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| {{Semireg polyhedra db|Semireg polyhedron stat table|grCO}}
| | Lilla Spafford is what's written on my little birth certificate although it's not at all the name on my birth card. He works for a bookkeeper. To play basketball is there isn't a he loves most. Puerto Rico is where my property is. Go to my website to recognise more: http://usmerch.co.uk/accessories/<br><br>Feel free to surf to my web-site ... [http://usmerch.co.uk/accessories/ fast n' loud] |
| In [[geometry]], the '''truncated cuboctahedron''' is an [[Archimedean solid]]. It has 12 [[Square (geometry)|square]] faces, 8 regular [[hexagon]]al faces, 6 regular [[octagon]]al faces, 48 vertices and 72 edges. Since each of its faces has [[point symmetry]] (equivalently, 180° [[rotation]]al symmetry), the truncated cuboctahedron is a [[zonohedron]].
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| ==Cartesian coordinates==
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| The [[Cartesian coordinates]] for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all [[permutation]]s of:
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| :(±1, ±(1+√2), ±(1+2√2))
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| ==Area and volume==
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| The area ''A'' and the volume ''V'' of the truncated cuboctahedron of edge length ''a'' are:
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| :<math>A = 12\left(2+\sqrt{2}+\sqrt{3}\right) a^2 \approx 61.7551724a^2</math>
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| :<math>V = \left(22+14\sqrt{2}\right) a^3 \approx 41.7989899a^3.</math>
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| ==Vertices==
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| To derive the number of vertices, we note that each vertex is the meeting point of a square, hexagon, and octagon. | |
| *Each of the 12 squares with their 4 vertices contribute 48 vertices because 12 × 4 = 48.
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| *Each of the 8 hexagons with their 6 vertices contribute 48 vertices because 8 × 6 = 48.
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| *Each of the 6 octagons with their 8 vertices contribute 48 vertices because 6 × 8 = 48.
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| Therefore, there may seem to exist {{nowrap|48 + 48 + 48 {{=}} 144}} vertices. However, we have over-counted the vertices thrice since a square, hexagon, and octagon meet at each vertex. Consequently, we divide 144 by 3 to correct for our over-counting: {{nowrap|144 / 3 {{=}} 48.}}
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| ==Dual==
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| If the original truncated cuboctahedron has edge length 1, its dual [[disdyakis dodecahedron]] has edge lengths <math>\tfrac{2}{7}\scriptstyle{\sqrt{30-3\sqrt{2}}}</math>, <math>\tfrac{3}{7}\scriptstyle{\sqrt{6(2+\sqrt{2})}}</math> and <math>\tfrac{2}{7}\scriptstyle{\sqrt{6(10+\sqrt{2})}}</math>.
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| ==Uniform colorings==
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| There is only one [[uniform coloring]] of the faces of this polyhedron, one color for each face type.
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| A 2-uniform coloring also exists with alternately colored hexagons.
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| ==Other names==
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| Alternate interchangeable names are:
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| *''Truncated cuboctahedron'' ([[Johannes Kepler]])
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| *''Rhombitruncated cuboctahedron'' ([[Magnus Wenninger]]<ref>{{Citation |last1=Wenninger |first1=Magnus |author1-link=Magnus Wenninger |title=Polyhedron Models |publisher=[[Cambridge University Press]] |isbn=978-0-521-09859-5 |id={{MathSciNet|id=0467493}} |year=1974}} (Model 15, p. 29)</ref>)
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| *''Great rhombicuboctahedron'' (Robert Williams<ref>{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9, p. 82)</ref>)
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| *''Great rhombcuboctahedron'' (Peter Cromwell<ref>Cromwell, P.; [http://books.google.com/books?id=OJowej1QWpoC&lpg=PP1&pg=PA82#v=onepage&q=&f=false ''Polyhedra''], CUP hbk (1997), pbk. (1999). (p. 82) </ref>)
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| *''Omnitruncated cube'' or ''cantitruncated cube'' ([[Norman Johnson (mathematician)|Norman Johnson]])
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| The name ''truncated cuboctahedron'', given originally by [[Johannes Kepler]], is a little misleading. If you [[truncation (geometry)|truncate]] a [[cuboctahedron]] by cutting the corners off, you do ''not'' get this uniform figure: some of the faces will be [[rectangle]]s. However, the resulting figure is [[topologically]] equivalent to a truncated cuboctahedron and can always be deformed until the faces are regular.
