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{{short description|Archimedean solid in geometry}}
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In [[Geometry|geometry]], the '''truncated cuboctahedron''' or '''great rhombicuboctahedron''' is an [[Archimedean solid]], named by Kepler as a [[Truncation (geometry)|truncation]] of a [[Cuboctahedron|cuboctahedron]]. It has 12 square faces, 8 regular [[Hexagon|hexagon]]al faces, 6 regular [[Octagon|octagon]]al faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry (equivalently, 180° [[Physics:Rotation|rotation]]al symmetry), the truncated cuboctahedron is a '''9'''-[[Zonohedron|zonohedron]]. The truncated cuboctahedron can tessellate with the [[Octagonal prism|octagonal prism]].

==Names==
{|
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The name ''truncated cuboctahedron'', given originally by [[Biography:Johannes Kepler|Johannes Kepler]], is misleading: an actual [[Truncation (geometry)|truncation]] of a [[Cuboctahedron|cuboctahedron]] has [[Rectangle|rectangle]]s instead of [[Square|square]]s; however, this nonuniform polyhedron is topologically equivalent to the Archimedean solid unrigorously named truncated cuboctahedron.

Alternate interchangeable names are:
*''Truncated cuboctahedron'' ([[Biography:Johannes Kepler|Johannes Kepler]]),
*''Rhombitruncated cuboctahedron'' (Magnus Wenninger<ref>{{Citation |last1=Wenninger |first1=Magnus |title=Polyhedron Models |publisher={{wipe|Cambridge University Press}} |isbn=978-0-521-09859-5 |mr=0467493 |year=1974}} (Model 15, p. 29)</ref>),
*''Great rhombicuboctahedron'' ([[Biography:Robert Williams (geometer)|Robert Williams]]<ref>{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9, p. 82)</ref>),
*''Great rhombcuboctahedron'' (Peter Cromwell<ref>Cromwell, P.; [https://books.google.com/books?id=OJowej1QWpoC&pg=PA82 ''Polyhedra''], CUP hbk (1997), pbk. (1999). (p. 82)</ref>),
*''[[Omnitruncation|Omnitruncated]] cube'' or ''cantitruncated cube'' ([[Biography:Norman Johnson (mathematician)|Norman Johnson]]),
* ''Beveled cube'' ([[Conway polyhedron notation]]).
|style="padding-left:50px;"| <!-- right table cell -->
{{multiple image
 | align = left  | width = 130
 | image1 = Polyhedron 6-8 max.png
 | image2 = Polyhedron nonuniform truncated 6-8 max.png
 | footer = Cuboctahedron and its truncation
}}
|}

There is a nonconvex uniform polyhedron with a similar name: the [[Nonconvex great rhombicuboctahedron|nonconvex great rhombicuboctahedron]].

==Cartesian coordinates==
The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all the [[Permutation|permutation]]s of:
<math display=block>\Bigl(\pm 1, \quad \pm\left(1 + \sqrt 2\right), \quad \pm\left(1 + 2\sqrt 2\right) \Bigr).</math>

==Area and volume==
The area ''A'' and the volume ''V'' of the truncated cuboctahedron of edge length ''a'' are:
:<math>\begin{align}
A &= 12\left(2+\sqrt{2}+\sqrt{3}\right) a^2 &&\approx 61.755\,1724~a^2, \\
V &= \left(22+14\sqrt{2}\right) a^3 &&\approx 41.798\,9899~a^3. \end{align}</math>

== Dissection ==
{{multiple image
 | align = left  | total_width = 300
 | image1 = Small in great rhombi 6-8, davinci small with cubes.png |width1=3944|height1=3876 
 | image2 = Small in great rhombi 6-8, davinci.png |width2=3944|height2=3876
}}
The truncated cuboctahedron is the [[Convex hull|convex hull]] of a [[Rhombicuboctahedron|rhombicuboctahedron]] with cubes above its 12 squares on 2-fold symmetry axes. The rest of its space can be dissected into 6 [[Square cupola|square cupola]]s below the octagons, and 8 [[Triangular cupola|triangular cupola]]s below the hexagons.

