View source for Cantellated 7-simplexes

{| class=wikitable align=right
|- align=center
|[[File:7-simplex t0.svg|150px]]<BR>[[7-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
|[[File:7-simplex t02.svg|150px]]<BR>Cantellated 7-simplex<BR>{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node|3|node}}
|[[File:7-simplex t13.svg|150px]]<BR>Bicantellated 7-simplex<BR>{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node|3|node}}
|[[File:7-simplex t24.svg|150px]]<BR>Tricantellated 7-simplex<BR>{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node}}
|- align=center
|[[File:7-simplex t2.svg|150px]]<BR>[[Birectified 7-simplex]]<BR>{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}
|[[File:7-simplex t012.svg|150px]]<BR>Cantitruncated 7-simplex<BR>{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}
|[[File:7-simplex t123.svg|150px]]<BR>Bicantitruncated 7-simplex<BR>{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}
|[[File:7-simplex t234.svg|150px]]<BR>Tricantitruncated 7-simplex<BR>{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}
|-
!colspan=4|[[Orthogonal projection]]s in A<sub>7</sub> [[Coxeter plane]]
|}

In seven-dimensional [[geometry]], a '''cantellated 7-simplex''' is a convex [[uniform 7-polytope]], being a [[cantellation]] of the regular [[7-simplex]].

There are unique 6 degrees of cantellation for the 7-simplex, including [[truncation (geometry)|truncation]]s.

== Cantellated 7-simplex==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Cantellated 7-simplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 7-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| rr{3,3,3,3,3,3}<BR>or <math>r\left\{\begin{array}{l}3, 3, 3, 3, 3\\3\end{array}\right\}</math>
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node|3|node}}<BR>or {{CDD|node|split1|nodes_11||3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
|-
|bgcolor=#e7dcc3|6-faces||
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||1008 
|-
|bgcolor=#e7dcc3|Vertices||168 
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||5-simplex prism
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||A<sub>7</sub>, [3,3,3,3,3,3]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Small rhombated octaexon (acronym: saro) (Jonathan Bowers)<ref>Klitizing, (x3o3x3o3o3o3o - saro)</ref>

=== Coordinates ===
The vertices of the ''cantellated 7-simplex'' can be most simply positioned in 8-space as [[permutation]]s of (0,0,0,0,0,1,1,2). This construction is based on [[Facet (geometry)|facets]] of the [[cantellated 8-orthoplex]].

=== Images ===

{{7-simplex Coxeter plane graphs|t02|150}}

== Bicantellated 7-simplex ==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
! style="background:#e7dcc3;" colspan="2"|Bicantellated 7-simplex
|-
| style="background:#e7dcc3;"|Type||[[uniform 7-polytope]]
|-
| style="background:#e7dcc3;"|[[Schläfli symbol]]|| r2r{3,3,3,3,3,3}<BR>or <math>r\left\{\begin{array}{l}3, 3, 3, 3\\3, 3\end{array}\right\}</math>
|-
| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node|3|node}}<BR>or {{CDD|node|split1|nodes_11|3ab|nodes|3a|nodea|3a|nodea}}
|-
| style="background:#e7dcc3;"|6-faces||
|-
| style="background:#e7dcc3;"|5-faces||
|-
| style="background:#e7dcc3;"|4-faces||
|-
| style="background:#e7dcc3;"|Cells||
|-
| style="background:#e7dcc3;"|Faces||
|-
| style="background:#e7dcc3;"|Edges||2520 
|-
| style="background:#e7dcc3;"|Vertices||420 
|-
| style="background:#e7dcc3;"|[[Vertex figure]]||
|-
| style="background:#e7dcc3;"|[[Coxeter group]]s||A<sub>7</sub>, [3,3,3,3,3,3]
|-
| style="background:#e7dcc3;"|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)<ref>Klitizing, (o3x3o3x3o3o3o - sabro)</ref>

=== Coordinates ===
The vertices of the ''bicantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). This construction is based on [[Facet (geometry)|facets]] of the [[bicantellated 8-orthoplex]].

=== Images ===

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

== Tricantellated 7-simplex==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
! style="background:#e7dcc3;" colspan="2"|Tricantellated 7-simplex
|-
| style="background:#e7dcc3;"|Type||[[uniform 7-polytope]]
|-
| style="background:#e7dcc3;"|[[Schläfli symbol]]|| r3r{3,3,3,3,3,3}<BR>or <math>r\left\{\begin{array}{l}3, 3, 3\\3, 3, 3\end{array}\right\}</math>
|-
| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node}}<BR>or {{CDD|node|split1|nodes_11|3ab|nodes|3ab|nodes}}
|-
| style="background:#e7dcc3;"|6-faces||
|-
| style="background:#e7dcc3;"|5-faces||
|-
| style="background:#e7dcc3;"|4-faces||
|-
| style="background:#e7dcc3;"|Cells||
|-
| style="background:#e7dcc3;"|Faces||
|-
| style="background:#e7dcc3;"|Edges||3360 
|-
| style="background:#e7dcc3;"|Vertices||560 
|-
| style="background:#e7dcc3;"|[[Vertex figure]]||
|-
| style="background:#e7dcc3;"|[[Coxeter group]]s||A<sub>7</sub>, [3,3,3,3,3,3]
|-
| style="background:#e7dcc3;"|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)<ref>Klitizing, (o3o3x3o3x3o3o - stiroh)</ref>

=== Coordinates ===
The vertices of the ''tricantellated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,2). This construction is based on [[Facet (geometry)|facets]] of the [[tricantellated 8-orthoplex]].

