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| {{Semireg polyhedra db|Semireg polyhedron stat table|lrCO}}
| | I'm Aimee and I live in a seaside city in northern Great Britain, Crumlin. I'm 24 and I'm will soon finish my study at Greek and Roman Culture.<br><br>Here is my site; [http://www.futurecol.co.nz/assets/cheap-nike.asp nike air max] |
| 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. In the book "[[De divina proportione]]", this shape was given the Latin name "Vigintisexbasium Planum Vacuum" meaning regular solid with twenty-six faces.<ref>Da divina proportione, page XXXVI</ref>
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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]].
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| ==Geometric relations==
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| [[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 either an [[Expansion (geometry)|expanded]] cube (the blue faces) or an expanded [[octahedron]] (the red faces).]]
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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]].
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| ===Orthogonal projections===
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| 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.
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| {|class=wikitable width=640
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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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| |-
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| {| class=wikitable
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| |+ [[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:Cube t02 f4b.png|100px]]
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| |[[File:3-cube t02 B2.svg|100px]]
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| |[[File:3-cube t02.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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| |[2]
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| |[4]
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| |[6]
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| |}
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| == Pyritohedral symmetry ==
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| A half symmetry form of the rhombicuboctahedron, {{CDD|node_1|3|node|4|node_1}}, exists with [[pyritohedral symmetry]], [4,3<sup>+</sup>], (3*2) as [[Coxeter diagram]] {{CDD|node_h|3|node_h|4|node_1}}, [[Schläfli symbol]] s<sub>2</sub>{3,4}, and can be called a ''cantic snub octahedron''. This form can be visualized by alternatingly coloring the edges of the 6 [[square]]s. These squares can then be distorted into [[rectangle]]s, while the 8 triangles remain equilateral. The 12 diagonal square faces will become [[isosceles trapezoid]]s. In the limit, the rectangles can be reduced to edges, and the trapezoids become triangles, and a [[icosahedron]] is formed, by a ''snub octahedron'' construction, {{CDD|node_h|3|node_h|4|node}}, s{3,4}.
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| {| class=wikitable
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| |[[File:Rhombicuboctahedron uniform edge coloring.png|160px]]<BR>Uniform geometry<BR>{{CDD|node_h|3|node_h|4|node_1}}
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| |[[File:Cantic snub octahedron.png|160px]]<BR>Nonuniform geometry
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| |[[File:Rhombicuboctahedron pyritohedral.png|160px]]<BR>Nonuniform geometry
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| |[[File:Rhombicuboctahedron pyritohedral2.png|160px]]<BR>In the limit, an [[icosahedron]]<BR>Snub octahedron, {{CDD|node_h|3|node_h|4|node}}
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| |[[File:Rhombicuboctahedron pyritohedral compound.png|160px]]<BR>[[Compound of two icosahedra]]<BR>from alternate positions
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| |}
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| ==Algebraic properties==
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| === Cartesian coordinates ===
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| [[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>\left(\pm1, \pm1, \pm(1+\sqrt{2})\right).\ </math>
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| If the original rhombicuboctahedron has unit edge length, its dual [[strombic icositetrahedron]] has edge lengths
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| :<math>\frac{2}{7}\sqrt{10-\sqrt{2}}</math> and <math>\sqrt{4-2\sqrt{2}}.\ </math>
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| ===Area and volume===
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| The area ''A'' and the volume ''V'' of the rhombicuboctahedron of edge length ''a'' are:
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| :<math>A = \left(18+2\sqrt{3}\right)a^2 \approx 21.4641016a^2</math>
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| :<math>V = \frac{1}{3} \left(12+10\sqrt{2}\right)a^3 \approx 8.71404521a^3.</math>
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| ===Close-Packing Packing Fraction===
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| The packing-fraction of the close-packed crystal formed by rhombicuboctahedra is given by:
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| :<math> \eta = \frac{4}{3} \left( 4\sqrt{2} - 5 \right) </math>
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| It was proven that this close-packed value is assumed in a Bravais-type lattice by {{harvs|txt|last=de Graaf|year=2011}}, who also described the lattice. The proof is conditionally dependent on {{harvs|txt|last=Hales|year=2005}} proof of the [[Kepler conjecture|Kepler Conjecture]] and the proof of the inscribed-sphere upper bound for the packing of particles by {{harvs|txt|last=Torquato|year=2009}}. The packing and lattice were originally found by {{harvs|txt|last1=Betke|last2=Henk|year=2000}}, who did not prove its close-packed nature.
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| ==In the arts==
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| {| class=wikitable align=right width=357
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| |[[Image:Pacioli.jpg|200px]]<BR>Rhombicuboctahedron in top left of 1495 ''[[Portrait of Luca Pacioli]]''.<ref>[http://www.ritrattopacioli.it/texting.htm RitrattoPacioli.it]</ref>
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| |[[Image:Leonardo polyhedra.png|157px]]<BR>[[Leonardo da Vinci]]'s rhombicuboctahedron from [[Divina Proportione]], 1509.
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| |}
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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 1509 ''[[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==
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| [[Image:Rubiksnake ball.png|150px|left|thumb|[[Rubik's 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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| ''[[Sonic the Hedgehog 3]]'''s Icecap Zone features pillars topped with rhombicuboctahedra.
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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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| == Related polyhedra==
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| The rhombicuboctahedron 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 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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| {{Expanded4 table}}
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| === 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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| ==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 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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| * {{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.|bibcode = 1954RSPTA.246..401C }}
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| * {{Citation | last1=de Graaf | first1=J. | last2=van Roij | first2=R. | last3=Dijkstra | first3=M. | title=Dense Regular Packings of Irregular Nonconvex Particles | doi=10.1103/PhysRevLett.107.155501 | year=2011 | journal=Phys. Rev. Lett. | volume=107 | pages=155501 | bibcode=2011PhRvL.107o5501D|arxiv = 1107.0603 }}
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| * {{Citation | last1=Betke | first1=U. | last2=Henk | first2=M. | title=Densest Lattice Packings of 3-Polytopes | doi=10.1016/S0925-7721(00)00007-9 | year=2000 | journal=Comput. Geom. | volume=16 | pages=157 }}
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| * {{Citation | last1=Torquato | first1=S. | last2=Jiao | first2=Y. | title=Dense packings of the Platonic and Archimedean solids | doi=10.1038/nature08239 | year=2009 | journal=Nature | volume=460 | pages=876 | pmid=19675649|arxiv = 0908.4107 |bibcode = 2009Natur.460..876T }}
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| * {{Citation | last1=Hales | first1=Thomas C. | title=A proof of the Kepler conjecture | doi=10.4007/annals.2005.162.1065 | year=2005 | journal=Annals of Mathematics | volume=162 | pages=1065 }}
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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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