View source for Cantellated 6-simplexes

{| class=wikitable align=right width=540 style="margin-left:1em;"
|- align=center
|[[File:6-simplex_t0.svg|180px]]<BR>[[6-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node}}
|[[File:6-simplex_t02.svg|180px]]<BR>Cantellated 6-simplex<BR>{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node}}
|[[File:6-simplex_t13.svg|180px]]<BR>Bicantellated 6-simplex<BR>{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node}}
|- align=center
|[[File:6-simplex_t2.svg|180px]]<BR>[[Birectified 6-simplex]]<BR>{{CDD|node|3|node|3|node_1|3|node|3|node|3|node}}
|[[File:6-simplex_t012.svg|180px]]<BR>Cantitruncated 6-simplex<BR>{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}
|[[File:6-simplex_t123.svg|180px]]<BR>Bicantitruncated 6-simplex<BR>{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}
|-
!colspan=3|[[Orthogonal projection]]s in A<sub>6</sub> [[Coxeter plane]]
|}

In six-dimensional [[geometry]], a '''cantellated 6-simplex''' is a convex [[uniform 6-polytope]], being a [[cantellation]] of the regular [[6-simplex]].

There are unique 4 degrees of cantellation for the 6-simplex, including truncations.
{{clear}}

== Cantellated 6-simplex==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Cantellated 6-simplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| rr{3,3,3,3,3}<BR>or <math>r\left\{\begin{array}{l}3, 3, 3, 3\\3\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node}}<BR>{{CDD|node|split1|nodes_11|3a|nodea|3a|nodea|3a|nodea}}
|-
|bgcolor=#e7dcc3|5-faces||35
|-
|bgcolor=#e7dcc3|4-faces||210
|-
|bgcolor=#e7dcc3|Cells||560
|-
|bgcolor=#e7dcc3|Faces||805
|-
|bgcolor=#e7dcc3|Edges||525
|-
|bgcolor=#e7dcc3|Vertices||105
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[5-cell prism]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||A<sub>6</sub>, [3<sup>5</sup>], order 5040
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Small rhombated heptapeton (Acronym: sril) (Jonathan Bowers)<ref>Klitizing, (x3o3x3o3o3o - sril)</ref>

=== Coordinates ===
The vertices of the ''cantellated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,2). This construction is based on [[Facet (geometry)|facets]] of the [[cantellated 7-orthoplex]].

=== Images ===
{{6-simplex Coxeter plane graphs|t02|150}}<ref>Klitzing, (x3o3x3o3o3o - sril)</ref>

==Bicantellated 6-simplex==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Bicantellated 6-simplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 2rr{3,3,3,3,3}<BR>or <math>r\left\{\begin{array}{l}3, 3, 3\\3, 3\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node}}<BR>{{CDD|node|split1|nodes_11|3ab|nodes|3a|nodea}}
|-
|bgcolor=#e7dcc3|5-faces||49
|-
|bgcolor=#e7dcc3|4-faces||329
|-
|bgcolor=#e7dcc3|Cells||980
|-
|bgcolor=#e7dcc3|Faces||1540
|-
|bgcolor=#e7dcc3|Edges||1050
|-
|bgcolor=#e7dcc3|Vertices||210
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||A<sub>6</sub>, [3<sup>5</sup>], order 5040
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Small prismated heptapeton (Acronym: sabril) (Jonathan Bowers)<ref>Klitzing, (o3x3o3x3o3o - sabril)</ref>

=== Coordinates ===
The vertices of the ''bicantellated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,2). This construction is based on [[Facet (geometry)|facets]] of the [[bicantellated 7-orthoplex]].

=== Images ===

{{6-simplex Coxeter plane graphs|t13|150}}

==Cantitruncated 6-simplex==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|cantitruncated 6-simplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| tr{3,3,3,3,3}<BR>or <math>t\left\{\begin{array}{l}3, 3, 3, 3\\3\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}<BR>{{CDD|node_1|split1|nodes_11|3a|nodea|3a|nodea|3a|nodea}}
|-
|bgcolor=#e7dcc3|5-faces||35 
|-
|bgcolor=#e7dcc3|4-faces||210 
|-
|bgcolor=#e7dcc3|Cells||560 
|-
|bgcolor=#e7dcc3|Faces||805 
|-
|bgcolor=#e7dcc3|Edges||630 
|-
|bgcolor=#e7dcc3|Vertices||210 
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||A<sub>6</sub>, [3<sup>5</sup>], order 5040
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Great rhombated heptapeton (Acronym: gril) (Jonathan Bowers)<ref>Klitzing, (x3x3x3o3o3o - gril)</ref>

=== Coordinates ===
The vertices of the ''cantitruncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,3). This construction is based on [[Facet (geometry)|facets]] of the [[cantitruncated 7-orthoplex]].

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

==Bicantitruncated 6-simplex==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|bicantitruncated 6-simplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 2tr{3,3,3,3,3}<BR>or <math>t\left\{\begin{array}{l}3, 3, 3\\3, 3\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}<BR>{{CDD|node_1|split1|nodes_11|3ab|nodes|3a|nodea}}
|-
|bgcolor=#e7dcc3|5-faces||49 
|-
|bgcolor=#e7dcc3|4-faces||329 
|-
|bgcolor=#e7dcc3|Cells||980 
|-
|bgcolor=#e7dcc3|Faces||1540 
|-
|bgcolor=#e7dcc3|Edges||1260 
|-
|bgcolor=#e7dcc3|Vertices||420 
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||A<sub>6</sub>, [3<sup>5</sup>], order 5040
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Great birhombated heptapeton (Acronym: gabril) (Jonathan Bowers)<ref>Klitzing, (o3x3x3x3o3o - gabril)</ref>

=== Coordinates ===
The vertices of the ''bicantitruncated 6-simplex'' can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,3). This construction is based on [[Facet (geometry)|facets]] of the [[bicantitruncated 7-orthoplex]].

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

== Related uniform 6-polytopes ==
The truncated 6-simplex is one of 35 [[Uniform 6-polytope#The A6 .5B3.2C3.2C3.2C3.2C3.5D family .286-simplex.29|uniform 6-polytopes]] based on the [3,3,3,3,3] [[Coxeter group]], all shown here in A<sub>6</sub> [[Coxeter plane]] [[orthographic projection]]s.

{{Heptapeton 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)}} x3o3x3o3o3o - sril, o3x3o3x3o3o - sabril, x3x3x3o3o3o - gril, o3x3x3x3o3o - gabril

== 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]]