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Runcinated 7-cubes#Biruncinated 7-cube

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

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Runcinated 7-cube
{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node|3|node}}

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Biruncinated 7-cube
{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node|3|node}}

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Runcitruncated 7-cube
{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node|3|node}}

|160px
Biruncitruncated 7-cube
{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node}}

|160px
Runcicantellated 7-cube
{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}

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|160px
Biruncicantellated 7-cube
{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node}}

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Runcicantitruncated 7-cube
{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}

|160px
Biruncicantitruncated 7-cube
{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}}

colspan=3|Orthogonal projections in B7 Coxeter plane

In seven-dimensional geometry, a runcinated 7-cube is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-cube.

There are 16 unique runcinations of the 7-cube with permutations of truncations, and cantellations. 8 are more simply constructed from the 7-orthoplex.

These polytopes are among 127 uniform 7-polytopes with B7 symmetry.

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Runcinated 7-cube

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! style="background:#e7dcc3;" colspan="2"|Runcinated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,3{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges33600
style="background:#e7dcc3;"|Vertices4480
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Small prismated hepteract (acronym: spesa) (Jonathan Bowers)Klitzing, (x3o3o3x3o3o4o - spesa)

= Images =

{{B7 Coxeter plane graphs|t03|150}}

Biruncinated 7-cube

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! style="background:#e7dcc3;" colspan="2"|Biruncinated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt1,4{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges67200
style="background:#e7dcc3;"|Vertices8960
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Small biprismated hepteract (Acronym sibposa) (Jonathan Bowers)Klitzing, (o3x3o3o3x3o4o - sibposa)

= Images =

{{B7 Coxeter plane graphs|t14|150}}

Runcitruncated 7-cube

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! style="background:#e7dcc3;" colspan="2"|Runcitruncated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,3{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges73920
style="background:#e7dcc3;"|Vertices13440
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Prismatotruncated hepteract (acronym: petsa) (Jonathan Bowers)Klitzing, (x3x3o3x3o3o4o - petsa)

= Images =

{{B7 Coxeter plane graphs|t013|150}}

== Biruncitruncated 7-cube ==

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! style="background:#e7dcc3;" colspan="2"|Biruncitruncated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt1,2,4{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges134400
style="background:#e7dcc3;"|Vertices26880
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Biprismatotruncated hepteract (acronym: biptesa) (Jonathan Bowers)Klitzing, (o3x3x3o3x3o4o - biptesa)

= Images =

{{B7 Coxeter plane graphs|t124|150}}

Runcicantellated 7-cube

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! style="background:#e7dcc3;" colspan="2"|Runcicantellated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,2,3{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges53760
style="background:#e7dcc3;"|Vertices13440
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Prismatorhombated hepteract (acronym: parsa) (Jonathan Bowers)Klitzing, (x3o3x3x3o3o4o - parsa)

= Images =

{{B7 Coxeter plane graphs|t023|150}}

Biruncicantellated 7-cube

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! style="background:#e7dcc3;" colspan="2"|biruncicantellated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt1,3,4{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges120960
style="background:#e7dcc3;"|Vertices26880
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Biprismatorhombated hepteract (acronym: bopresa) (Jonathan Bowers)Klitzing, (o3o3x3x3o3x4o - bopresa)

= Images =

{{B7 Coxeter plane graphs|t134|150}}

Runcicantitruncated 7-cube

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! style="background:#e7dcc3;" colspan="2"|Runcicantitruncated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt0,1,2,3{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges94080
style="background:#e7dcc3;"|Vertices26880
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Great prismated hepteract (acronym: gapsa) (Jonathan Bowers)Klitzing, (x3x3x3x3o3o4o - gapsa)

= Images =

{{B7 Coxeter plane graphs|t0123|150}}

Biruncicantitruncated 7-cube

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! style="background:#e7dcc3;" colspan="2"|biruncicantitruncated 7-cube

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt1,2,3,4{4,35}
style="background:#e7dcc3;"|Coxeter-Dynkin diagrams{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}}
style="background:#e7dcc3;"|6-faces
style="background:#e7dcc3;"|5-faces
style="background:#e7dcc3;"|4-faces
style="background:#e7dcc3;"|Cells
style="background:#e7dcc3;"|Faces
style="background:#e7dcc3;"|Edges188160
style="background:#e7dcc3;"|Vertices53760
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,35]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names =

  • Great biprismated hepteract (acronym: gibposa) (Jonathan Bowers)Klitzing, (o3x3x3x3x3o3o - gibposa)

= Images =

{{B7 Coxeter plane graphs|t1234|150}}

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. (1966)
  • {{KlitzingPolytopes|polyexa.htm|7D|uniform polytopes (polyexa)}} x3o3o3x3o3o4o - spo, o3x3o3o3x3o4o - sibpo, x3x3o3x3o3o4o - patto, o3x3x3o3x3o4o - bipto, x3o3x3x3o3o4o - paro, x3x3x3x3o3o4o - gapo, o3x3x3x3x3o3o- gibpo

External links

  • [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
  • [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:7-polytopes