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Truncated 7-cubes#Truncated 7-cube

{{Short description|Uniform 7- polytope}}

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

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

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

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

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

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

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

colspan=4|Orthogonal projections in B7 Coxeter plane

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

There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the square faces of the 7-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 7-cube. The final three truncations are best expressed relative to the 7-orthoplex.

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Truncated 7-cube

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

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Schläfli symbolt{4,35}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|4|node_1|3|node|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|Edges3136
bgcolor=#e7dcc3|Vertices896
bgcolor=#e7dcc3|Vertex figureElongated 5-simplex pyramid
bgcolor=#e7dcc3|Coxeter groupsB7, [35,4]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names=

  • Truncated hepteract (Jonathan Bowers)Klitizing (x3x3o3o3o3o4o - taz)

= Coordinates =

Cartesian coordinates for the vertices of a truncated 7-cube, centered at the origin, are all sign and coordinate permutations of

: (1,1+√2,1+√2,1+√2,1+√2,1+√2,1+√2)

= Images =

{{7-cube Coxeter plane graphs|t01|150}}

= Related polytopes =

The truncated 7-cube, is sixth in a sequence of truncated hypercubes:

{{Truncated hypercube polytopes}}

Bitruncated 7-cube

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

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Schläfli symbol2t{4,35}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node|3|node}}
{{CDD|nodes_11|split2|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|Edges9408
bgcolor=#e7dcc3|Vertices2688
bgcolor=#e7dcc3|Vertex figure{ }v{3,3,3}
bgcolor=#e7dcc3|Coxeter groupsB7, [35,4]
D7, [34,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names=

  • Bitruncated hepteract (Jonathan Bowers)Klitizing (o3x3x3o3o3o4o - botaz)

= Coordinates =

Cartesian coordinates for the vertices of a bitruncated 7-cube, centered at the origin, are all sign and coordinate permutations of

: (±2,±2,±2,±2,±2,±1,0)

= Images =

{{7-cube Coxeter plane graphs|t12|150}}

= Related polytopes =

The bitruncated 7-cube is fifth in a sequence of bitruncated hypercubes:

{{Bitruncated hypercube polytopes}}

Tritruncated 7-cube

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

bgcolor=#e7dcc3|Typeuniform 7-polytope
bgcolor=#e7dcc3|Schläfli symbol3t{4,35}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}
{{CDD|nodes|split2|node_1|3|node_1|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|Edges13440
bgcolor=#e7dcc3|Vertices3360
bgcolor=#e7dcc3|Vertex figure{4}v{3,3}
bgcolor=#e7dcc3|Coxeter groupsB7, [35,4]
D7, [34,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names=

  • Tritruncated hepteract (Jonathan Bowers)Klitizing (o3o3x3x3o3o4o - totaz)

= Coordinates =

Cartesian coordinates for the vertices of a tritruncated 7-cube, centered at the origin, are all sign and coordinate permutations of

: (±2,±2,±2,±2,±1,0,0)

= Images =

{{7-cube Coxeter plane graphs|t23|150}}

Notes

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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)}} o3o3o3o3o3x4x - taz, o3o3o3o3x3x4o - botaz, o3o3o3x3x3o4o - totaz

External links

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

{{polytopes}}

{{DEFAULTSORT:Truncated 7-Cube}}

Category:7-polytopes