View source for Stericated 6-orthoplexes

{| class=wikitable align=right width=480 style="margin-left:1em"
|- align=center valign=top
|[[File:6-cube t5.svg|160px]]<BR><small>[[6-orthoplex|6-orthoplex]]</small><BR>{{CDD|node_1|3|node|3|node|3|node|3|node|4|node}}
|[[File:6-cube t15.svg|160px]]<BR><small>Stericated 6-orthoplex</small><BR>{{CDD|node_1|3|node|3|node|3|node|3|node_1|4|node}}
|[[File:6-cube t145.svg|160px]]<BR><small>Steritruncated 6-orthoplex</small><BR>{{CDD|node_1|3|node_1|3|node|3|node|3|node_1|4|node}}
|- align=center valign=top
|[[File:6-cube t135.svg|160px]]<BR><small>Stericantellated 6-orthoplex</small><BR>{{CDD|node_1|3|node|3|node_1|3|node|3|node_1|4|node}}
|[[File:6-cube t1345.svg|160px]]<BR><small>Stericantitruncated 6-orthoplex</small><BR>{{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1|4|node}}
|[[File:6-cube t125.svg|160px]]<BR><small>Steriruncinated 6-orthoplex</small><BR>{{CDD|node_1|3|node|3|node|3|node_1|3|node_1|4|node}}
|- align=center valign=top
|[[File:6-cube t1245.svg|160px]]<BR><small>Steriruncitruncated 6-orthoplex</small><BR>{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|4|node}}
|[[File:6-cube t1235.svg|160px]]<BR><small>Steriruncicantellated 6-orthoplex</small><BR>{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|4|node}}
|[[File:6-cube t12345.svg|160px]]<BR><small>Steriruncicantitruncated 6-orthoplex</small><BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|4|node}}
|-
!colspan=3|Orthogonal projections in B<sub>6</sub> Coxeter plane
|}
In six-dimensional [[Geometry|geometry]], a '''stericated 6-orthoplex''' is a convex [[Uniform 6-polytope|uniform 6-polytope]], constructed as a sterication (4th order truncation) of the regular [[6-orthoplex|6-orthoplex]].

There are 16 unique sterications for the 6-orthoplex with permutations of truncations, cantellations, and runcinations. Eight are better represented from the stericated 6-cube.

== Stericated 6-orthoplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Stericated 6-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 2r2r{3,3,3,3,4}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||{{CDD|node_1|3|node|3|node|3|node|3|node_1|4|node}}<BR>{{CDD|node|split1|nodes|3ab|nodes_11|4a|nodea}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells|| 
|-
|bgcolor=#e7dcc3|Faces|| 
|-
|bgcolor=#e7dcc3|Edges||5760
|-
|bgcolor=#e7dcc3|Vertices||960
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [4,3,3,3,3]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Small cellated hexacontatetrapeton (Acronym: scag) (Jonathan Bowers)<ref>Klitzing, (x3o3o3o3x4o - scag)</ref>

=== Images ===

{{6-cube Coxeter plane graphs|t15|150}}

==Steritruncated 6-orthoplex==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Steritruncated 6-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,1,4</sub>{3,3,3,3,4}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||{{CDD|node_1|3|node_1|3|node|3|node|3|node_1|4|node}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces|| 
|-
|bgcolor=#e7dcc3|Cells|| 
|-
|bgcolor=#e7dcc3|Faces|| 
|-
|bgcolor=#e7dcc3|Edges||19200
|-
|bgcolor=#e7dcc3|Vertices||3840
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [4,3,3,3,3]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Cellitruncated hexacontatetrapeton (Acronym: catog) (Jonathan Bowers)<ref>Klitzing, (x3x3o3o3x4o - catog)</ref>

=== Images ===
{{6-cube Coxeter plane graphs|t145|150}}

==Stericantellated 6-orthoplex==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Stericantellated 6-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||t<sub>0,2,4</sub>{3<sup>4</sup>,4}<BR>rr2r{3,3,3,3,4}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||{{CDD|node_1|3|node|3|node_1|3|node|3|node_1|4|node}}{{CDD|node_1|split1|nodes|3ab|nodes_11|4a|nodea}}
|-
|bgcolor=#e7dcc3|5-faces|| 
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces|| 
|-
|bgcolor=#e7dcc3|Edges||28800
|-
|bgcolor=#e7dcc3|Vertices||5760
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [4,3,3,3,3]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Cellirhombated hexacontatetrapeton (Acronym: crag) (Jonathan Bowers)<ref>Klitzing, (x3o3x3o3x4o - crag)</ref>

=== Images ===
{{6-cube Coxeter plane graphs|t135|150}}

==Stericantitruncated 6-orthoplex==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Stericantitruncated 6-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,1,2,4</sub>{3,3,3,3,4}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells|| 
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||46080 
|-
|bgcolor=#e7dcc3|Vertices||11520
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [4,3,3,3,3]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Celligreatorhombated hexacontatetrapeton (Acronym: cagorg) (Jonathan Bowers)<ref>Klitzing, (x3x3x3o3x4o - cagorg)</ref>

