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Rectified 7-cubes#Rectified 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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Rectified 7-cube
{{CDD|node|4|node_1|3|node|3|node|3|node|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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Trirectified 7-cube
{{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node}}

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

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

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

colspan=4|Orthogonal projections in B7 Coxeter plane

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

There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Vertices of the rectified 7-cube are located at the edge-centers of the 7-ocube. Vertices of the birectified 7-cube are located in the square face centers of the 7-cube. Vertices of the trirectified 7-cube are located in the cube cell centers of the 7-cube.

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Rectified 7-cube

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

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Schläfli symbolr{4,3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node}}
{{CDD|nodes_11|split2|node|3|node|3|node|3|node|3|node}}
bgcolor=#e7dcc3|6-faces128 + 14
bgcolor=#e7dcc3|5-faces896 + 84
bgcolor=#e7dcc3|4-faces2688 + 280
bgcolor=#e7dcc3|Cells4480 + 560
bgcolor=#e7dcc3|Faces4480 + 672
bgcolor=#e7dcc3|Edges2688
bgcolor=#e7dcc3|Vertices448
bgcolor=#e7dcc3|Vertex figure5-simplex prism
bgcolor=#e7dcc3|Coxeter groupsB7, [3,3,3,3,3,4]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names=

  • rectified hepteract (Acronym rasa) (Jonathan Bowers)Klitzing, (o3o3o3o3o3x4o - rasa)

= Images =

{{7-cube Coxeter plane graphs|t1|150}}

= Cartesian coordinates =

Cartesian coordinates for the vertices of a rectified 7-cube, centered at the origin, edge length \sqrt{2}\ are all permutations of:

: (±1,±1,±1,±1,±1,±1,0)

Birectified 7-cube

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

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Coxeter symbol0411
bgcolor=#e7dcc3|Schläfli symbol2r{4,3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node}}
{{CDD|nodes|split2|node_1|3|node|3|node|3|node|3|node}}
bgcolor=#e7dcc3|6-faces128 + 14
bgcolor=#e7dcc3|5-faces448 + 896 + 84
bgcolor=#e7dcc3|4-faces2688 + 2688 + 280
bgcolor=#e7dcc3|Cells6720 + 4480 + 560
bgcolor=#e7dcc3|Faces8960 + 4480
bgcolor=#e7dcc3|Edges6720
bgcolor=#e7dcc3|Vertices672
bgcolor=#e7dcc3|Vertex figure{3}x{3,3,3}
bgcolor=#e7dcc3|Coxeter groupsB7, [3,3,3,3,3,4]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names=

  • Birectified hepteract (Acronym bersa) (Jonathan Bowers)Klitzing, (o3o3o3o3x3o4o - bersa)

= Images =

{{7-cube Coxeter plane graphs|t2|150}}

= Cartesian coordinates =

Cartesian coordinates for the vertices of a birectified 7-cube, centered at the origin, edge length \sqrt{2}\ are all permutations of:

: (±1,±1,±1,±1,±1,0,0)

Trirectified 7-cube

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

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Schläfli symbol3r{4,3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node}}
{{CDD|nodes|split2|node|3|node_1|3|node|3|node|3|node}}
bgcolor=#e7dcc3|6-faces128 + 14
bgcolor=#e7dcc3|5-faces448 + 896 + 84
bgcolor=#e7dcc3|4-faces672 + 2688 + 2688 + 280
bgcolor=#e7dcc3|Cells3360 + 6720 + 4480
bgcolor=#e7dcc3|Faces6720 + 8960
bgcolor=#e7dcc3|Edges6720
bgcolor=#e7dcc3|Vertices560
bgcolor=#e7dcc3|Vertex figure{3,3}x{3,3}
bgcolor=#e7dcc3|Coxeter groupsB7, [3,3,3,3,3,4]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names=

  • Trirectified hepteract
  • Trirectified 7-orthoplex
  • Trirectified heptacross (Acronym sez) (Jonathan Bowers)Klitzing, (o3o3o3x3o3o4o - sez)

= Images =

{{7-cube Coxeter plane graphs|t3|150}}

= Cartesian coordinates =

Cartesian coordinates for the vertices of a trirectified 7-cube, centered at the origin, edge length \sqrt{2}\ are all permutations of:

: (±1,±1,±1,±1,0,0,0)

= Related polytopes=

{{2-isotopic_uniform_hypercube_polytopes}}

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|polyexa.htm|7D|uniform polytopes (polyexa)}} o3o3o3x3o3o4o - sez, o3o3o3o3x3o4o - bersa, o3o3o3o3o3x4o - rasa

External links

  • [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:7-polytopes