View source for Cantellated 6-cubes

{| class=wikitable align=right width=360 style="margin-left:1em;"
|- align=center
|colspan=4|[[File:6-cube t0.svg|120px]]<BR>[[6-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|3|node}}
|colspan=4|[[File:6-cube t02.svg|120px]]<BR>'''Cantellated 6-cube'''<BR>{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node}}
|colspan=4|[[File:6-cube t13.svg|120px]]<BR>'''Bicantellated 6-cube'''<BR>{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node}}
|- align=center
|colspan=4|[[File:6-cube t5.svg|120px]]<BR>[[6-orthoplex]]<BR>{{CDD|node|4|node|3|node|3|node|3|node|3|node_1}}
|colspan=4|[[File:6-cube t35.svg|120px]]<BR>[[Cantellated 6-orthoplex]]<BR>{{CDD|node|4|node|3|node|3|node_1|3|node|3|node_1}}
|colspan=4|[[File:6-cube t24.svg|120px]]<BR>[[Bicantellated 6-orthoplex]]<BR>{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node}}
|- align=center
|colspan=3|[[File:6-cube t012.svg|90px]]<BR>'''Cantitruncated 6-cube'''<BR>{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node}}
|colspan=3|[[File:6-cube t123.svg|90px]]<BR>'''Bicantitruncated 6-cube'''<BR>{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node}}
|colspan=3|[[File:6-cube t234.svg|90px]]<BR>[[Bicantitruncated 6-orthoplex]]<BR>{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node}}
|colspan=3|[[File:6-cube t345.svg|90px]]<BR>[[Cantitruncated 6-orthoplex]]<BR>{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node_1}}
|-
!colspan=12|[[Orthogonal projection]]s in B<sub>6</sub> [[Coxeter plane]]
|}
In six-dimensional [[geometry]], a '''cantellated 6-cube''' is a convex [[uniform 6-polytope]], being a [[cantellation]] of the regular [[6-cube]].

There are 8 cantellations for the 6-cube, including truncations. Half of them are more easily constructed from the dual [[5-orthoplex]].

{{TOC left}}
{{Clear}}

== Cantellated 6-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Cantellated 6-cube
|-
|bgcolor=#e7dcc3|Type||[[uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| rr{4,3,3,3,3}<BR>or <math>r\left\{\begin{array}{l}3, 3, 3, 3\\4\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node}}<BR>{{CDD|node|split1-43|nodes_11|3b|nodeb|3b|nodeb|3b|nodeb}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||4800
|-
|bgcolor=#e7dcc3|Vertices||960
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [3,3,3,3,4]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Cantellated hexeract
* Small rhombated hexeract (acronym: srox) (Jonathan Bowers)<ref>Klitzing, (o3o3o3x3o4x - srox)</ref>

===Images===
{{6-cube Coxeter plane graphs|t02|150}}

== Bicantellated 6-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Cantellated 6-cube
|-
|bgcolor=#e7dcc3|Type||[[uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 2rr{4,3,3,3,3}<BR>or <math>r\left\{\begin{array}{l}3, 3, 3\\3, 4\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node}}<BR>{{CDD|node|split1|nodes_11|4a3b|nodes|3b|nodeb}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||
|-
|bgcolor=#e7dcc3|Vertices||
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [3,3,3,3,4]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Bicantellated hexeract
* Small birhombated hexeract (acronym: saborx) (Jonathan Bowers)<ref>Klitzing, (o3o3x3o3x4o - saborx)</ref>

===Images===
{{6-cube Coxeter plane graphs|t13|150}}

== Cantitruncated 6-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Cantellated 6-cube
|-
|bgcolor=#e7dcc3|Type||[[uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| tr{4,3,3,3,3}<BR>or <math>t\left\{\begin{array}{l}3, 3, 3, 3\\4\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node}}<BR>{{CDD|node_1|split1-43|nodes_11|3b|nodeb|3b|nodeb|3b|nodeb}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||
|-
|bgcolor=#e7dcc3|Vertices||
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [3,3,3,3,4]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Cantitruncated hexeract
* Great rhombihexeract (acronym: grox) (Jonathan Bowers)<ref>Klitzing, (o3o3o3x3x4x - grox)</ref>

===Images===
{{6-cube Coxeter plane graphs|t012|150}}

It is fourth in a series of cantitruncated hypercubes:
{{Cantitruncated hypercube polytopes}}

== Bicantitruncated 6-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Cantellated 6-cube
|-
|bgcolor=#e7dcc3|Type||[[uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 2tr{4,3,3,3,3}<BR>or <math>t\left\{\begin{array}{l}3, 3, 3\\3, 4\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node}}<BR>{{CDD|node_1|split1|nodes_11|4a3b|nodes|3b|nodeb}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||
|-
|bgcolor=#e7dcc3|Vertices||
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [3,3,3,3,4]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Bicantitruncated hexeract
* Great birhombihexeract (acronym: gaborx) (Jonathan Bowers)<ref>Klitzing, (o3o3x3x3x4o - gaborx)</ref>

===Images===
{{6-cube Coxeter plane graphs|t123|150}}

== Related polytopes==

These polytopes are part of a set of 63 [[uniform 6-polytope]]s generated from the B<sub>6</sub> [[Coxeter plane]], including the regular [[6-cube]] or [[6-orthoplex]].

{{Hexeract family}}

== Notes==
{{reflist}}

== References==
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]: 
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 
** '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10]
*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. 
* {{KlitzingPolytopes|polypeta.htm|6D|uniform polytopes (polypeta)}} o3o3o3x3o4x - srox, o3o3x3o3x4o - saborx, o3o3o3x3x4x - grox, o3o3x3x3x4o - gaborx

== External links ==
* [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{polytopes}}

[[Category:6-polytopes]]