Editing Rhombicuboctahedron

{{short description|Archimedean solid with 26 faces}}
{{redirect|Expanded octahedron|the tensegrity structure|Jessen's icosahedron}}
{{Infobox polyhedron
 | name = Rhombicuboctahedron
 | image = Rhombicuboctahedron.jpg
 | type = [[Archimedean solid|Archimedean]]<br>[[Uniform polyhedron]]
 | faces = 8 [[equilateral triangle]]s<br>18 [[Square (geometry)|square]]s
 | edges = 48
 | vertices = 24
 | symmetry = [[Octahedral symmetry]] <math> \mathrm{O}_\mathrm{h} </math><br>[[Pyritohedral symmetry]] <math> \mathrm{T}_\mathrm{h} </math>
 | schläfli = <math> r \begin{Bmatrix} 3 \\ 4 \end{Bmatrix} </math>
 | angle = square-to-square: 135° <br> square-to-triangle: 144.7°
 | vertex_figure = Polyhedron small rhombi 6-8 vertfig.svg
 | vertex_config = <math> 24 (3 \cdot 4^3) </math>
 | dual = [[Deltoidal icositetrahedron]]
 | net = Polyhedron small rhombi 6-8 net.svg
}}
In geometry, the '''rhombicuboctahedron''' is an [[Archimedean solid]] with 26 faces, consisting of 8 equilateral triangles and 18 squares. It was named by [[Johannes Kepler]] in his 1618 [[Harmonices Mundi]], being short for ''truncated cuboctahedral rhombus'', with cuboctahedral rhombus being his name for a [[rhombic dodecahedron]].<ref>{{multiref
|{{harvp|Kepler|1997|p=[https://books.google.com/books?id=rEkLAAAAIAAJ&pg=PA119 119]}}
|{{harvp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/83/mode/1up 83]}}
}}</ref>

The rhombicuboctahedron is an [[Archimedean solid]], and its [[dual polyhedron|dual]] is a [[Catalan solid]], the [[deltoidal icositetrahedron]]. The [[elongated square gyrobicupola]] is a polyhedron that is similar to a rhombicuboctahedron, but it is not an Archimedean solid because it is not [[vertex-transitive]].  The rhombicuboctahedron is found in diverse cultures in architecture, toys, the arts, and elsewhere.

== Construction ==
The rhombicuboctahedron may be constructed from a [[Cube (geometry)|cube]] by drawing a smaller one in the middle of each face, parallel to the cube's edges. After removing the edges of a cube, the squares may be joined by adding more squares adjacent between them, and the corners may be filled by the [[equilateral triangle]]s. Another way to construct the rhombicuboctahedron is by attaching two regular [[square cupola]]s into the bases of a regular [[octagonal prism]].<ref>{{multiref
|{{harvp|Hartshorne|2000|p=[https://books.google.com/books?id=EJCSL9S6la0C&pg=PA463 463]}}
|{{harvp|Berman|1971|p=336|loc=See table IV, the Properties of regular-faced convex polyhedra, line 13. Here, <math> P_8 </math> represents the octagonal prism and <math> M_5 </math> represents the square cupola.}}
}}</ref>

[[File:P2-A5-P3.gif|left|thumb|Process of expanding the rhombicuboctahedron.]]
A rhombicuboctahedron may also be known as an ''expanded octahedron'' or ''expanded cube''. This is because the rhombicuboctahedron may also be constructed by separating and pushing away the faces of a cube or a [[regular octahedron]] from their centroid (in blue or red, respectively, in the animation), and filling between them with the squares and equilateral triangles. This construction process is known as [[Expansion (geometry)|expansion]].{{sfnp|Viana|Xavier|Aires|Campos|2019|p=1123|loc=See Fig. 6}} By using all of these methods above, the rhombicuboctahedron has 8 equilateral triangles and 16 squares as its faces.<ref>{{multiref
|{{harvp|Cockram|2020|p=[https://books.google.com/books?id=jrITEAAAQBAJ&pg=PA52 52]}}
|{{harvp|Berman|1971|p=336|loc=See table IV, the Properties of regular-faced convex polyhedra, line 13.}}
}}</ref> Relatedly, the rhombicuboctahedron may also be constructed by cutting all edges and vertices of either cube or a regular octahedron, a process known as [[Cantellation (geometry)|cantellation]].{{sfnp|Linti|2013|p=[https://books.google.com/books?id=_4C7oid1kQQC&pg=RA7-PA41 41]}}