=== Images ===

{{7-simplex Coxeter plane graphs|t24|150}}

== Cantitruncated 7-simplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
! style="background:#e7dcc3;" colspan="2"|Cantitruncated 7-simplex
|-
| style="background:#e7dcc3;"|Type||[[uniform 7-polytope]]
|-
| style="background:#e7dcc3;"|[[Schläfli symbol]]|| tr{3,3,3,3,3,3}<BR>or <math>t\left\{\begin{array}{l}3, 3, 3, 3, 3\\3\end{array}\right\}</math>
|-
| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}<BR>{{CDD|node_1|split1|nodes_11||3a|nodea|3a|nodea|3a|nodea|3a|nodea}}
|-
| style="background:#e7dcc3;"|6-faces||
|-
| style="background:#e7dcc3;"|5-faces||
|-
| style="background:#e7dcc3;"|4-faces||
|-
| style="background:#e7dcc3;"|Cells||
|-
| style="background:#e7dcc3;"|Faces||
|-
| style="background:#e7dcc3;"|Edges||1176 
|-
| style="background:#e7dcc3;"|Vertices||336 
|-
| style="background:#e7dcc3;"|[[Vertex figure]]||
|-
| style="background:#e7dcc3;"|[[Coxeter group]]s||A<sub>7</sub>, [3,3,3,3,3,3]
|-
| style="background:#e7dcc3;"|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Great rhombated octaexon (acronym: garo) (Jonathan Bowers)<ref>Klitizing, (x3x3x3o3o3o3o - garo)</ref>

=== Coordinates ===
The vertices of the ''cantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,3). This construction is based on [[Facet (geometry)|facets]] of the [[cantitruncated 8-orthoplex]].

=== Images ===

{{7-simplex Coxeter plane graphs|t012|150}}

== Bicantitruncated 7-simplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
! style="background:#e7dcc3;" colspan="2"|Bicantitruncated 7-simplex
|-
| style="background:#e7dcc3;"|Type||[[uniform 7-polytope]]
|-
| style="background:#e7dcc3;"|[[Schläfli symbol]]|| t2r{3,3,3,3,3,3}<BR>or <math>t\left\{\begin{array}{l}3, 3, 3, 3\\3, 3\end{array}\right\}</math>
|-
| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}<BR>or {{CDD|node_1|split1|nodes_11|3ab|nodes|3a|nodea|3a|nodea}}
|-
| style="background:#e7dcc3;"|6-faces||
|-
| style="background:#e7dcc3;"|5-faces||
|-
| style="background:#e7dcc3;"|4-faces||
|-
| style="background:#e7dcc3;"|Cells||
|-
| style="background:#e7dcc3;"|Faces||
|-
| style="background:#e7dcc3;"|Edges||2940 
|-
| style="background:#e7dcc3;"|Vertices||840 
|-
| style="background:#e7dcc3;"|[[Vertex figure]]||
|-
| style="background:#e7dcc3;"|[[Coxeter group]]s||A<sub>7</sub>, [3,3,3,3,3,3]
|-
| style="background:#e7dcc3;"|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)<ref>Klitizing, (o3x3x3x3o3o3o - gabro)</ref>

=== Coordinates ===

The vertices of the ''bicantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,3). This construction is based on [[Facet (geometry)|facets]] of the [[bicantitruncated 8-orthoplex]].

=== Images ===

{{7-simplex Coxeter plane graphs|t123|150}}

== Tricantitruncated 7-simplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
! style="background:#e7dcc3;" colspan="2"|Tricantitruncated 7-simplex
|-
| style="background:#e7dcc3;"|Type||[[uniform 7-polytope]]
|-
| style="background:#e7dcc3;"|[[Schläfli symbol]]|| t3r{3,3,3,3,3,3}<BR>or <math>t\left\{\begin{array}{l}3, 3, 3\\3, 3, 3\end{array}\right\}</math>
|-
| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}<BR>or {{CDD|node_1|split1|nodes_11|3ab|nodes|3ab|nodes}}
|-
| style="background:#e7dcc3;"|6-faces||
|-
| style="background:#e7dcc3;"|5-faces||
|-
| style="background:#e7dcc3;"|4-faces||
|-
| style="background:#e7dcc3;"|Cells||
|-
| style="background:#e7dcc3;"|Faces||
|-
| style="background:#e7dcc3;"|Edges||3920 
|-
| style="background:#e7dcc3;"|Vertices||1120 
|-
| style="background:#e7dcc3;"|[[Vertex figure]]||
|-
| style="background:#e7dcc3;"|[[Coxeter group]]s||A<sub>7</sub>, [3,3,3,3,3,3]
|-
| style="background:#e7dcc3;"|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)<ref>Klitizing, (o3o3x3x3x3o3o - gatroh)</ref>

=== Coordinates ===

The vertices of the ''tricantitruncated 7-simplex'' can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on [[Facet (geometry)|facets]] of the [[tricantitruncated 8-orthoplex]].

=== Images ===

{{7-simplex2 Coxeter plane graphs|t234|150}}

== Related polytopes ==

This polytope is one of 71 [[uniform 7-polytope]]s with A<sub>7</sub> symmetry.
{{Octaexon family}}

==See also==
*[[List of A7 polytopes]]

== 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|polyexa.htm|7D| uniform polytopes (polyexa)}} x3o3x3o3o3o3o - saro, o3x3o3x3o3o3o - sabro, o3o3x3o3x3o3o - stiroh, x3x3x3o3o3o3o - garo, o3x3x3x3o3o3o - gabro, o3o3x3x3x3o3o - gatroh

== External links ==
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
{{Polytopes}}

[[Category:7-polytopes]]