space|2-covers]] the [[tetrahemihexahedron]],<ref name="richter">{{Harv|Richter}}</ref> which accordingly has the same [[abstract polytope|abstract]] [[vertex figure]] (two triangles and two squares: 3.4.3.4) and half the vertices, edges, and faces. (The actual vertex figure of the tetrahemihexahedron is 3.4.3/2.4, with the ''a''/2 factor due to the cross.)
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| {|class="wikitable" width="400" style="vertical-align:top;text-align:center"
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| |[[Image:cuboctahedron.png|100px]]<br>Cuboctahedron
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| |[[Image:tetrahemihexahedron.png|100px]]<br>[[Tetrahemihexahedron]]
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| |}
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| ==Related polyhedra==
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| The cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.
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| {{Tetrahedron family}}
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| {{Octahedral truncations}}
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| The cuboctahedron can be seen in a sequence of [[quasiregular polyhedron]]s and tilings:
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| {{Quasiregular figure table}}
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| {{Quasiregular4 table}}
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| This polyhedron is topologically related as a part of sequence of [[Cantellation (geometry)|cantellated]] polyhedra with vertex figure (3.4.n.4), and continues as tilings of the [[Hyperbolic space|hyperbolic plane]]. These [[vertex-transitive]] figures have (*n32) reflectional [[Orbifold notation|symmetry]].
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| {{Expanded table}}
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| ==Related polytopes==
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| The cuboctahedron can be decomposed into a regular [[octahedron]] and eight irregular but equal octahedra in the shape of the convex hull of a cube with two opposite vertices removed. This decomposition of the cuboctahedron corresponds with the cell-first parallel projection of the [[24-cell]] into three dimensions. Under this projection, the cuboctahedron forms the projection envelope, which can be decomposed into six square faces, a regular octahedron, and eight irregular octahedra. These elements correspond with the images of six of the octahedral cells in the 24-cell, the nearest and farthest cells from the 4D viewpoint, and the remaining eight pairs of cells, respectively.
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| ==Cultural occurrences==
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| [[File:Two cuboctahedrons 2.jpg|200px|thumbnail|Two cuboctahedra on a chimney. [[Israel]].]]
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| *In the ''[[Star Trek]]'' episode "[[By Any Other Name]]", aliens seize the [[Starship Enterprise|Enterprise]] by transforming crew members into inanimate cuboctahedra.
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| *The "Geo Twister" fidget toy [http://www.trainerswarehouse.com/prodinfo.asp?number=FIGEO] is a flexible cuboctahedron.
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| ==See also==
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| *[[Icosidodecahedron]]
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| *[[Rhombicuboctahedron]]
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| *[[Truncated cuboctahedron]]
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| *{{cite book|last=Ghyka|first=Matila|title=The geometry of art and life.|year=1977|publisher=Dover Publications|location=New York|isbn=9780486235424|pages=51–56, 81–84|edition=[Nachdr.]}}
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| *{{citation |ref={{harvid|Richter}} |first=David A. |last=Richter |url=http://homepages.wmich.edu/~drichter/rptwo.htm |title=Two Models of the Real Projective Plane}}
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| *{{cite encyclopedia|title=Cuboctahedron|last=Weisstein|first=Eric W.|encyclopedia=CRC Concise Encyclopedia of Mathematics.|year=2002|publisher=CRC Press|location=Hoboken|isbn=9781420035223|pages=620–621|edition=2nd}}
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| *{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
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| * Cromwell, P. ''Polyhedra'', CUP hbk (1997), pbk. (1999). Ch.2 p.79-86 ''Archimedean solids''
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| {{refend}}
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| ==External links==
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| *[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
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| *[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
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| *{{mathworld2 |urlname=Cuboctahedron |title=Cuboctahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
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| *{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|o3x4o - co}}
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| *[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=dYI7PStK037OedkFHPiwHK2fbnxjxylGZCjWfNh0UwMgy82zEaWFzVL3PBfYB9SDz8RMvhNkpb8sS9R&name=Cuboctahedron#applet Editable printable net of a Cuboctahedron with interactive 3D view]
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| {{Archimedean solids}}
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| {{Polyhedron navigator}}
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| [[Category:Archimedean solids]]
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| [[Category:Quasiregular polyhedra]]
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Hi there. Allow me start by introducing the writer, her name is Sophia. Alaska is exactly where he's always been residing. For years she's been operating as a travel agent. To perform lacross is the factor I love most of all.
Here is my website: online psychic (linked internet site)