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Steric 6-cubes#Runcinated 6-demicube

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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}}

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

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

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

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

colspan=3|Orthogonal projections in D5 Coxeter plane

In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube.

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Steric 6-cube

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,3{3,33,1}
h4{4,34}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node}}
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges3360
bgcolor=#e7dcc3|Vertices480
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsD6, [33,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Runcinated demihexeract
  • Runcinated 6-demicube
  • Small prismated hemihexeract (Acronym: sophax) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/sophax.htm (x3o3o *b3o3x3o - sophax)]}}

= Cartesian coordinates =

The Cartesian coordinates for the 480 vertices of a steric 6-cube centered at the origin are coordinate permutations:

: (±1,±1,±1,±1,±1,±3)

with an odd number of plus signs.

= Images =

{{6-demicube Coxeter plane graphs|t03|150}}

= Related polytopes=

{{Steric cube table}}

Stericantic 6-cube

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,1,3{3,33,1}
h2,4{4,34}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node}}
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges12960
bgcolor=#e7dcc3|Vertices2880
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsD6, [33,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Runcitruncated demihexeract
  • Runcitruncated 6-demicube
  • Prismatotruncated hemihexeract (Acronym: pithax) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/pithax.htm (x3x3o *b3o3x3o - pithax)]}}

= Cartesian coordinates =

The Cartesian coordinates for the 2880 vertices of a stericantic 6-cube centered at the origin are coordinate permutations:

: (±1,±1,±1,±3,±3,±5)

with an odd number of plus signs.

= Images =

{{6-demicube Coxeter plane graphs|t013|150}}

Steriruncic 6-cube

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,2,3{3,33,1}
h3,4{4,34}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node}}
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges7680
bgcolor=#e7dcc3|Vertices1920
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsD6, [33,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Runcicantellated demihexeract
  • Runcicantellated 6-demicube
  • Prismatorhombated hemihexeract (Acronym: prohax) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/prohax.htm (x3o3o *b3x3x3o - prohax)]}}

= Cartesian coordinates =

The Cartesian coordinates for the 1920 vertices of a steriruncic 6-cube centered at the origin are coordinate permutations:

: (±1,±1,±1,±1,±3,±5)

with an odd number of plus signs.

= Images =

{{6-demicube Coxeter plane graphs|t023|150}}

Steriruncicantic 6-cube

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

bgcolor=#e7dcc3|Typeuniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,1,2,3{3,32,1}
h2,3,4{4,34}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node}}
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges17280
bgcolor=#e7dcc3|Vertices5760
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsD6, [33,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Runcicantitruncated demihexeract
  • Runcicantitruncated 6-demicube
  • Great prismated hemihexeract (Acronym: gophax) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/gophax.htm (x3x3o *b3x3x3o - gophax)]}}

= Cartesian coordinates =

The Cartesian coordinates for the 5760 vertices of a steriruncicantic 6-cube centered at the origin are coordinate permutations:

: (±1,±1,±1,±3,±5,±7)

with an odd number of plus signs.

= Images =

{{6-demicube Coxeter plane graphs|t0123|150}}

Related polytopes

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 *b3o3x3o - sophax, x3x3o *b3o3x3o - pithax, x3o3o *b3x3x3o - prohax, x3x3o *b3x3x3o - gophax {{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