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Runcinated 6-cubes#Runcicantitruncated 6-cube

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

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Runcinated 6-cube
{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node}}

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Biruncinated 6-cube
{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node}}

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Runcinated 6-orthoplex
{{CDD|node|4|node|3|node_1|3|node|3|node|3|node_1}}

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6-orthoplex
{{CDD|node|4|node|3|node|3|node|3|node|3|node_1}}

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Runcitruncated 6-cube
{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node}}

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Biruncitruncated 6-cube
{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node}}

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Runcicantellated 6-orthoplex
{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node_1}}

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Runcicantellated 6-cube
{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node}}

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Biruncitruncated 6-orthoplex
{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node}}

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Runcitruncated 6-orthoplex
{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node_1}}

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Runcicanti-truncated 6-cube
{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node}}

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Biruncicanti-truncated 6-cube
{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node}}

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Runcicanti-truncated 6-orthoplex
{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node_1}}

colspan=5|Orthogonal projections in B6 Coxeter plane

In six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube.

There are 12 unique runcinations of the 6-cube with permutations of truncations, and cantellations. 5 are expressed relative to the dual 6-orthoplex.

Runcinated 6-cube

class="wikitable" align="right" style="margin-left:10px" width="250"
bgcolor=#e7dcc3 align=center colspan=2|Runcinated 6-cube
bgcolor=#e7dcc3|TypeUniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,3{4,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node}}
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges7680
bgcolor=#e7dcc3|Vertices1280
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupB6 [4,3,3,3,3]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Small prismated hexeract (spox) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/spox.htm (o3o3x3o3o4x - spox)]}}

= Images =

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

Biruncinated 6-cube

class="wikitable" align="right" style="margin-left:10px" width="250"
bgcolor=#e7dcc3 align=center colspan=2|Biruncinated 6-cube
bgcolor=#e7dcc3|TypeUniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt1,4{4,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node}}
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges11520
bgcolor=#e7dcc3|Vertices1920
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupB6 [4,3,3,3,3]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Small biprismated hexeractihexacontatetrapeton (sobpoxog) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/sobpoxog.htm (o3x3o3o3x4o - sobpoxog)]}}

= Images =

{{6-cube Coxeter plane graphs|t14|150}}

Runcitruncated 6-cube

class="wikitable" align="right" style="margin-left:10px" width="250"
bgcolor=#e7dcc3 align=center colspan=2|Runcitruncated 6-cube
bgcolor=#e7dcc3|TypeUniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,1,3{4,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node}}
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges17280
bgcolor=#e7dcc3|Vertices3840
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupB6 [4,3,3,3,3]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Prismatotruncated hexeract (potax) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/potax.htm (o3o3x3o3x4x - potax)]}}

= Images =

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

Biruncitruncated 6-cube

class="wikitable" align="right" style="margin-left:10px" width="250"
bgcolor=#e7dcc3 align=center colspan=2|Biruncitruncated 6-cube
bgcolor=#e7dcc3|TypeUniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt1,2,4{4,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node}}
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges23040
bgcolor=#e7dcc3|Vertices5760
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupB6 [4,3,3,3,3]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Biprismatotruncated hexeract (boprag) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/boprag.htm (o3x3o3x3x4o - boprag)]}}

= Images =

{{6-cube Coxeter plane graphs|t124|150}}

Runcicantellated 6-cube

class="wikitable" align="right" style="margin-left:10px" width="250"
bgcolor=#e7dcc3 align=center colspan=2|Runcicantellated 6-cube
bgcolor=#e7dcc3|TypeUniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,2,3{4,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node}}
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges13440
bgcolor=#e7dcc3|Vertices3840
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupB6 [4,3,3,3,3]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Prismatorhombated hexeract (prox) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/prox.htm (o3o3x3x3o4x - prox)]}}

= Images =

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

Runcicantitruncated 6-cube

class="wikitable" align="right" style="margin-left:10px" width="250"
bgcolor=#e7dcc3 align=center colspan=2|Runcicantitruncated 6-cube
bgcolor=#e7dcc3|TypeUniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt0,1,2,3{4,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node}}
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges23040
bgcolor=#e7dcc3|Vertices7680
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupB6 [4,3,3,3,3]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Great prismated hexeract (gippox) (Jonathan Bowers)Klitzing, (o3o3x3x3x4x - gippox)

= Images =

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

Biruncicantitruncated 6-cube

class="wikitable" align="right" style="margin-left:10px" width="250"
bgcolor=#e7dcc3 align=center colspan=2|Biruncicantitruncated 6-cube
bgcolor=#e7dcc3|TypeUniform 6-polytope
bgcolor=#e7dcc3|Schläfli symbolt1,2,3,4{4,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagram{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node}}
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges23040
bgcolor=#e7dcc3|Vertices5760
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupB6 [4,3,3,3,3]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Great biprismated hexeractihexacontitetrapeton (gobpoxog) (Jonathan Bowers)Klitzing, (o3x3x3x3x4o - gobpoxog)

= Images =

{{6-cube Coxeter plane graphs|t1234|150}}

Related polytopes

These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

{{Hexeract family}}

Notes

{{reflist}}

References

{{sfn whitelist|CITEREFKlitzing}}

  • 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}} o3o3x3o3o4x - spox, o3x3o3o3x4o - sobpoxog, o3o3x3o3x4x - potax, o3x3o3x3x4o - boprag, o3o3x3x3o4x - prox, o3o3x3x3x4x - gippox, o3x3x3x3x4o - gobpoxog

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