A dissected truncated cuboctahedron can create a genus 5, 7, or 11 Stewart toroid by removing the central rhombicuboctahedron, and either the 6 square cupolas, the 8 triangular cupolas, or the 12 cubes respectively. Many other lower symmetry toroids can also be constructed by removing the central rhombicuboctahedron, and a subset of the other dissection components. For example, removing 4 of the triangular cupolas creates a genus 3 toroid; if these cupolas are appropriately chosen, then this toroid has tetrahedral symmetry.<ref>B. M. Stewart, ''[[Adventures Among the Toroids]]'' (1970) {{isbn|978-0-686-11936-4}}</ref><ref>{{cite web|url=http://www.doskey.com/polyhedra/Stewart05.html|title=Adventures Among the Toroids - Chapter 5 - Simplest (R)(A)(Q)(T) Toroids of genus p=1|first=Alex|last=Doskey|website=www.doskey.com}}</ref>

{| class="wikitable collapsible collapsed"
!colspan="4"| Stewart toroids
|-
!Genus 3
!Genus 5
!Genus 7
!Genus 11
|-
|[[File:Excavated truncated cuboctahedron4.png|160px]]
|[[File:Excavated truncated cuboctahedron2.png|160px]]
|[[File:Excavated truncated cuboctahedron3.png|160px]]
|[[File:Excavated truncated cuboctahedron.png|160px]]
|}

==Uniform colorings==
There is only one [[Uniform coloring|uniform coloring]] of the faces of this polyhedron, one color for each face type.

A 2-uniform coloring, with [[Tetrahedral symmetry|tetrahedral symmetry]], exists with alternately colored hexagons.

==Orthogonal projections==
{{multiple image
 | align = right | width = 120 | direction=vertical
 | image1 = Polyhedron great rhombi 6-8 from blue max.png
 | image2 = Polyhedron great rhombi 6-8 from yellow max.png
 | image3 = Polyhedron great rhombi 6-8 from red max.png
}}
The truncated cuboctahedron has two special orthogonal projections in the A<sub>2</sub> and B<sub>2</sub> Coxeter planes with [6] and [8] projective symmetry, and numerous [2] symmetries can be constructed from various projected planes relative to the polyhedron elements.
{|class=wikitable
|+ Orthogonal projections
|-
!Centered by
!Vertex
!Edge<br>4-6
!Edge<br>4-8
!Edge<br>6-8
!Face normal<BR>4-6
|-
!Image
|[[File:Cube_t012_v.png|100px]]
|[[File:Cube_t012_e46.png|100px]]
|[[File:Cube_t012_e48.png|100px]]
|[[File:Cube_t012_e68.png|100px]]
|[[File:Cube_t012_f46.png|100px]]
|- align=center
!Projective<br>symmetry
|[2]<sup>+</sup>
|[2]
|[2]
|[2]
|[2]
|-
!Centered by
!Face normal<BR>Square
!Face normal<BR>Octagon
!Face<br>Square
!Face<br>Hexagon
!Face<br>Octagon
|-
!Image
|[[File:Cube_t012_af4.png|100px]]
|[[File:Cube_t012_af8.png|100px]]
|[[File:Cube_t012_f4.png|100px]]
|[[File:3-cube_t012.svg|100px]]
|[[File:3-cube_t012_B2.svg|100px]]
|- align=center
!Projective<br>symmetry
|[2]
|[2]
|[2]
|[6]
|[4]
|}