[[Cartesian coordinate]]s of a rhombicuboctahedron with an edge length 2 are the permutations of <math> \left(\pm \left(1 + \sqrt{2}\right), \pm 1, \pm 1 \right)</math>. {{sfnp|Shepherd|1954}}

== Properties ==
=== Measurement and metric properties ===
The surface area of a rhombicuboctahedron <math> A </math> can be determined by adding the area of all faces: 8 equilateral triangles and 18 squares. The volume of a rhombicuboctahedron <math> V </math> can be determined by slicing it into two square cupolas and one octagonal prism. Given that the edge length <math> a </math>, its surface area and volume is:{{sfnp|Berman|1971|p=336|loc=See table IV, the Properties of regular-faced convex polyhedra, line 13.}}
<math display="block"> \begin{align}
 A &= \left(18+2\sqrt{3}\right)a^2 &\approx 21.464a^2,\\
 V &= \frac{12+10\sqrt{2}}{3}a^3 &\approx 8.714a^3.
\end{align} </math>

The optimal [[Packing density|packing fraction]] of rhombicuboctahedra is given by
<math display="block"> \eta = \frac{4}{3} \left( 4\sqrt{2} - 5 \right). </math>
It was noticed that this optimal value is obtained in a [[Bravais lattice]] by {{harvtxt|de Graaf|van Roij|Dijkstra|2011}}.{{sfnp|de Graaf|van Roij|Dijkstra|2011}} Since the rhombicuboctahedron is contained in a [[rhombic dodecahedron]] whose [[inscribed sphere]] is identical to its inscribed sphere, the value of the optimal packing fraction is a corollary of the [[Kepler conjecture]]: it can be achieved by putting a rhombicuboctahedron in each cell of the [[rhombic dodecahedral honeycomb]], and it cannot be surpassed, since otherwise the optimal packing density of spheres could be surpassed by putting a sphere in each rhombicuboctahedron of the hypothetical packing which surpasses it.{{cn|date=May 2024}}

The [[dihedral angle]] of a rhombicuboctahedron can be determined by adding the dihedral angle of a square cupola and an octagonal prism:{{sfnp|Johnson|1966}}
* the dihedral angle of a rhombicuboctahedron between two adjacent squares on both the top and bottom is that of a square cupola 135°. The dihedral angle of an octagonal prism between two adjacent squares is the internal angle of a [[regular octagon]] 135°. The dihedral angle between two adjacent squares on the edge where a square cupola is attached to an octagonal prism is the sum of the dihedral angle of a square cupola square-to-octagon and the dihedral angle of an octagonal prism square-to-octagon 45° + 90° = 135°. Therefore, the dihedral angle of a rhombicuboctahedron for every two adjacent squares is 135°.
* the dihedral angle of a rhombicuboctahedron square-to-triangle is that of a square cupola between those 144.7°. The dihedral angle between square-to-triangle, on the edge where a square cupola is attached to an octagonal prism is the sum of the dihedral angle of a square cupola triangle-to-octagon and the dihedral angle of an octagonal prism square-to-octagon 54.7° + 90° = 144.7°. Therefore, the dihedral angle of a rhombicuboctahedron for every square-to-triangle is 144.7°.

A rhombicuboctahedron has the [[Rupert property]], meaning there is a polyhedron with the same or larger size that can pass through its hole.<ref>{{multiref
|{{harvp|Hoffmann|2019}}
|{{harvp|Chai|Yuan|Zamfirescu|2018}}
}}</ref>

