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Runcic 6-cubes#Runcic 6-cube

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|160px
6-demicube
{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node}}

|160px
Runcic 6-cube
{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node}}

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Runcicantic 6-cube
{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node}}

colspan=4|Orthogonal projections in D6 Coxeter plane

In six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube.

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Runcic 6-cube

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!bgcolor=#e7dcc3 colspan=2|Runcic 6-cube

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,2{3,33,1}
h3{4,34}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node}}
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges3840
bgcolor=#e7dcc3|Vertices640
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsD6, [33,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Cantellated 6-demicube
  • Cantellated demihexeract
  • Small rhombated hemihexeract (Acronym: sirhax) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/sirhax.htm (x3o3o *b3x3o3o - sirhax)]}}

= Cartesian coordinates =

The Cartesian coordinates for the vertices of a runcic 6-cube centered at the origin are coordinate permutations:

: (±1,±1,±1,±3,±3,±3)

with an odd number of plus signs.

= Images =

{{6-demicube Coxeter plane graphs|t02|200}}

= Related polytopes =

{{Runcic cube table}}

Runcicantic 6-cube

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!bgcolor=#e7dcc3 colspan=2|Runcicantic 6-cube

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,1,2{3,33,1}
h2,3{4,34}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node}}
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges5760
bgcolor=#e7dcc3|Vertices1920
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsD6, [33,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Cantitruncated 6-demicube
  • Cantitruncated demihexeract
  • Great rhombated hemihexeract (Acronym: girhax) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/girhax.htm (x3x3o *b3x3o3o - girhax)]}}

= Cartesian coordinates =

The Cartesian coordinates for the vertices of a runcicantic 6-cube centered at the origin are coordinate permutations:

: (±1,±1,±3,±5,±5,±5)

with an odd number of plus signs.

= Images =

{{6-demicube Coxeter plane graphs|t012|200}}

Related polytopes

This polytope is based on the 6-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:

{{Demihexeract_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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 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|polypeta.htm|6D uniform polytopes (polypeta) with acronyms}} x3o3o *b3x3o3o, x3x3o *b3x3o3o {{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:6-polytopes