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Runcic 5-cubes#Cantellated 5-demicube

{{Short description|Concept in geometry}}

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5-cube
{{CDD|node_1|4|node|3|node|3|node}}

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Runcic 5-cube
{{CDD|nodes_10ru|split2|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node}}

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5-demicube
{{CDD|nodes_10ru|split2|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node}}

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Runcicantic 5-cube
{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node}}

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!colspan=4|Orthogonal projections in B5 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

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

bgcolor=#e7dcc3|Typeuniform 5-polytope
bgcolor=#e7dcc3|Schläfli symbolh3{4,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodes_10ru|split2|node|3|node_1|3|node}}
{{CDD|node_h1|4|node|3|node|3|node_1|3|node}}
bgcolor=#e7dcc3|4-faces42
bgcolor=#e7dcc3|Cells360
bgcolor=#e7dcc3|Faces880
bgcolor=#e7dcc3|Edges720
bgcolor=#e7dcc3|Vertices160
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsD5, [32,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Cantellated 5-demicube/demipenteract
  • Small rhombated hemipenteract (sirhin) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/sirhin.htm (x3o3o *b3x3o - sirhin)]}}

= 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:

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Runcic 5-cube

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Runcinated 5-cube

{{Runcic cube table}}

Runcicantic 5-cube

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

bgcolor=#e7dcc3|Typeuniform 5-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,1,2{3,32,1}
h3{4,33}
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-faces42
bgcolor=#e7dcc3|Cells360
bgcolor=#e7dcc3|Faces1040
bgcolor=#e7dcc3|Edges1200
bgcolor=#e7dcc3|Vertices480
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsD5, [32,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Cantitruncated 5-demicube/demipenteract
  • Great rhombated hemipenteract (girhin) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/girhin.htm (x3x3o *b3x3o - girhin)]}}

= 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:

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Runcicantic 5-cube

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Runcicantellated 5-cube

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 5-polytopes that can be constructed from the D5 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

  • 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, [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html wiley.com], {{isbn|978-0-471-01003-6}}
  • (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 Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
  • {{KlitzingPolytopes|polytera.htm|5D uniform polytopes (polytera) with acronyms}} x3o3o *b3x3o - sirhin, x3x3o *b3x3o - girhin {{sfn whitelist| CITEREFKlitzing}}

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