View source for Pentic 6-cubes

{| class=wikitable align=right width=480 style="margin-left:1em;"
|- align=center valign=top
|[[File:6-demicube t0 D5.svg|160px]]<BR>[[6-demicube|6-demicube]]<BR>(half 6-cube)<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node}}
|[[File:6-demicube t04 D5.svg|160px]]<BR>Pentic 6-cube<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1}}
|[[File:6-demicube t014 D5.svg|160px]]<BR>Penticantic 6-cube<BR>{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1}}
|- align=center valign=top
|[[File:6-demicube t024 D5.svg|160px]]<BR>Pentiruncic 6-cube<BR>{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1}}
|[[File:6-demicube t0124 D5.svg|160px]]<BR>Pentiruncicantic 6-cube<BR>{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1}}
|[[File:6-demicube t034 D5.svg|160px]]<BR>Pentisteric 6-cube<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1}}
|- align=center valign=top
|[[File:6-demicube t0134 D5.svg|160px]]<BR>Pentistericantic 6-cube<BR>{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1}}
|[[File:6-demicube t0234 D5.svg|160px]]<BR>Pentisteriruncic 6-cube<BR>{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1}}
|[[File:6-demicube t01234 D5.svg|160px]]<BR>Pentisteriruncicantic 6-cube<BR>{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1}}
|-
!colspan=3|Orthogonal projections in D<sub>6</sub> Coxeter plane
|}
In six-dimensional [[Geometry|geometry]], a '''pentic 6-cube''' is a convex [[Uniform 6-polytope|uniform 6-polytope]].

There are 8 pentic forms of the 6-cube.

{{TOC left}}
{{-}}
==Pentic 6-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Pentic 6-cube
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,4</sub>{3,3<sup>4,1</sup>}<BR>h<sub>5</sub>{4,3<sup>4</sup>}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||1440
|-
|bgcolor=#e7dcc3|Vertices||192
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D<sub>6</sub>, [3<sup>3,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

The ''pentic 6-cube'', {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1}}, has half of the vertices of a pentellated 6-cube, {{CDD|node_1|4|node|3|node|3|node|3|node|3|node_1}}.
=== Alternate names ===
* Stericated 6-demicube/demihexeract
* Small cellated hemihexeract (Acronym: sochax) (Jonathan Bowers)<ref>Klitzing, (x3o3o *b3o3x3o3o - sochax)</ref>

=== Cartesian coordinates ===
The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:
: (±1,±1,±1,±1,±1,±3)
with an odd number of plus signs.

===Images===

==Penticantic 6-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Penticantic 6-cube
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,1,4</sub>{3,3<sup>4,1</sup>}<BR>h<sub>2,5</sub>{4,3<sup>4</sup>}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|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||9600
|-
|bgcolor=#e7dcc3|Vertices||1920
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D<sub>6</sub>, [3<sup>3,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The ''penticantic 6-cube'', {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1}}, has half of the vertices of a penticantellated 6-cube, {{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node_1}}.
=== Alternate names ===
* Steritruncated 6-demicube/demihexeract
* cellitruncated hemihexeract (Acronym: cathix) (Jonathan Bowers)<ref>Klitzing, (x3x3o *b3o3x3o3o - cathix)</ref>

=== Cartesian coordinates ===
The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:
: (±1,±1,±3,±3,±3,±5)
with an odd number of plus signs.

===Images===

==Pentiruncic 6-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Pentiruncic 6-cube
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,2,4</sub>{3,3<sup>4,1</sup>}<BR>h<sub>3,5</sub>{4,3<sup>4</sup>}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||10560
|-
|bgcolor=#e7dcc3|Vertices||1920
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D<sub>6</sub>, [3<sup>3,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The ''pentiruncic 6-cube'', {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1}}, has half of the vertices of a pentiruncinated 6-cube (penticantellated 6-orthoplex), {{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node_1}}.
=== Alternate names ===
* Stericantellated 6-demicube/demihexeract
* cellirhombated hemihexeract (Acronym: crohax) (Jonathan Bowers)<ref>Klitzing, (x3o3o *b3x3x3o3o - crohax)</ref>

=== Cartesian coordinates ===
The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:
: (±1,±1,±1,±3,±3,±5)
with an odd number of plus signs.

===Images===

==Pentiruncicantic 6-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Pentiruncicantic 6-cube
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,1,2,4</sub>{3,3<sup>2,1</sup>}<BR>h<sub>2,3,5</sub>{4,3<sup>4</sup>}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||20160
|-
|bgcolor=#e7dcc3|Vertices||5760
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D<sub>6</sub>, [3<sup>3,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The ''pentiruncicantic 6-cube'', {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1}}, has half of the vertices of a pentiruncicantellated 6-cube or (pentiruncicantellated 6-orthoplex), {{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node_1}}
=== Alternate names ===
* Stericantitruncated demihexeract, stericantitruncated 7-demicube
* Great cellated hemihexeract (Acronym: cagrohax) (Jonathan Bowers)<ref>Klitzing, (x3x3o *b3x3x3o3o - cagrohax)</ref>

=== Cartesian coordinates ===
The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:
: (±1,±1,±3,±3,±5,±7)
with an odd number of plus signs.

