View source for Truncated cube

{{Short description|Archimedean solid with 14 regular faces}}
{{Semireg polyhedra db|Semireg polyhedron stat table|tC}}
[[File:Truncated cube.stl|thumb|3D model of a truncated cube]]
In [[geometry]], the '''truncated cube''', or '''truncated hexahedron''', is an [[Archimedean solid]]. It has 14 regular faces (6 [[octagon]]al and 8 [[triangle (geometry)|triangular]]), 36 edges, and 24 vertices.

If the truncated cube has unit edge length, its dual [[triakis octahedron]] has edges of lengths {{math|2}} and {{math|''δ<sub>S</sub>'' +1}}, 
where  ''δ<sub>S</sub>'' is the silver ratio, {{sqrt|2}} +1.

==Area and volume==
The area ''A'' and the [[volume]] ''V'' of a truncated cube of edge length ''a'' are:
:<math>\begin{align}
A &= 2\left(6+6\sqrt{2}+\sqrt{3}\right)a^2 &&\approx 32.434\,6644a^2 \\
V &= \frac{21+14\sqrt{2}}{3}a^3 &&\approx 13.599\,6633a^3. \end{align}</math>

==Orthogonal projections==
The ''truncated cube'' has five special [[orthogonal projection]]s, centered, on a vertex, on two types of edges, and two types of faces: triangles, and octagons. The last two correspond to the B<sub>2</sub> and A<sub>2</sub> [[Coxeter plane]]s.
{|class=wikitable width=640
|+ Orthogonal projections
|-
! Centered by
!Vertex
!Edge<br>3-8
!Edge<br>8-8
!Face<br>Octagon
!Face<br>Triangle
|-
! Solid
|
|
|[[File:Polyhedron truncated 6 from blue max.png|70px|center]]
|[[File:Polyhedron truncated 6 from red max.png|100px]]
|[[File:Polyhedron truncated 6 from yellow max.png|100px]]
|-
! Wireframe
|[[File:Cube t01 v.png|100px]]
|[[File:Cube t01 e38.png|100px]]
|[[File:Cube t01 e88.png|100px]]
|[[File:3-cube t01_B2.svg|100px]]
|[[File:3-cube t01.svg|100px]]
|-
! Dual
|[[File:Dual truncated cube t01 v.png|100px]]
|[[File:Dual truncated cube t01 e8.png|100px]]
|[[File:Dual truncated cube t01 e88.png|100px]]
|[[File:Dual truncated cube t01_B2.png|100px]]
|[[File:Dual truncated cube t01.png|100px]]
|- align=center
!Projective<BR>symmetry
|[2]
|[2]
|[2]
|[4]
|[6]
|}

==Spherical tiling==
The truncated cube can also be represented as a [[spherical tiling]], and projected onto the plane via a [[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-t01.png|160px]]
|[[Image:truncated cube stereographic projection octagon.png|160px]]<br>[[octagon]]-centered
|[[Image:truncated cube stereographic projection triangle.png|160px]]<br>[[triangle]]-centered
|-
![[Orthographic projection]]
!colspan=2|[[Stereographic projection]]s
|}

==Cartesian coordinates==
[[File:Icosidecahedron_in_truncated_cube.png|thumb|A truncated cube with its octagonal faces [[pyritohedral symmetry|pyritohedrally]] dissected with a central vertex into triangles and pentagons, creating a topological [[icosidodecahedron]]]]
[[Cartesian coordinates]] for the vertices of a [[Truncation (geometry)|truncated]] [[hexahedron]] centered at the origin with edge length 2{{sfrac|1|''δ<sub>S</sub>''}} are all the permutations of

:(±{{sfrac|1|''δ<sub>S</sub>''}}, ±1, ±1),

where '''δ<sub>S</sub>'''={{sqrt|2}}+1.

If we let a parameter ''ξ''= {{sfrac|1|''δ<sub>S</sub>''}}, in the case of a Regular Truncated Cube, then the parameter ''ξ'' can be varied between ±1. A value of 1 produces a [[cube]], 0 produces a [[cuboctahedron]], and negative values produces self-intersecting [[Octagram#Other octagrams|octagrammic]] faces.
:[[File:Truncated_cube_sequence.png|640px]]

If the self-intersected portions of the octagrams are removed, leaving squares, and truncating the triangles into hexagons, [[truncated octahedron|truncated octahedra]] are produced, and the sequence ends with the central squares being reduced to a point, and creating an [[octahedron]].

== Dissection ==
[[File:Dissected_truncated_cube.png|thumb|Dissected truncated cube, with elements expanded apart]]
The truncated cube can be dissected into a central [[cube]], with six [[square cupola]]e around each of the cube's faces, and 8 regular tetrahedra in the corners. This dissection can also be seen within the [[runcic cubic honeycomb]], with [[cube]], [[tetrahedron]], and [[rhombicuboctahedron]] cells.