=== Images ===
{{6-cube Coxeter plane graphs|t1345|150}}

==Steriruncinated 6-orthoplex==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Steriruncinated 6-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,3,4</sub>{3,3,3,3,4}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||{{CDD|node_1|3|node|3|node|3|node_1|3|node_1|4|node}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||15360
|-
|bgcolor=#e7dcc3|Vertices||3840
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [4,3,3,3,3]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Celliprismated hexacontatetrapeton (Acronym: copog) (Jonathan Bowers)<ref>Klitzing, (x3o3o3x3x4o - copog)</ref>

=== Images ===
{{6-cube Coxeter plane graphs|t125|150}}

==Steriruncitruncated 6-orthoplex==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Steriruncitruncated 6-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 2t2r{3,3,3,3,4}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|4|node}}<BR>{{CDD|node|split1|nodes_11|3ab|nodes_11|4a|nodea}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||40320
|-
|bgcolor=#e7dcc3|Vertices||11520
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [4,3,3,3,3]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Celliprismatotruncated hexacontatetrapeton (Acronym: captog) (Jonathan Bowers)<ref>Klitzing, (x3x3o3x3x4o - captog)</ref>

=== Images ===
{{6-cube Coxeter plane graphs|t1245|150}}

==Steriruncicantellated 6-orthoplex==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Steriruncicantellated 6-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,2,3,4</sub>{3,3,3,3,4}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|4|node}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells|| 
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||40320
|-
|bgcolor=#e7dcc3|Vertices||11520
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [4,3,3,3,3]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Celliprismatorhombated hexacontatetrapeton (Acronym: coprag) (Jonathan Bowers)<ref>Klitzing, (x3o3x3x3x4o - coprag)</ref>

=== Images ===
{{6-cube Coxeter plane graphs|t1235|150}}

==Steriruncicantitruncated 6-orthoplex==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Steriuncicantitruncated 6-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||t<sub>0,1,2,3,4</sub>{3<sup>4</sup>,4}<BR>tr2r{3,3,3,3,4}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|4|node}}{{CDD|node_1|split1|nodes_11|3ab|nodes_11|4a|nodea}}
|-
|bgcolor=#e7dcc3|5-faces||536:<br>12 [[Runcinated 5-orthoplexes#Runcicantitruncated 5-orthoplex|t<sub>0,1,2,3</sub>{3,3,3,4}]][[File:5-cube t1234.svg|40px]]<br>60 {}×t<sub>0,1,2</sub>{3,3,4} 40px×40px<br>160 {6}×t<sub>0,1,2</sub>{3,3} 40px×40px<br>240 {4}×t<sub>0,1,2</sub>{3,3} 40px×40px<br>64 t<sub>0,1,2,3,4</sub>{3<sup>4</sup>}[[File:5-simplex t01234.svg|40px]]<br>
|-
|bgcolor=#e7dcc3|4-faces||8216
|-
|bgcolor=#e7dcc3|Cells||38400
|-
|bgcolor=#e7dcc3|Faces||76800
|-
|bgcolor=#e7dcc3|Edges||69120
|-
|bgcolor=#e7dcc3|Vertices||23040
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||irregular [[5-simplex|5-simplex]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [4,3,3,3,3]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Great cellated hexacontatetrapeton (Acronym: gocog) (Jonathan Bowers)<ref>Klitzing, (x3x3x3x3x4o - gocog)</ref>

=== Images ===
{{6-cube Coxeter plane graphs|t12345|150}}

=== Snub 6-demicube ===

The '''snub 6-demicube''' defined as an [[Alternation (geometry)|alternation]] of the omnitruncated 6-demicube is not uniform, but it can be given Coxeter diagram {{CDD|nodes_hh|split2|node_h|3|node_h|3|node_h|3|node_h}} or {{CDD|node|4|node_h|3|node_h|3|node_h|3|node_h|3|node_h}} and symmetry [3<sup>2,1,1,1</sup>]<sup>+</sup> or [4,(3,3,3,3)<sup>+</sup>], and constructed from 12 snub 5-demicubes, 64 [[Runcinated 5-cell#Full snub 5-simplex|snub 5-simplex]]es, 60 snub 24-cell antiprisms, 160 3-s{3,4} duoantiprisms, 240 2-sr{3,3} duoantiprisms, and 11520 irregular [[5-simplex|5-simplex]]es filling the gaps at the deleted vertices.

== Related polytopes==

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


== Notes==
{{reflist}}

== References==
* [[Biography: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]
* [[Biography: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)}}

== 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]]

{{Sourceattribution|Stericated 6-orthoplexes}}