==Spherical tiling==
The truncated cuboctahedron can also be represented as a spherical tiling, and projected onto the plane via a [[Stereographic projection|stereographic projection]]. This projection is [[Conformal map|conformal]], preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
{|class=wikitable
|- align=center valign=top
|[[Image:Uniform tiling 432-t012.png|155px]]
|[[Image:truncated cuboctahedron stereographic projection square.png|165px]]
|[[Image:truncated cuboctahedron stereographic projection hexagon.png|152px]]
|[[Image:truncated cuboctahedron stereographic projection octagon.png|160px]]
|-
!rowspan=2|Orthogonal projection
![[Square|square]]-centered||[[Hexagon|hexagon]]-centered||[[Octagon|octagon]]-centered
|-
!colspan=3|[[Stereographic projection]]s
|}

==Full octahedral group==
[[File:Full octahedral group elements in truncated cuboctahedron; JF.png|thumb|right]]
Like many other solids the truncated octahedron has full [[Octahedral symmetry|octahedral symmetry]] - but its relationship with the full octahedral group is closer than that: Its 48 vertices correspond to the elements of the group, and each face of [[Disdyakis dodecahedron|its dual]] is a [[Fundamental domain|fundamental domain]] of the group.

The image on the right shows the 48 permutations in the group applied to an example object (namely the light JF compound on the left). The 24 light elements are rotations, and the dark ones are their reflections.

The edges of the solid correspond to the 9 reflections in the group:
* Those between octagons and squares correspond to the 3 reflections between opposite octagons.
* Hexagon edges correspond to the 6 reflections between opposite squares.
* (There are no reflections between opposite hexagons.)

The subgroups correspond to solids that share the respective vertices of the truncated octahedron.<br>
E.g. the 3 subgroups with 24 elements correspond to a nonuniform [[Snub cube|snub cube]] with chiral octahedral symmetry, a nonuniform [[Rhombicuboctahedron|rhombicuboctahedron]] with pyritohedral symmetry (the [[Cantic snub octahedron|cantic snub octahedron]]) and a nonuniform [[Truncated octahedron|truncated octahedron]] with full tetrahedral symmetry. The unique subgroup with 12 elements is the [[Alternating group|alternating group]] A<sub>4</sub>. It corresponds to a nonuniform [[Icosahedron|icosahedron]] with chiral tetrahedral symmetry.

{| class="wikitable collapsible" style="text-align: center;"
!colspan="5"| Subgroups and corresponding solids
|- valign=top
!Truncated cuboctahedron<BR>{{CDD|node_1|4|node_1|3|node_1}}<BR>tr{4,3}
![[Snub cube]]<BR>{{CDD|node_h|4|node_h|3|node_h}}<BR>sr{4,3}
![[Rhombicuboctahedron#Pyritohedral symmetry|Rhombicuboctahedron]]<BR>{{CDD|node_1|4|node_h|3|node_h}}<BR>s<sub>2</sub>{3,4}
![[Truncated octahedron]]<BR>{{CDD|node_h|4|node_1|3|node_1}}<BR>h<sub>1,2</sub>{4,3}
![[Regular icosahedron#Symmetries|Icosahedron]]<BR>{{CDD|node_h|2|4|2|node_h|3|node_h}}
|-
![4,3]<BR>[[Octahedral symmetry#Subgroups of full octahedral symmetry|Full octahedral]]
![4,3]<sup>+</sup><BR>Chiral octahedral
![4,3<sup>+</sup>]<BR>[[Tetrahedral symmetry#Subgroups of pyritohedral symmetry|Pyritohedral]]
![1<sup>+</sup>,4,3] = [3,3]<BR>[[Tetrahedral symmetry#Subgroups of achiral tetrahedral symmetry|Full tetrahedral]]
![1<sup>+</sup>,4,3<sup>+</sup>] = [3,3]<sup>+</sup><BR>Chiral tetrahedral
|-
| [[File:Polyhedron great rhombi 6-8 max.png|150px]]
| [[File:Polyhedron great rhombi 6-8 subsolid snub right maxmatch.png|150px]]
| [[File:Polyhedron great rhombi 6-8 subsolid pyritohedral maxmatch.png|150px]]
| [[File:Polyhedron great rhombi 6-8 subsolid tetrahedral maxmatch.png|150px]]
| [[File:Polyhedron great rhombi 6-8 subsolid 20 maxmatch.png|150px]]
|-
| all 48 vertices
|colspan="3"| 24 vertices
| 12 vertices
|}