This dissection can be used to create a [[Stewart toroid]] with all regular faces by removing two square cupolae and the central cube. This '''excavated cube''' has 16 [[triangle]]s, 12 [[square]]s, and 4 [[octagon]]s.<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}}</ref>
:[[File:Excavated_truncated_cube.png|240px]]

== Vertex arrangement==
It shares the [[vertex arrangement]] with three [[nonconvex uniform polyhedra]]:
{|class="wikitable" width="400" style="vertical-align:top;text-align:center"
|[[Image:Truncated hexahedron.png|100px]]<br>'''Truncated cube'''
|[[Image:Uniform great rhombicuboctahedron.png|100px]]<br>[[Nonconvex great rhombicuboctahedron]]
|[[Image:Great cubicuboctahedron.png|100px]]<br>[[Great cubicuboctahedron]]
|[[Image:Great rhombihexahedron.png|100px]]<br>[[Great rhombihexahedron]]
|}

==Related polyhedra==
The truncated cube is related to other polyhedra and tilings in symmetry.

The truncated cube is one of a family of uniform polyhedra related to the cube and regular octahedron.
{{Octahedral truncations}}

=== Symmetry mutations===
This polyhedron is topologically related as a part of sequence of uniform [[Truncation (geometry)|truncated]] polyhedra with [[vertex configuration]]s (3.2''n''.2''n''), and [''n'',3] [[Coxeter group]] symmetry, and a series of polyhedra and tilings ''n''.8.8.
{{Truncated figure1 small table}}
{{Truncated figure4 table}}

=== Alternated truncation===
{{multiple image
 | align = right  | total_width = 400
 | image1 = Polyhedron 4a.png
 | image2 = Polyhedron chamfered 4a.png
 | image3 = Polyhedron truncated 6.png
 | footer = Tetrahedron, its edge truncation, and the truncated cube
}}
Truncating alternating vertices of the cube gives the [[chamfered tetrahedron]], i.e. the edge truncation of the tetrahedron.

The [[truncated triangular trapezohedron]] is another polyhedron which can be formed from cube edge truncation.

== Related polytopes ==
The ''[[Truncation (geometry)|truncated]] [[cube]]'', is second in a sequence of truncated [[hypercube]]s:
{{Truncated hypercube polytopes}}

== Truncated cubical graph ==
{{Infobox graph
 | name = Truncated cubical graph
 | image = [[File:Truncated cubic graph.png|240px]]
 | image_caption = 4-fold symmetry [[Schlegel diagram]]
 | namesake = 
 | vertices = 24
 | edges = 36
 | automorphisms = 48
 | radius = 
 | diameter = 
 | girth = 
 | chromatic_number = 3
 | chromatic_index = 
 | fractional_chromatic_index = 
 | properties = [[Cubic graph|Cubic]], [[hamiltonian graph|Hamiltonian]], [[regular graph|regular]], [[Zero-symmetric graph|zero-symmetric]]
}}
In the [[mathematics|mathematical]] field of [[graph theory]], a '''truncated cubical graph''' is the [[1-skeleton|graph of vertices and edges]] of the ''truncated cube'', one of the [[Archimedean solid]]s. It has 24 [[Vertex (graph theory)|vertices]] and 36 edges, and is a [[cubic graph|cubic]] [[Archimedean graph]].<ref>{{citation|last1=Read|first1=R. C.|last2=Wilson|first2=R. J.|title=An Atlas of Graphs|publisher=[[Oxford University Press]]|year= 1998|page=269}}</ref>

{| class=wikitable
|- align=center
|[[File:3-cube t01.svg|200px]]<BR>Orthographic
|}
{{-}}

== See also ==
* [[:Image:Truncatedhexahedron.gif|Spinning truncated cube]]
* [[Cube-connected cycles]], a family of graphs that includes the [[skeleton (topology)|skeleton]] of the truncated cube
* [[Chamfered cube]], obtained by replacing the edges of a cube with non-uniform hexagons

==References==
{{reflist}}
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
* Cromwell, P. ''Polyhedra'', CUP hbk (1997), pbk. (1999). Ch.2 p.&nbsp;79-86 ''Archimedean solids''

==External links==
*{{mathworld2 |urlname=TruncatedCube |title=Truncated cube |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
**{{mathworld |urlname=TruncatedCubicalGraph |title=Truncated cubical graph}}
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|o3x4x - tic}}
*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=621wh65c7Ey8v4cRpEVhGs0pPxZ5raM9uNf8HcBUgOyrp6acSwZGvkvEcL6m06RDKxmSAduYsvTvoCvEDokvHrjyVEqlGVdIH8WamnxFO1qnGpUtgt7K0ZD57RlX&name=Truncated+Cube#applet Editable printable net of a truncated cube with interactive 3D view]
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] www.georgehart.com: The Encyclopedia of Polyhedra
**[[VRML]] [http://www.georgehart.com/virtual-polyhedra/vrml/truncated_cube.wrl model]
**[http://www.georgehart.com/virtual-polyhedra/conway_notation.html Conway Notation for Polyhedra] Try: "tC"

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

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