View source for Steric 5-cubes

{{Use Oxford spelling|date=May 2022}}
{| class=wikitable align=right width=480 style="margin-left:1em;"
|- align=center valign=top
|{{ubl|[[File:5-cube t0.svg|160px]]|[[5-cube|5-cube]]|{{CDD|node_1|4|node|3|node|3|node|3|node}}}}
|{{ubl|[[File:5-demicube t03 B5.svg|160px]]|{{small|Steric 5-cube}}|{{CDD|nodes_10ru|split2|node|3|node|3|node_1}}|{{CDD|node_h1|4|node|3|node|3|node|3|node_1}}}}
|{{ubl|[[File:5-demicube t013 B5.svg|160px]]|{{small|Stericantic 5-cube}}|{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1}}|{{CDD|node_h1|4|node|3|node_1|3|node|3|node_1}}}}
|- align=center valign=top
|{{ubl|[[File:5-demicube t0 B5.svg|160px]]|{{small|Half 5-cube}}|{{CDD|nodes_10ru|split2|node|3|node|3|node}}|{{CDD|node_h1|4|node|3|node|3|node|3|node}}}}
|{{ubl|[[File:5-demicube t023 B5.svg|160px]]|{{small|Steriruncic 5-cube}}|{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1}}|{{CDD|node_h1|4|node|3|node|3|node_1|3|node_1}}}}
|{{ubl|[[File:5-demicube t0123 B5.svg|160px]]|{{small|Steriruncicantic 5-cube}}|{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1}}|{{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1}}}}
|-
!colspan=3|Orthogonal projections in B{{sub|5}} Coxeter plane
|}

In five-dimensional [[Geometry|geometry]], a '''steric 5-cube''' or ('''steric 5-demicube''' or '''sterihalf 5-cube''') is a convex [[Uniform 5-polytope|uniform 5-polytope]]. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half the vertices of [[Stericated 5-cubes|stericated 5-cubes]].

{{TOC left}}
{{-}}
==Steric 5-cube==

{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Steric 5-cube
|-
|bgcolor=#e7dcc3|Type||uniform polyteron
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {{ubl|t{{sub|0,3}}{3,3{{sup|2,1}}}|h{{sub|4}}{4,3,3,3}}}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{ubl|{{CDD|nodes_10ru|split2|node|3|node|3|node_1}}|{{CDD|node_h1|4|node|3|node|3|node|3|node_1}}}}
|-
|bgcolor=#e7dcc3|4-faces||82
|-
|bgcolor=#e7dcc3|Cells||480
|-
|bgcolor=#e7dcc3|Faces||720
|-
|bgcolor=#e7dcc3|Edges||400
|-
|bgcolor=#e7dcc3|Vertices||80
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||{3,3}-t{{sub|1}}{3,3} antiprism
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D{{sub|5}}, [3{{sup|2,1,1}}]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Steric penteract, runcinated demipenteract
* Small prismated hemipenteract (siphin) (Jonathan Bowers){{refn|name="Klitzing"|{{KlitzingPolytopes|polytera.htm|5D|uniform polytopes (polytera)}}}}{{rp|(x3o3o *b3o3x - siphin)}}

=== Cartesian coordinates ===
The Cartesian coordinates for the 80 vertices of a steric 5-cube centered at the origin are the permutations of
: (±1,±1,±1,±1,±3)
with an odd number of plus signs.

===Images===
{{5-demicube Coxeter plane graphs|t03|200}}
=== Related polytopes===

==Stericantic 5-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Stericantic 5-cube
|-
|bgcolor=#e7dcc3|Type||uniform polyteron
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {{ubl|t{{sub|0,1,3}}{3,3{{sup|2,1}}}|h{{sub|2,4}}{4,3,3,3}}}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{ubl|{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1}}|{{CDD|node_h1|4|node|3|node_1|3|node|3|node_1}}}}
|-
|bgcolor=#e7dcc3|4-faces||82
|-
|bgcolor=#e7dcc3|Cells||720
|-
|bgcolor=#e7dcc3|Faces||1840
|-
|bgcolor=#e7dcc3|Edges||1680
|-
|bgcolor=#e7dcc3|Vertices||480
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D{{sub|5}}, [3{{sup|2,1,1}}]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
=== Alternate names ===
* Prismatotruncated hemipenteract (pithin) (Jonathan Bowers){{r|Klitzing|p=(x3x3o *b3o3x - pithin)}}

=== Cartesian coordinates ===
The Cartesian coordinates for the 480 vertices of a stericantic 5-cube centered at the origin are coordinate permutations:
: (±1,±1,±3,±3,±5)
with an odd number of plus signs.

===Images===
{{5-demicube Coxeter plane graphs|t013|200}}

==Steriruncic 5-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Steriruncic 5-cube
|-
|bgcolor=#e7dcc3|Type||uniform polyteron
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {{ubl|t{{sub|0,2,3}}{3,3{{sup|2,1}}}|h{{sub|3,4}}{4,3,3,3}}}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{ubl|{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1}}|{{CDD|node_h1|4|node|3|node|3|node_1|3|node_1}}}}
|-
|bgcolor=#e7dcc3|4-faces||82
|-
|bgcolor=#e7dcc3|Cells||560
|-
|bgcolor=#e7dcc3|Faces||1280
|-
|bgcolor=#e7dcc3|Edges||1120
|-
|bgcolor=#e7dcc3|Vertices||320
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D{{sub|5}}, [3{{sup|2,1,1}}]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
=== Alternate names ===
* Prismatorhombated hemipenteract (pirhin) (Jonathan Bowers){{r|Klitzing|p=(x3o3o *b3x3x - pirhin)}}

=== Cartesian coordinates ===
The Cartesian coordinates for the 320 vertices of a steriruncic 5-cube centered at the origin are coordinate permutations:
: (±1,±1,±1,±3,±5)
with an odd number of plus signs.

