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Truncated 8-cubes#Truncated 8-cube

{{Short description|Convex uniform 8-polytope in 8-dimensional geometry}}

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

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Truncated 8-cube
{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}

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Bitruncated 8-cube
{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node|3|node|3|node}}

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Quadritruncated 8-cube
{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node|4|node}}

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Tritruncated 8-cube
{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node|3|node|3|node}}

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Tritruncated 8-orthoplex
{{CDD|node|4|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node}}

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Bitruncated 8-orthoplex
{{CDD|node|4|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node}}

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Truncated 8-orthoplex
{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1}}

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

colspan=5|Orthogonal projections in B8 Coxeter plane

In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.

There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edge of the 8-cube. Vertices of the bitruncated 8-cube are located on the square faces of the 8-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 8-cube. The final truncations are best expressed relative to the 8-orthoplex.

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Truncated 8-cube

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!bgcolor=#e7dcc3 colspan=2|Truncated 8-cube

bgcolor=#e7dcc3|Typeuniform 8-polytope
bgcolor=#e7dcc3|Schläfli symbolt{4,3,3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|4|node_1|3|node|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|Edges
bgcolor=#e7dcc3|Vertices
bgcolor=#e7dcc3|Vertex figure( )v{3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter groupsB8, [3,3,3,3,3,3,4]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Truncated octeract (acronym tocto) (Jonathan Bowers)Klitizing, (o3o3o3o3o3o3x4x – tocto)

= Coordinates =

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of

: (±2,±2,±2,±2,±2,±2,±1,0)

= Images =

{{8-cube Coxeter plane graphs|t01|200}}

= Related polytopes =

The truncated 8-cube, is seventh in a sequence of truncated hypercubes:

{{Truncated hypercube polytopes}}

Bitruncated 8-cube

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!bgcolor=#e7dcc3 colspan=2|Bitruncated 8-cube

bgcolor=#e7dcc3|Typeuniform 8-polytope
bgcolor=#e7dcc3|Schläfli symbol2t{4,3,3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|4|node_1|3|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|Edges
bgcolor=#e7dcc3|Vertices
bgcolor=#e7dcc3|Vertex figure{ }v{3,3,3,3}
bgcolor=#e7dcc3|Coxeter groupsB8, [3,3,3,3,3,3,4]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Bitruncated octeract (acronym bato) (Jonathan Bowers)Klitizing, (o3o3o3o3o3x3x4o – bato)

= Coordinates =

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of

: (±2,±2,±2,±2,±2,±1,0,0)

= Images =

{{8-cube Coxeter plane graphs|t12|200}}

= Related polytopes =

The bitruncated 8-cube is sixth in a sequence of bitruncated hypercubes:

{{Bitruncated hypercube polytopes}}

Tritruncated 8-cube

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!bgcolor=#e7dcc3 colspan=2|Tritruncated 8-cube

bgcolor=#e7dcc3|Typeuniform 8-polytope
bgcolor=#e7dcc3|Schläfli symbol3t{4,3,3,3,3,3,3}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|4|node|3|node_1|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|Edges
bgcolor=#e7dcc3|Vertices
bgcolor=#e7dcc3|Vertex figure{4}v{3,3,3}
bgcolor=#e7dcc3|Coxeter groupsB8, [3,3,3,3,3,3,4]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Tritruncated octeract (acronym tato) (Jonathan Bowers)Klitizing, (o3o3o3o3x3x3o4o – tato)

= Coordinates =

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of

: (±2,±2,±2,±2,±1,0,0,0)

= Images =

{{8-cube Coxeter plane graphs|t23|200}}

Quadritruncated 8-cube

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!bgcolor=#e7dcc3 colspan=2|Quadritruncated 8-cube

bgcolor=#e7dcc3|Typeuniform 8-polytope
bgcolor=#e7dcc3|Schläfli symbol4t{3,3,3,3,3,3,4}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node|3|node|3|node}}

{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node|split1|nodes}}

bgcolor=#e7dcc3|6-faces
bgcolor=#e7dcc3|5-faces
bgcolor=#e7dcc3|4-faces
bgcolor=#e7dcc3|Cells
bgcolor=#e7dcc3|Faces
bgcolor=#e7dcc3|Edges
bgcolor=#e7dcc3|Vertices
bgcolor=#e7dcc3|Vertex figure{3,4}v{3,3}
bgcolor=#e7dcc3|Coxeter groupsB8, [3,3,3,3,3,3,4]
D8, [35,1,1]
bgcolor=#e7dcc3|Propertiesconvex

= Alternate names =

  • Quadritruncated octeract (acronym oke) (Jonathan Bowers)Klitizing, (o3o3o3x3x3o3o4o – oke)

= Coordinates =

Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of

: (±2,±2,±2,±2,±1,0,0,0)

= Images =

{{8-cube Coxeter plane graphs|t34|200}}

= 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, {{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|polyzetta.htm|8D|uniform polytopes (polyzetta)}} o3o3o3o3o3o3x4x – tocto, o3o3o3o3o3x3x4o – bato, o3o3o3o3x3x3o4o – tato, o3o3o3x3x3o3o4o – oke

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:8-polytopes