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| The alternative name ''great rhombicuboctahedron'' refers to the fact that the 12 square faces lie in the same planes as the 12 faces of the [[rhombic dodecahedron]] which is dual to the cuboctahedron. Compare to [[small rhombicuboctahedron]].
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| One unfortunate point of confusion: There is a [[nonconvex uniform polyhedron]] by the same name. See [[nonconvex great rhombicuboctahedron]].
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| ==Orthogonal projections==
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| The truncated cuboctahedron has two special [[orthogonal projection]]s in the A<sub>2</sub> and B<sub>2</sub> [[Coxeter plane]]s with [6] and [8] projective symmetry, and numerous [2] symmtries can be constructed from various projected planes relative to the polyhedron elements.
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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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| !Vertex
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| !Edge<br>4-6
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| !Edge<br>4-8
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| !Edge<br>6-8
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| !Face normal<BR>4-6
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| |-
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| !Image
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| |[[File:Cube_t012_v.png|100px]]
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| |[[File:Cube_t012_e46.png|100px]]
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| |[[File:Cube_t012_e48.png|100px]]
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| |[[File:Cube_t012_e68.png|100px]]
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| |[[File:Cube_t012_f46.png|100px]]
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| |- align=center
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| !Projective<br>symmetry
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| |[2]<sup>+</sup>
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| |[2]
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| |[2]
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| |[2]
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| |[2]
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| |-
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| !Centered by
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| !Face normal<BR>Square
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| !Face normal<BR>Octagon
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| !Face<br>Square
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| !Face<br>Hexagon
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| !Face<br>Octagon
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| |-
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| !Image
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| |[[File:Cube_t012_af4.png|100px]]
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| |[[File:Cube_t012_af8.png|100px]]
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| |[[File:Cube_t012_f4.png|100px]]
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| |[[File:3-cube_t012.svg|100px]]
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| |[[File:3-cube_t012_B2.svg|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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| |[6]
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| |[8]
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| |}
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| == Related polyhedra==
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| The truncated cuboctahedron 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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| 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]]s), 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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| {{Omnitruncated table}}
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| {{Omnitruncated4 table}}
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| ==See also==
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| {{Commons category|Truncated cuboctahedron}}
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| *[[cube (geometry)|Cube]]
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| *[[Cuboctahedron]]
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| *[[Octahedron]]
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| *[[Truncated icosidodecahedron]]
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| *[[Truncated octahedron]] – truncated tetratetrahedron
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| ==References==
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| {{Reflist}}
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| *{{cite book|author=Cromwell, P.|year=1997|title=Polyhedra|location=United Kingdom|publisher=Cambridge|pages=79-86 ''Archimedean solids''|isbn=0-521-55432-2}}
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| ==External links==
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| *{{mathworld2 |urlname=GreatRhombicuboctahedron |title=Great rhombicuboctahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
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| *{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x4x - girco}}
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| *[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=K1RB4IgXRIYFsRCy4aHC6dnfSy7rq2q4f3v4yl4nRAbEtZul3CvUo1k8TrBjkgTqbZvaXjrwBrfylwUDWC8lfzMM1h1E0NDKFJh7udiMaOsp1DWdM1xR11UHzHVHzVTY4gCc1nRXW879CvTzcUwX9wUb4vJUXWfCEBFZvcR1dJrJyE4FpNhyWBHvw6Qyuph3rT9jgDZRKNn8FaEqBr5dZ1o9SBpnEObkMMx31INNrEf75EZOWGdEXwvDNeJIuKbtorvRJHy4iYcMHnYO1v1axRBQudr31fS0np5Jn7nuzR0Vx&name=Truncated+Cuboctahedron#applet Editable printable net of a truncated cuboctahedron with interactive 3D view]
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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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| *[http://www.faust.fr.bw.schule.de/mhb/flechten/grko/indexeng.htm great Rhombicuboctahedron: paper strips for plaiting]
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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:Zonohedra]]
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Lilla Spafford is what's written on my little birth certificate although it's not at all the name on my birth card. He works for a bookkeeper. To play basketball is there isn't a he loves most. Puerto Rico is where my property is. Go to my website to recognise more: http://usmerch.co.uk/accessories/
Feel free to surf to my web-site ... fast n' loud