===Images===

==Pentisteric 6-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Pentisteric 6-cube
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,3,4</sub>{3,3<sup>4,1</sup>}<BR>h<sub>4,5</sub>{4,3<sup>4</sup>}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||5280
|-
|bgcolor=#e7dcc3|Vertices||960
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D<sub>6</sub>, [3<sup>3,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The ''pentisteric 6-cube'', {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1}}, has half of the vertices of a pentistericated 6-cube (pentitruncated 6-orthoplex), {{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node_1}}
=== Alternate names ===
* Steriruncinated 6-demicube/demihexeract
* Small cellipriamated hemihexeract (Acronym: cophix) (Jonathan Bowers)<ref>Klitzing, (x3o3o *b3o3x3x3x - cophix)</ref>

=== Cartesian coordinates ===
The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:
: (±1,±1,±1,±1,±3,±5)
with an odd number of plus signs.

===Images===

==Pentistericantic 6-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Pentistericantic 6-cube
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,1,3,4</sub>{3,3<sup>4,1</sup>}<BR>h<sub>2,4,5</sub>{4,3<sup>4</sup>}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||23040
|-
|bgcolor=#e7dcc3|Vertices||5760
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D<sub>6</sub>, [3<sup>3,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The ''pentistericantic 6-cube'', {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1}}, has half of the vertices of a pentistericantellated 6-cube (pentiruncitruncated 6-orthoplex), {{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node_1}}.
=== Alternate names ===
* Steriruncitruncated demihexeract/7-demicube
* cellitruncated hemihexeract (Acronym: capthix) (Jonathan Bowers)<ref>Klitzing, (x3x3o *b3o3x3x3x - capthix)</ref>

=== Cartesian coordinates ===
The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:
: (±1,±1,±3,±3,±5,±7)
with an odd number of plus signs.

===Images===

==Pentisteriruncic 6-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Pentisteriruncic 6-cube
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,2,3,4</sub>{3,3<sup>4,1</sup>}<BR>h<sub>3,4,5</sub>{4,3<sup>4</sup>}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1}}
|-
|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||D<sub>6</sub>, [3<sup>3,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The ''pentisteriruncic 6-cube'', {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1}}, has half of the vertices of a pentisteriruncinated 6-cube (penticantitruncated 6-orthoplex), {{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node_1}}.
=== Alternate names ===
* Steriruncicantellated 6-demicube/demihexeract
* Celliprismatorhombated hemihexeract (Acronym: caprohax) (Jonathan Bowers)<ref>Klitzing, (x3o3o *b3x3x3x3x - caprohax)</ref>

=== Cartesian coordinates ===
The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:
: (±1,±1,±1,±3,±5,±7)
with an odd number of plus signs.

===Images===

==Pentisteriruncicantic 6-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Pentisteriruncicantic 6-cube
|-
|bgcolor=#e7dcc3|Type||[[Uniform 6-polytope|uniform 6-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,1,2,3,4</sub>{3,3<sup>2,1</sup>}<BR>h<sub>2,3,4,5</sub>{4,3<sup>4</sup>}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||34560
|-
|bgcolor=#e7dcc3|Vertices||11520
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D<sub>6</sub>, [3<sup>3,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The ''pentisteriruncicantic 6-cube'', {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1}}, has half of the vertices of a pentisteriruncicantellated 6-cube (pentisteriruncicantitruncated 6-orthoplex), {{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node_1}}.
=== Alternate names ===
* Steriruncicantitruncated 6-demicube/demihexeract
* Great cellated hemihexeract (Acronym: gochax) (Jonathan Bowers)<ref>Klitzing, (x3x3o *b3x3x3x3o - gochax)</ref>

=== Cartesian coordinates ===
The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:
: (±1,±1,±3,±3,±5,±7)
with an odd number of plus signs.

===Images===

== Related polytopes ==

There are 47 uniform polytopes with D<sub>6</sub> symmetry, 31 are shared by the B<sub>6</sub> symmetry, and 16 are unique:

== 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)}} x3o3o *b3o3x3o3o - sochax, x3x3o *b3o3x3o3o - cathix, x3o3o *b3x3x3o3o - crohax, x3x3o *b3x3x3o3o - cagrohax, x3o3o *b3o3x3x3x - cophix, x3x3o *b3o3x3x3x - capthix, x3o3o *b3x3x3x3x - caprohax, x3x3o *b3x3x3x3o - gochax

== External links ==
* {{MathWorld|title=Hypercube|urlname=Hypercube}}
* [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|Pentic 6-cubes}}