View source for Runcic 5-cubes

{{Short description|Concept in geometry}}
{| class=wikitable align=right width=320
|- align=center valign=top
|[[File:5-cube t0.svg|160px]]<BR>[[5-cube]]<BR>{{CDD|node_1|4|node|3|node|3|node}}
|[[File:5-demicube t02 B5.svg|160px]]<BR>Runcic 5-cube<BR>{{CDD|nodes_10ru|split2|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node}}
|- align=center valign=top
|[[File:5-demicube t0 B5.svg|160px]]<BR>[[5-demicube]]<BR>{{CDD|nodes_10ru|split2|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node}}
|[[File:5-demicube t012 B5.svg|160px]]<BR>Runcicantic 5-cube<BR>{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node}}
|- align=center valign=top
!colspan=4|[[Orthogonal projection]]s in B<sub>5</sub> [[Coxeter plane]]
|}
In six-dimensional [[geometry]], a '''runcic 5-cube''' or ('''runcic 5-demicube''', '''runcihalf 5-cube''') is a convex [[uniform 5-polytope]]. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of [[runcinated 5-cubes]].
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== Runcic 5-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Runcic 5-cube
|-
|bgcolor=#e7dcc3|Type||[[uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| h<sub>3</sub>{4,3,3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodes_10ru|split2|node|3|node_1|3|node}}<BR>{{CDD|node_h1|4|node|3|node|3|node_1|3|node}}
|-
|bgcolor=#e7dcc3|4-faces||42
|-
|bgcolor=#e7dcc3|Cells||360
|-
|bgcolor=#e7dcc3|Faces||880
|-
|bgcolor=#e7dcc3|Edges||720
|-
|bgcolor=#e7dcc3|Vertices||160
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D<sub>5</sub>, [3<sup>2,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
=== Alternate names ===
* Cantellated 5-demicube/demipenteract
* Small rhombated hemipenteract (sirhin) (Jonathan Bowers)<ref>Klitzing, (x3o3o *b3x3o - sirhin)</ref>
=== Cartesian coordinates ===
The [[Cartesian coordinates]] for the 960 vertices of a runcic 5-cubes centered at the origin are coordinate permutations:
: (±1,±1,±1,±3,±3)
with an odd number of plus signs.
===Images===
{{5-demicube Coxeter plane graphs|t02|200}}
=== Related polytopes ===
It has half the vertices of the [[runcinated 5-cube]], as compared here in the B5 Coxeter plane projections:
{| class=wikitable
|- align=center
|[[File:5-demicube t02 B5.svg|240px]]<BR>Runcic 5-cube
|[[File:5-cube t03.svg|240px]]<BR>Runcinated 5-cube
|}
{{Runcic cube table}}
== Runcicantic 5-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Runcicantic 5-cube
|-
|bgcolor=#e7dcc3|Type||[[uniform 5-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>0,1,2</sub>{3,3<sup>2,1</sup>}<BR>h<sub>3</sub>{4,3<sup>3</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node}}{{CDD|node_h1|4|node|3|node_1|3|node_1|3|node}}
|-
|bgcolor=#e7dcc3|4-faces||42
|-
|bgcolor=#e7dcc3|Cells||360
|-
|bgcolor=#e7dcc3|Faces||1040
|-
|bgcolor=#e7dcc3|Edges||1200
|-
|bgcolor=#e7dcc3|Vertices||480
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D<sub>5</sub>, [3<sup>2,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
=== Alternate names ===
* Cantitruncated 5-demicube/demipenteract
* Great rhombated hemipenteract (girhin) (Jonathan Bowers)<ref>Klitzing, (x3x3o *b3x3o - girhin)</ref>
=== Cartesian coordinates ===
The [[Cartesian coordinates]] for the 480 vertices of a runcicantic 5-cube centered at the origin are coordinate permutations:
: (±1,±1,±3,±5,±5)
with an odd number of plus signs.
===Images===
{{5-demicube Coxeter plane graphs|t012|200}}
=== Related polytopes ===
It has half the vertices of the [[runcicantellated 5-cube]], as compared here in the B5 Coxeter plane projections:
{| class=wikitable
|- align=center
|[[File:5-demicube t012 B5.svg|240px]]<BR>Runcicantic 5-cube
|[[File:5-cube t023.svg|240px]]<BR>Runcicantellated 5-cube
|}
== Related polytopes ==
This polytope is based on the [[5-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 23 [[uniform 5-polytope]]s that can be constructed from the D<sub>5</sub> symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the [[5-cube]] family.
{{Demipenteract 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|polytera.htm|5D|uniform polytopes (polytera)}} x3o3o *b3x3o - sirhin, x3x3o *b3x3o - girhin
== 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:5-polytopes]]