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| {{Semireg polyhedra db|Semireg polyhedron stat table|lrCO}}
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
| In [[geometry]], the '''rhombicuboctahedron''', or '''small rhombicuboctahedron''', is an [[Archimedean solid]] with eight [[triangular]] and eighteen [[square (geometry)|square]] faces. There are 24 identical vertices, with one triangle and three squares meeting at each. (Note that six of the squares only share vertices with the triangles while the other twelve share an edge.) The [[polyhedron]] has [[octahedral symmetry]], like the [[Cube (geometry)|cube]] and [[octahedron]]. Its [[dual polyhedron|dual]] is called the [[deltoidal icositetrahedron]] or trapezoidal icositetrahedron, although its faces are not really true [[trapezoid]]s.
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| The name ''rhombicuboctahedron'' refers to the fact that twelve of the square faces lie in the same planes as the twelve faces of the [[rhombic dodecahedron]] which is dual to the [[cuboctahedron]]. ''Great rhombicuboctahedron'' is an alternative name for a [[truncated cuboctahedron]], whose faces are parallel to those of the (small) rhombicuboctahedron.
| | If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]] |
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| It can also be called an ''[[Expansion (geometry)|expanded]] cube'' or ''[[Cantellation (geometry)|cantellated]] cube'' or a ''cantellated octahedron'' from truncation operations of the [[uniform polyhedron]].
| | Registered users will be able to choose between the following three rendering modes: |
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| If the original rhombicuboctahedron has unit edge length, its dual [[strombic icositetrahedron]] has edge lengths
| | '''MathML''' |
| :<math>\frac{2}{7}\sqrt{10-\sqrt{2}}</math> and <math>\sqrt{4-2\sqrt{2}}.\ </math> | | :<math forcemathmode="mathml">E=mc^2</math> |
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| ==Area and volume==
| | <!--'''PNG''' (currently default in production) |
| The area ''A'' and the volume ''V'' of the rhombicuboctahedron of edge length ''a'' are:
| | :<math forcemathmode="png">E=mc^2</math> |
| :<math>A = (18+2\sqrt{3})a^2 \approx 21.4641016a^2</math>
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| :<math>V = \frac{1}{3} (12+10\sqrt{2})a^3 \approx 8.71404521a^3.</math> | |
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| ==Orthogonal projections==
| | '''source''' |
| The ''rhombicuboctahedron'' has six special [[orthogonal projection]]s, centered, on a vertex, on two types of edges, and three types of faces: triangles, and two squares. The last two correspond to the B<sub>2</sub> and A<sub>2</sub> [[Coxeter plane]]s.
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| |+ Orthogonal projections
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| !Centered by
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| !Vertex
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| !Edge<br>3-4
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| !Edge<br>4-4
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| !Face<br>Square-1
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| !Face<br>Square-2
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| !Face<br>Triangle
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| {| class=wikitable
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
| |+ [[Orthographic projection]]s
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| !Image
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| |[[File:Cube t02 v.png|100px]]
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| |[[File:Cube t02 e34.png|100px]]
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| |[[File:Cube t02 e44.png|100px]]
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| |[[File:3-cube t02.svg|100px]]
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| !Projective<BR>symmetry
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| |[4]
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| |[6]
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| == Cartesian coordinates == | | ==Demos== |
| [[Cartesian coordinates]] for the vertices of a rhombicuboctahedron centred at the origin, with edge length 2 units, are all permutations of
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| :<math>(\pm1, \pm1, \pm(1+\sqrt{2})).\ </math>
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| ==Geometric relations==
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
| [[Image:Exploded rhombicuboctahedron.png|160px|left|thumb|Rhombicuboctahedron dissected into two [[square cupola]]e and a central [[octagonal prism]]. A rotation of one cupola creates the ''pseudo­rhombi­cubocta­hedron''. Both of these polyhedra have the same vertex figure: 3.4.4.4'']]
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| There are three pairs of parallel planes that each intersect the rhombicuboctahedron in a regular octagon. The rhombicuboctahedron may be divided along any of these to obtain an octagonal prism with regular faces and two additional polyhedra called square [[cupola (geometry)|cupolae]], which count among the [[Johnson solid]]s; it is thus an ''elongated square ortho[[bicupola]]''. These pieces can be reassembled to give a new solid called the [[elongated square gyrobicupola]] or ''pseudorhombicuboctahedron'', with the symmetry of a square antiprism. In this the vertices are all locally the same as those of a rhombicuboctahedron, with one triangle and three squares meeting at each, but are not all identical with respect to the entire polyhedron, since some are closer to the symmetry axis than others.