=== Symmetry and its classification family ===
[[File:Rhombicuboctahedron.stl|thumb|3D model of a rhombicuboctahedron]]
The rhombicuboctahedron has the same symmetry as a cube and regular octahedron, the [[octahedral symmetry]] <math> \mathrm{O}_\mathrm{h} </math>.<ref>{{multiref
|{{harvp|Koca|Koca|2013|p=[https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA48 48]}}
|{{harvp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/377/mode/1up 377]}}. See Figure 10.12.
}}</ref> 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 [[pyritohedral symmetry]] <math> \mathrm{T}_\mathrm{h} </math>, so they are invariant under the same rotations as the tetrahedron but different reflections.{{sfnp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/386/mode/1up 386]. See Table 10.21, Classes of vertex-transitive polyhedra.}} It is [[centrosymmetric]], meaning its symmetric is interchangeable by the appearance of [[inversion center]]. It is also non-[[Chirality (mathematics)|chiral]]; that is, it is congruent to its own mirror image.<ref>{{multiref
|{{harvp|O'Keeffe|Hyde|2020|p=[https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA54 54]}}
|{{harvp|Koca|Koca|2013|p=[https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA48 48]}}
}}</ref>

The rhombicuboctahedron is an [[Archimedean solid]], meaning it is a highly symmetric and semi-regular polyhedron, and two or more different [[Regular polygon|regular polygonal]] faces meet in a vertex.{{sfnp|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]}} The polygonal faces that meet for every vertex are one equilateral triangle and three squares, and the [[vertex figure]] is denoted as <math> 3 \cdot 4^3 </math>. Its dual is [[deltoidal icositetrahedron]], a [[Catalan solid]], shares the same symmetry as the rhombicuboctahedron.{{sfnp|Williams|1979|p=[https://archive.org/details/geometricalfound00will/page/80/mode/1up?view=theater 80]}}

The [[elongated square gyrobicupola]] is the only polyhedron resembling the rhombicuboctahedron. The difference is that the elongated square gyrobicupola is constructed by twisting one of its cupolae. It was once considered as the 14th Archimedean solid, until it was discovered that it is not [[vertex-transitive]], categorizing it as the [[Johnson solid]] instead.<ref>{{multiref
|{{harvp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/91/mode/1up 91]}}
|{{harvp|Grünbaum|2009}}
|{{harvp|Lando|Zvonkin|2004}}
}}</ref>

== Graph ==
[[File:Rhombicuboctahedral graph.png|thumb|The graph of a rhombicuboctahedron]]
The [[Skeleton (topology)|skeleton]] of a rhombicuboctahedron can be described as a [[polyhedral graph]], meaning a [[Graph (discrete mathematics)|graph]] that is [[Planar graph|planar]] and [[Vertex connectivity|3-vertex-connected]]. In other words, the edges of a graph are not crossed while being drawn, and removing any two of its vertices leaves a connected subgraph.

The rhombicuboctahedral graph has 24 [[Vertex (graph theory)|vertices]] and 48 edges. It is [[quartic graph|quartic]], meaning each of its vertices is connected to four others. This graph is classified as [[Archimedean graph]], because it resembles the graph of Archimedean solid.{{sfnp|Read|Wilson|1998|p=269}}
{{-}}

== Appearances ==
{{multiple image|total_width=400
 |image1 = Národní knihovna, Minsk - panoramio.jpg
 |image2 = Diamond cube.jpg
 |image3 = Pacioli.jpg
 |image4 = De divina proportione - Vigintisex Basium Planum Vacuum.jpg
 |perrow = 2
 |footer = Many rhombicuboctahedral objects such as [[National Library of Belarus|National Library in Minsk]] in the commemorative image (top left) and [[Rubik's cube]] variation (top right). The rhombicuboctahedron may also appear in art, as in ''[[Portrait of Luca Pacioli]]'' (bottom left) and [[Leonardo da Vinci]]'s 1509 illustration in ''[[Divina proportione]]'' (bottom right).}}
The rhombicuboctahedron sometimes appears in architecture, with an example being the building of the [[National Library of Belarus|National Library located at Minsk]].<ref>{{multiref
|{{harvp|Gan|2020|p=[https://books.google.com/books?id=9xynDwAAQBAJ&pg=PA14 14]}}
|{{harvp|Cockram|2020|p=[https://books.google.com/books?id=jrITEAAAQBAJ&pg=PA52 52]}}
}}</ref> The Wilson House by [[Bruce Goff]] is another example of a rhombicuboctahedral building, although its module was depicted as a truncated cube in which the edges are all cut off. It was built during the [[Second World War]] and [[Operation Breakthrough (housing program)|Operation Breakthrough]] in the 1960s.{{sfnp|Gabriel|1997|p=[https://books.google.com/books?id=FkM0945nFV8C&pg=PA105 105&ndash;109]}}

