View source for Pentic 7-cubes

{| class=wikitable align=right width=480
|- align=center valign=top
|[[File:7-demicube_t0 D7.svg|120px]]<BR>[[7-demicube]]<BR>(half 7-cube, h{4,3<sup>5</sup>})<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node}}<BR>{{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node}}
|[[File:7-demicube_t04 D7.svg|120px]]<BR>Pentic 7-cube<BR>h<sub>5</sub>{4,3<sup>5</sup>}<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node}}<BR>{{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1|3|node}}
|[[File:7-demicube_t014 D7.svg|120px]]<BR>Penticantic 7-cube<BR>h<sub>2,5</sub>{4,3<sup>5</sup>}<BR>{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1|3|node}}<BR>{{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1|3|node}}
|- align=center valign=top
|[[File:7-demicube_t024 D7.svg|120px]]<BR>Pentiruncic 7-cube<BR>h<sub>3,5</sub>{4,3<sup>5</sup>}<BR>{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1|3|node}}<BR>{{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1|3|node}}
|[[File:7-demicube_t0124 D7.svg|120px]]<BR>Pentiruncicantic 7-cube<BR>h<sub>2,3,5</sub>{4,3<sup>5</sup>}<BR>{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1|3|node}}<BR>{{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}
|[[File:7-demicube_t034 D7.svg|120px]]<BR>Pentisteric 7-cube<BR>h<sub>4,5</sub>{4,3<sup>5</sup>}<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1|3|node}}<BR>{{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1|3|node}}
|- align=center valign=top
|[[File:7-demicube_t0134 D7.svg|120px]]<BR>Pentistericantic 7-cube<BR>h<sub>2,4,5</sub>{4,3<sup>5</sup>}<BR>{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1|3|node}}<BR>{{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}
|[[File:7-demicube_t0234 D7.svg|120px]]<BR>Pentisteriruncic 7-cube<BR>h<sub>3,4,5</sub>{4,3<sup>5</sup>}<BR>{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1|3|node}}<BR>{{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}
|[[File:7-demicube_t01234 D7.svg|120px]]<BR>Penticsteriruncicantic 7-cube<BR>h<sub>2,3,4,5</sub>{4,3<sup>5</sup>}<BR>{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1|3|node}}<BR>{{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}
|-
!colspan=3|[[Orthogonal projection]]s in D<sub>7</sub> [[Coxeter plane]]
|}
In seven-dimensional [[geometry]], a '''pentic 7-cube''' is a convex [[uniform 7-polytope]], related to the uniform [[7-demicube]]. There are 8 unique forms.
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== Pentic 7-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Pentic 7-cube
|-
|bgcolor=#e7dcc3|Type||[[uniform 7-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,4</sub>{3,3<sup>4,1</sup>}<BR>h<sub>5</sub>{4,3<sup>5</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node}}<br>{{CDD|node_h|4|node|3|node|3|node|3|node|3|node_1|3|node}}
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||13440
|-
|bgcolor=#e7dcc3|Vertices||1344
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D<sub>7</sub>, [3<sup>4,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
=== Cartesian coordinates ===
The [[Cartesian coordinates]] for the vertices of a pentic 7-cube centered at the origin are coordinate permutations:
: (±1,±1,±1,±1,±1,±3,±3)
with an odd number of plus signs.
===Images===
{{7-demicube Coxeter plane graphs|t04|120}}
=== Related polytopes===
{{Pentic cube table}}
== Penticantic 7-cube==
===Images===
{{7-demicube Coxeter plane graphs|t014|120}}
== Pentiruncic 7-cube==
===Images===
{{7-demicube Coxeter plane graphs|t024|120}}
== Pentiruncicantic 7-cube==
===Images===
{{7-demicube Coxeter plane graphs|t0124|120}}
== Pentisteric 7-cube==
===Images===
{{7-demicube Coxeter plane graphs|t034|120}}
== Pentistericantic 7-cube==
===Images===
{{7-demicube Coxeter plane graphs|t0134|120}}
== Pentisteriruncic 7-cube==
===Images===
{{7-demicube Coxeter plane graphs|t0234|120}}
== Pentisteriruncicantic 7-cube==
===Images===
{{7-demicube Coxeter plane graphs|t01234|120}}
== Related polytopes ==
This polytope is based on the [[7-demicube]], a part of a dimensional family of [[uniform polytope]]s called [[demihypercube]]s for being [[Alternation (geometry)|alternation]] of the [[hypercube]] family.
There are 95 uniform polytopes with D<sub>7</sub> symmetry, 63 are shared by the BC<sub>7</sub> symmetry, and 32 are unique:
{{Demihepteract_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|polyexa.htm|7D|uniform polytopes (polyexa)}}
== External links ==
* {{MathWorld|title=Hypercube|urlname=Hypercube}}
* [http://www.polytopes.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
{{Polytopes}}
[[Category:7-polytopes]]