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| {|class="wikitable" align="left" style="text-align:center;"
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| |[[Image:Small rhombicuboctahedron.png|75px]]
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| |Rhombicuboctahedron
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| |[[Image:Pseudorhombicuboctahedron.png|75px]]
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| |Pseudorhombicuboctahedron
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| |}
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| [[File:P2-A5-P3.gif||thumb|The rhombicuboctahedron can be seen as an [[Expansion (geometry)|expanded]] cube.]]
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| There are distortions of the rhombicuboctahedron that, while some of the faces are not regular polygons, are still vertex-uniform. Some of these can be made by taking a cube or octahedron and cutting off the edges, then trimming the corners, so the resulting polyhedron has six square and twelve rectangular faces. These have octahedral symmetry and form a continuous series between the cube and the octahedron, analogous to the distortions of the [[rhombicosidodecahedron]] or the tetrahedral distortions of the [[cuboctahedron]]. However, the rhombicuboctahedron also has a second set of distortions with six rectangular and sixteen trapezoidal faces, which do not have octahedral symmetry but rather T<sub>h</sub> symmetry, so they are invariant under the same rotations as the [[tetrahedron]] but different reflections.
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| The lines along which a [[Rubik's Cube]] can be turned are, projected onto a sphere, similar, [[topologically]] identical, to a rhombicuboctahedron's edges. In fact, variants using the Rubik's Cube mechanism have been produced which closely resemble the rhombicuboctahedron.
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| The rhombicuboctahedron is used in three [[Honeycomb (geometry)|uniform space-filling tessellations]]: the [[cantellated cubic honeycomb]], the [[runcitruncated cubic honeycomb]], and the [[runcinated alternated cubic honeycomb]].
| | * accessibility: |
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| == Related polyhedra== | | ==Test pages == |
| The rhombicuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.
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| {{Octahedral truncations}}
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| | *[[Displaystyle]] |
| | *[[MathAxisAlignment]] |
| | *[[Styling]] |
| | *[[Linebreaking]] |
| | *[[Unique Ids]] |
| | *[[Help:Formula]] |
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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]].
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| {{Expanded table}}
| | ==Bug reporting== |
| | | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| === Vertex arrangement===
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| It shares its vertex arrangement with three [[nonconvex uniform polyhedra]]: the [[stellated truncated hexahedron]], the [[small rhombihexahedron]] (having the triangular faces and six square faces in common), and the [[small cubicuboctahedron]] (having twelve square faces in common).
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| {|class="wikitable" width="400" style="vertical-align:top;text-align:center"
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| |[[Image:Small rhombicuboctahedron.png|100px]]<br>Rhombicuboctahedron
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| |[[Image:Small cubicuboctahedron.png|100px]]<br>[[Small cubicuboctahedron]]
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| |[[Image:Small rhombihexahedron.png|100px]]<br>[[Small rhombihexahedron]]
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| |[[Image:Stellated truncated hexahedron.png|100px]]<br>[[Stellated truncated hexahedron]]
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| |}
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| ==In the arts== | |
| [[Image:Pacioli.jpg|thumb|300px|Rhombicuboctahedron in top left of ''[[Portrait of Luca Pacioli]]''.<ref>[http://www.ritrattopacioli.it/texting.htm RitrattoPacioli.it]</ref> ]]
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| [[Image:Leonardo polyhedra.png|thumb|[[Leonardo da Vinci]]'s rhombicuboctahedron]]
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| The large polyhedron in the 1495 portrait of [[Luca Pacioli]], traditionally though controversially attributed to [[Jacopo de' Barbari]] is a glass rhombicuboctahedron half-filled with water.
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| The first printed version of the rhombicuboctahedron was by [[Leonardo da Vinci]] and appeared in his ''Divina Proportione''.