===Images===
{{5-demicube Coxeter plane graphs|t023|200}}

==Steriruncicantic 5-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Steriruncicantic 5-cube
|-
|bgcolor=#e7dcc3|Type||uniform polyteron
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {{ubl|t{{sub|0,1,2,3}}{3,3{{sup|2,1}}}|h{{sub|2,3,4}}{4,3,3,3}}}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagram||{{ubl|{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1}}|{{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1}}}}
|-
|bgcolor=#e7dcc3|4-faces||82
|-
|bgcolor=#e7dcc3|Cells||720
|-
|bgcolor=#e7dcc3|Faces||2080
|-
|bgcolor=#e7dcc3|Edges||2400
|-
|bgcolor=#e7dcc3|Vertices||960
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||D{{sub|5}}, [3{{sup|2,1,1}}]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
=== Alternate names ===
* Great prismated hemipenteract (giphin) (Jonathan Bowers){{r|Klitzing|p=(x3x3o *b3x3x - giphin)}}

=== Cartesian coordinates ===
The Cartesian coordinates for the 960 vertices of a steriruncicantic 5-cube centered at the origin are coordinate permutations:
: (±1,±1,±3,±5,±7)
with an odd number of plus signs.

===Images===
{{5-demicube Coxeter plane graphs|t0123|200}}

== Related polytopes ==

This polytope is based on the [[5-demicube|5-demicube]], a part of a dimensional family of uniform polytopes called [[Demihypercube|demihypercube]]s for being [[Alternation (geometry)|alternation]] of the [[Hypercube|hypercube]] family.

There are 23 uniform polytera (uniform 5-polytope) that can be constructed from the D{{sub|5}} symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the [[5-cube|5-cube]] family.

==References==
{{reflist}}

==Further reading==
{{refbegin}}
* {{Cite book |last=Coxeter |first=H. S. M. |url={{GBurl|id=iWvXsVInpgMC}} |title=Regular Polytopes |date=1973 |publisher=Dover |edition=3rd |location={{wipe|New York City}} |access-date=2022-05-19}}
* {{Cite book |last=Coxeter |first=H. S. M. |url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html |title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter |date=1995-05-17 |publisher={{wipe|John Wiley & Sons}} |isbn=978-0-471-01003-6 |editor-last1=Sherk |editor-first1=F. Arthur |series=Canadian Mathematical Society Series of Monographs and Advanced Texts |language=en-CA |lccn=94047368 |oclc=632987525 |ol=7598569M |access-date=2022-05-19 |editor-last2=McMullen |editor-first2=Peter |editor-last3=Thompson |editor-first3=Anthony C. |editor-last4=Weiss |editor-first4=Asia Ivić}}
* {{Cite journal |last=Coxeter |first=H. S. M. |date=1940-12-01 |title=Regular and Semi Regular Polytopes I |url=https://link.springer.com/article/10.1007/BF01181449 |journal=Mathematische Zeitschrift |language=en-CA |publisher=[[Company:Springer Nature|Springer Nature]] |volume=46 |pages=380–407 |doi=10.1007/BF01181449 |s2cid=186237114 |issn=1432-1823 |url-access=subscription |access-date=2022-05-19}}
* {{Cite journal |last=Coxeter |first=H. S. M. |date=1985-12-01 |title=Regular and Semi-Regular Polytopes II |url=https://link.springer.com/article/10.1007/BF01161657 |journal=Mathematische Zeitschrift |language=en-CA |publisher=[[Company:Springer Nature|Springer Nature]] |volume=188 |issue=4 |pages=559–591 |doi=10.1007/BF01161657 |s2cid=120429557 |issn=1432-1823 |url-access=subscription |access-date=2022-05-19}}
* {{Cite journal |last=Coxeter |first=H. S. M. |date=1988-03-01 |title=Regular and Semi-Regular Polytopes III |url=https://link.springer.com/article/10.1007/BF01161745 |journal=Mathematische Zeitschrift |language=en-CA |publisher=[[Company:Springer Nature|Springer Nature]] |volume=200 |issue=1 |pages=3–45 |doi=10.1007/BF01161745 |s2cid=186237142 |issn=1432-1823 |url-access=subscription |access-date=2022-05-19}}
* {{Cite thesis |last=Johnson |first=Norman W. |title=Uniform Polytopes |date=1991 |degree=Unfinished manuscript |language=en-CA |}}
* {{Cite thesis |last=Johnson |first=Norman W. |title=The Theory of Uniform Polytopes and Honeycombs |date=1966 |degree=PhD |publisher=[[Organization:University of Toronto|University of Toronto]] |url={{GBurl|id=PzOFswEACAAJ}} |language=en-CA |access-date=2022-05-19}}
{{refend}} 

== 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]]

{{Sourceattribution|Steric 5-cubes}}