== Related polyhedra ==
{| class=wikitable align=right width=320
|[[File:Conway_polyhedron_b3O.png|160px]] 
|[[File:Conway_polyhedron_b3C.png|160px]]
|-
|colspan=2|Bowtie tetrahedron and cube contain two trapezoidal faces in place of each square.<ref>[http://www.cgl.uwaterloo.ca/csk/papers/bridges2001.html Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons] Craig S. Kaplan</ref>
|}
The truncated cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.


This polyhedron can be considered a member of a sequence of uniform patterns with [[Vertex configuration|vertex configuration]] (4.6.2''p'') and Coxeter-Dynkin diagram {{CDD|node_1|p|node_1|3|node_1}}.  For ''p''&nbsp;<&nbsp;6, the members of the sequence are omnitruncated polyhedra ([[Zonohedron|zonohedron]]s), shown below as spherical tilings. For ''p''&nbsp;<&nbsp;6, they are tilings of the hyperbolic plane, starting with the [[Truncated triheptagonal tiling|truncated triheptagonal tiling]].

It is first in a series of cantitruncated hypercubes:

== Truncated cuboctahedral graph ==
{{Infobox graph
 | name = Truncated cuboctahedral graph
 | image = [[File:Truncated cuboctahedral graph.png|240px]]
 | image_caption = 4-fold symmetry
 | namesake = 
 | vertices = 48
 | edges = 72
 | automorphisms = 48
 | radius = 
 | diameter = 
 | girth = 
 | chromatic_number = 2
 | chromatic_index = 
 | fractional_chromatic_index = 
 | properties = [[Cubic graph|Cubic]], Hamiltonian, [[Regular graph|regular]], [[Zero-symmetric graph|zero-symmetric]]
}}
In the [[Mathematics|mathematical]] field of [[Graph theory|graph theory]], a '''truncated cuboctahedral graph''' (or '''great rhombcuboctahedral graph''') is the graph of vertices and edges of the truncated cuboctahedron, one of the [[Archimedean solid]]s. It has 48 [[Vertex (graph theory)|vertices]] and 72 edges, and is a [[Zero-symmetric graph|zero-symmetric]] and [[Cubic graph|cubic]] [[Archimedean graph]].<ref>{{citation|last1=Read|first1=R. C.|last2=Wilson|first2=R. J.|title=An Atlas of Graphs|publisher=[[Organization:Oxford University Press|Oxford University Press]]|year= 1998|page=269}}</ref>

{{Clear}}

==See also==
*Cube
*[[Cuboctahedron]]
*[[Octahedron]]
*[[Truncated icosidodecahedron]]
*[[Truncated octahedron]] – truncated tetratetrahedron
*[[Snub cube]]

==References==

{{Reflist}}
*{{cite book|author=Cromwell, P.|year=1997|title=Polyhedra|location=United Kingdom|publisher=Cambridge|pages=79–86 ''Archimedean solids''|isbn=0-521-55432-2}}

==External links==
*{{mathworld2 |urlname=GreatRhombicuboctahedron |title=Great rhombicuboctahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
**{{mathworld |urlname=GreatRhombicuboctahedralGraph |title=Great rhombicuboctahedral graph}}
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x4x - girco}}
*[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]
*[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.faust.fr.bw.schule.de/mhb/flechten/grko/indexeng.htm Great rhombicuboctahedron: paper strips for plaiting]

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{{Sourceattribution|Truncated cuboctahedron}}