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| A spherical 180×360° panorama can be projected onto any polyhedron; but the rhombicuboctahedron provides a good enough approximation of a sphere while being easy to build. This type of projection, called ''Philosphere'', is possible from some panorama assembly software. It consists of two images that are printed separately and cut with scissors while leaving some flaps for assembly with glue.<ref>[http://www.philohome.com/rhombicuboctahedron/rhombicuboctahedron.htm Philosphere]</ref>
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| ==Games and toys== | |
| [[Image:Rubiksnake ball.png|thumb|Snake in a [[ball]] solution: nonuniform concave rhombicuboctahedron.]]
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| The [[Freescape]] games ''[[Driller (video game)|Driller]]'' and ''[[Dark Side (video game)|Dark Side]]'' both had a game map in the form of a rhombicuboctahedron.
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| A level in the videogame ''[[Super Mario Galaxy]]'' has a planet in the shape of a rhombicuboctahedron.
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| During the [[Rubik's Cube]] craze of the 1980s, one combinatorial puzzle sold had the form of a rhombicuboctahedron (the mechanism was of course that of a [[Rubik's Cube]]).
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| The [[Rubik's Snake]] toy was usually sold in the shape of a stretched rhombicuboctahedron (12 of the squares being replaced with 1:√2 rectangles).
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| ==See also==
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| <div style="-moz-column-count:2; column-count:2;">
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| *[[Compound of five rhombicuboctahedra]]
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| *[[Cube (geometry)|Cube]]
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| *[[Cuboctahedron]]
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| *[[Elongated square gyrobicupola]]
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| *[[Moravian star]]
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| *[[Octahedron]]
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| *[[Rhombicosidodecahedron]]
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| *[[Rubik's Snake]] – puzzle that can form a Rhombicuboctahedron "ball"
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| *[[The National Library of Belarus]] – its architectural main component has the shape of a rhombicuboctahedron.
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| *[[Truncated cuboctahedron]] (great rhombicuboctahedron)
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| *[[Portrait of Luca Pacioli]]
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| *[[Nonconvex great rhombicuboctahedron]]
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| </div>
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| *{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
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| *{{cite journal |last=Coxeter |first=H.S.M. |authorlink=Harold Scott MacDonald Coxeter |title=Uniform Polyhedra |journal=Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences |volume=246 |date=May 13, 1954 |pages=401–450 |doi=10.1098/rsta.1954.0003 |issue=916 |last2=Longuet-Higgins |first2=M.S. |last3=Miller |first3=J.C.P.}}
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| ==External links==
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| *{{mathworld2 |urlname=SmallRhombicuboctahedron |title=Rhombicuboctahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
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| *{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3o4x - sirco}}
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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.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=D9tM7OPpTeb5fdRIwHgyGnGQuG8Er0hjmb3YtRZaHYQAAns6eVm2BFtWBcy9fTY8oFG11608c05WEHBOIwUgR9mWjE9aHectfR7dboSNgfU8YAeliUFlMaH63asg2zGgf6foy68Hxg4VxIsKuxmBG9yNlpLAO1A9euZZPJ7&name=Rhombicuboctahedron#applet Editable printable net of a rhombicuboctahedron with interactive 3D view]
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| *''[http://demonstrations.wolfram.com/RhombicuboctahedronStar/ Rhombicuboctahedron Star]'' by Sándor Kabai, [[Wolfram Demonstrations Project]].
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| *[http://www.hbmeyer.de/flechten/rhku/indexeng.htm 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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| [[ca:Petit rombicuboctàedre]]
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| [[cv:Ромбокубоктаэдр]]
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| [[de:Rhombenkuboktaeder]]
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| [[el:Ρομβοκυβοκτάεδρο]]
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| [[es:Rombicuboctaedro]]
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| [[eo:Rombokub-okedro]]
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| [[fr:Petit rhombicuboctaèdre]]
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| [[ko:부풀린 육팔면체]]
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| [[it:Rombicubottaedro]]
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| [[nl:Romboëdrisch kuboctaëder]]
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| [[ja:斜方立方八面体]]
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| [[no:Rombkuboktaeder]]
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| [[pl:Sześcio-ośmiościan rombowy mały]]
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| [[pt:Rombicuboctaedro]]
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| [[ru:Ромбокубоктаэдр]]
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| [[th:รอมบิคิวบอกทาฮีดรอน]]
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| [[zh:小斜方截半立方体]]
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