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Steric 5-cubes

{{Use Oxford spelling|date=May 2022}}

{{Use dmy dates|date=May 2022|cs1-dates=y}}

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|{{ubl|160px|5-cube|{{CDD|node_1|4|node|3|node|3|node|3|node}}}}

|{{ubl|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|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}}}}

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|{{ubl|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|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|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, a steric 5-cube or (steric 5-demicube or sterihalf 5-cube) is a convex uniform 5-polytope. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half the vertices of stericated 5-cubes.

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Steric 5-cube

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!bgcolor=#e7dcc3 colspan=2|Steric 5-cube

bgcolor=#e7dcc3|Typeuniform 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-faces82
bgcolor=#e7dcc3|Cells480
bgcolor=#e7dcc3|Faces720
bgcolor=#e7dcc3|Edges400
bgcolor=#e7dcc3|Vertices80
bgcolor=#e7dcc3|Vertex figure{3,3}-t{{sub|1}}{3,3} antiprism
bgcolor=#e7dcc3|Coxeter groupsD{{sub|5}}, [3{{sup|2,1,1}}]
bgcolor=#e7dcc3|Propertiesconvex

= 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=

{{Steric cube table}}

Stericantic 5-cube

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!bgcolor=#e7dcc3 colspan=2|Stericantic 5-cube

bgcolor=#e7dcc3|Typeuniform 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-faces82
bgcolor=#e7dcc3|Cells720
bgcolor=#e7dcc3|Faces1840
bgcolor=#e7dcc3|Edges1680
bgcolor=#e7dcc3|Vertices480
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsD{{sub|5}}, [3{{sup|2,1,1}}]
bgcolor=#e7dcc3|Propertiesconvex

= 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

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!bgcolor=#e7dcc3 colspan=2|Steriruncic 5-cube

bgcolor=#e7dcc3|Typeuniform 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-faces82
bgcolor=#e7dcc3|Cells560
bgcolor=#e7dcc3|Faces1280
bgcolor=#e7dcc3|Edges1120
bgcolor=#e7dcc3|Vertices320
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsD{{sub|5}}, [3{{sup|2,1,1}}]
bgcolor=#e7dcc3|Propertiesconvex

= 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

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!bgcolor=#e7dcc3 colspan=2|Steriruncicantic 5-cube

bgcolor=#e7dcc3|Typeuniform 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-faces82
bgcolor=#e7dcc3|Cells720
bgcolor=#e7dcc3|Faces2080
bgcolor=#e7dcc3|Edges2400
bgcolor=#e7dcc3|Vertices960
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsD{{sub|5}}, [3{{sup|2,1,1}}]
bgcolor=#e7dcc3|Propertiesconvex

= 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, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the 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 family.

{{Demipenteract family}}

References

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Further reading

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  • {{Cite book |last=Coxeter |first=H. S. M. |url={{GBurl|id=iWvXsVInpgMC}} |title=Regular Polytopes |date=1973 |publisher=Dover |edition=3rd |location=New York City |author-link=Harold Scott MacDonald Coxeter |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=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 |author-link=Harold Scott MacDonald Coxeter |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. |author-link=Harold Scott MacDonald Coxeter |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=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. |author-link=Harold Scott MacDonald Coxeter |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=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. |author-link=Harold Scott MacDonald Coxeter |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=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 |author-link=Norman Johnson (mathematician)}}
  • {{Cite thesis |last=Johnson |first=Norman W. |title=The Theory of Uniform Polytopes and Honeycombs |date=1966 |degree=PhD |publisher=University of Toronto |url={{GBurl|id=PzOFswEACAAJ}} |language=en-CA |author-link=Norman Johnson (mathematician) |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