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Cantellated 7-cubes#Bicantellated 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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Cantellated 7-cube
{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node|3|node}}

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

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

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

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

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

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

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Cantellated 7-orthoplex
{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node|4|node}}

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Bicantellated 7-orthoplex
{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node|4|node}}

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Cantitruncated 7-orthoplex
{{CDD|node_1|3|node_1|3|node_1|3|node|3|node|3|node|4|node}}

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Bicantitruncated 7-orthoplex
{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node|4|node}}

colspan=4|Orthogonal projections in B6 Coxeter plane

In seven-dimensional geometry, a cantellated 7-cube is a convex uniform 7-polytope, being a cantellation of the regular 7-cube.

There are 10 degrees of cantellation for the 7-cube, including truncations. 4 are most simply constructible from the dual 7-orthoplex.

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Cantellated 7-cube

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!bgcolor=#e7dcc3 colspan=2|Cantellated 7-cube

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Schläfli symbolrr{4,3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter diagram{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node|3|node}}
bgcolor=#e7dcc3|6-faces
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges16128
bgcolor=#e7dcc3|Vertices2688
bgcolor=#e7dcc3|Vertex figure
bgcolor=#e7dcc3|Coxeter groupsB7, [4,3,3,3,3,3]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names=

  • Small rhombated hepteract (acronym: sersa) (Jonathan Bowers)Klitizing, (x3o3x3o3o3o4o - sersa)

= Images =

{{B7 Coxeter plane graphs|t02|150}}

Bicantellated 7-cube

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

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolr2r{4,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter diagrams{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node|3|node}}
{{CDD|nodes_11|split2|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;"|Edges40320
style="background:#e7dcc3;"|Vertices6720
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Small birhombated hepteract (acronym: sibrosa) (Jonathan Bowers)Klitizing, (o3x3o3x3o3o4o - sibrosa)

= Images =

{{B7 Coxeter plane graphs|t13|150}}

Tricantellated 7-cube

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

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolr3r{4,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter diagrams{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node|3|node}}
{{CDD|nodes|split2|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;"|Edges47040
style="background:#e7dcc3;"|Vertices6720
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Small trirhombihepteractihecatonicosoctaexon (acronym: strasaz) (Jonathan Bowers)Klitizing, (o3o3x3o3x3o4o - strasaz)

= Images =

{{B7 Coxeter plane graphs|t24|150}}

Cantitruncated 7-cube

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

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symboltr{4,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter diagrams{{CDD|node_1|4|node_1|3|node_1|3|node|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;"|Edges18816
style="background:#e7dcc3;"|Vertices5376
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Great rhombated hepteract (acronym: gersa) (Jonathan Bowers)Klitizing, (x3x3x3o3o3o4o - gersa)

= Images =

{{B7 Coxeter plane graphs|t012|150}}

It is fifth in a series of cantitruncated hypercubes:

{{Cantitruncated hypercube polytopes}}

Bicantitruncated 7-cube

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

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolr2r{4,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter diagrams{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}
{{CDD|nodes_11|split2|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;"|Edges47040
style="background:#e7dcc3;"|Vertices13440
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Great birhombated hepteract (acronym: gibrosa) (Jonathan Bowers)Klitizing, (o3x3x3x3o3o4o - gibrosa)

= Images =

{{B7 Coxeter plane graphs|t123|150}}

Tricantitruncated 7-cube

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

style="background:#e7dcc3;"|Typeuniform 7-polytope
style="background:#e7dcc3;"|Schläfli symbolt3r{4,3,3,3,3,3}
style="background:#e7dcc3;"|Coxeter diagrams{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}
{{CDD|nodes|split2|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;"|Edges53760
style="background:#e7dcc3;"|Vertices13440
style="background:#e7dcc3;"|Vertex figure
style="background:#e7dcc3;"|Coxeter groupsB7, [4,3,3,3,3,3]
style="background:#e7dcc3;"|Propertiesconvex

= Alternate names=

  • Great trirhombihepteractihecatonicosoctaexon (acronym: gotrasaz) (Jonathan Bowers)Klitizing, (o3o3x3x3x3o4o - gotrasaz)

= Images =

{{B7 Coxeter plane graphs|t234|150|NOB7A6}}

Related polytopes

These polytopes are from a family of 127 uniform 7-polytopes with B7 symmetry.

See also

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, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
  • (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|polyexa.htm|7D| uniform polytopes (polyexa)}} x3o3x3o3o3o4o- sersa, o3x3o3x3o3o4o - sibrosa, o3o3x3o3x3o4o - strasaz, x3x3x3o3o3o4o - gersa, o3x3x3x3o3o4o - gibrosa, o3o3x3x3x3o4o - gotrasaz

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