The rhombicuboctahedron may also be found in toys. For example, if the lines along which a [[Rubik's Cube]] can be turned are projected onto a sphere, they are [[topologically]] identical to a rhombicuboctahedron's edges. Variants using the Rubik's Cube mechanism have been produced, which closely resemble the rhombicuboctahedron. During the Rubik's Cube craze of the 1980s, at least two twisty puzzles sold had the form of a rhombicuboctahedron (the mechanism was similar to that of a Rubik's Cube)<ref name="puzzleball">{{cite web |url=http://twistypuzzles.com/cgi-bin/puzzle.cgi?pkey=5070 |title=Soviet Puzzle Ball |website=TwistyPuzzles.com |access-date=23 December 2015}}</ref><ref name="diamondstyle">{{cite web |url=https://www.jaapsch.net/puzzles/diamstyl.htm |title=Diamond Style Puzzler |website=Jaap's Puzzle Page |access-date=31 May 2017}}</ref> Another example may be found in dice from [[Corfe Castle]], each of whose square faces have marks of pairs of letters and [[Pip (counting)|pips]].{{sfnp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/4/mode/1up 4&ndash;5]}}

The rhombicuboctahedron may also appear in art. An example is the 1495 ''[[Portrait of Luca Pacioli]]'', traditionally attributed to [[Jacopo de' Barbari]], which includes a glass rhombicuboctahedron half-filled with water, which may have been painted by [[Leonardo da Vinci]].<ref>{{cite journal |doi=10.2307/3619717 |title=The Portrait of Fra Luca Pacioli |journal=[[The Mathematical Gazette]] |volume=77 |issue=479 |page=143 |year=1993 |last1=MacKinnon |first1=Nick|jstor=3619717 |s2cid=195006163 }}</ref>
The first printed version of the rhombicuboctahedron was by Leonardo da Vinci and appeared in [[Pacioli]]'s ''[[Divina proportione]]'' (1509).

{{-}}

== References ==
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{{refend}}

==See also==
* [[Truncated rhombicuboctahedron]]
* [[Moravian star]]
* [[Chamfered cube]], obtained by [[augmention (geometry)|augmenting]] the triangles to obtain non-uniform hexagon faces

==Further reading==
{{refbegin}}
*{{cite book|author=Cromwell, P.|year=1997|title=Polyhedra|location=United Kingdom|publisher=Cambridge|pages=79–86 ''Archimedean solids''|isbn=0-521-55432-2}}
* {{cite journal |last1=Coxeter |first1=H.S.M. |author-link=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 |s2cid=202575183 }}
* {{Citation | last1=Betke | first1=U. | last2=Henk | first2=M. | title=Densest Lattice Packings of 3-Polytopes | doi=10.1016/S0925-7721(00)00007-9 | doi-access=free | year=2000 | journal=[[Computational Geometry (journal)|Computational Geometry]] | volume=16 | issue=3 | pages=157–186 | arxiv=math/9909172 }}
* {{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 | issue=7257 | pages=876–879 | pmid=19675649|arxiv = 0908.4107 |bibcode = 2009Natur.460..876T | s2cid=52819935 }}
* {{Citation | last1=Hales | first1=Thomas C. | authorlink1=Thomas Callister Hales | title=A proof of the Kepler conjecture | doi=10.4007/annals.2005.162.1065 | doi-access=free | year=2005 | journal=Annals of Mathematics | volume=162 | issue=3 | pages=1065–1185 | arxiv=math/9811078v2 }}
{{refend}}

==External links==
{{commons category}}
*{{mathworld2 |urlname=SmallRhombicuboctahedron |title=Rhombicuboctahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
**{{mathworld |urlname=SmallRhombicuboctahedralGraph |title=Small rhombicuboctahedral graph}}
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3o4x - sirco}}
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
*[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]
*''[http://demonstrations.wolfram.com/RhombicuboctahedronStar/ Rhombicuboctahedron Star]'' by Sándor Kabai, [[Wolfram Demonstrations Project]].
*[http://www.hbmeyer.de/flechten/rhku/indexeng.htm Rhombicuboctahedron: paper strips for plaiting]

{{Archimedean solids}}
{{Polyhedron navigator}}

[[Category:Uniform polyhedra]]
[[Category:Archimedean solids]]