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词条 Truncated 8-orthoplexes
释义

  1. Truncated 8-orthoplex

      Alternate names    Construction    Coordinates    Images  

  2. Bitruncated 8-orthoplex

      Alternate names    Coordinates    Images  

  3. Tritruncated 8-orthoplex

      Alternate names    Coordinates    Images  

  4. Notes

  5. References

  6. External links

8-orthoplex
{{CDD>node_1|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
Truncated 8-orthoplex
{{CDD>node_1|3|node_1|3|node|3|node|3|node|3|node|3|node|4|node}}
Bitruncated 8-orthoplex
{{CDD>node|3|node_1|3|node_1|3|node|3|node|3|node|3|node|4|node}}
Tritruncated 8-orthoplex
{{CDD>node|3|node|3|node|3|node_1|3|node_1|3|node|3|node|4|node}}
Quadritruncated 8-cube
{{CDD>node|3|node|3|node|3|node_1|3|node_1|3|node|3|node|4|node}}
Tritruncated 8-cube
{{CDD>node|3|node|3|node|3|node|3|node_1|3|node_1|3|node|4|node}}
Bitruncated 8-cube
{{CDD>node|3|node|3|node|3|node|3|node|3|node_1|3|node_1|4|node}}
Truncated 8-cube
{{CDD>node|3|node|3|node|3|node|3|node|3|node|3|node_1|4|node_1}}
8-cube
{{CDD>node|3|node|3|node|3|node|3|node|3|node|3|node|4|node_1}}
Orthogonal projections in B8 Coxeter plane

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

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

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Truncated 8-orthoplex

Truncated 8-orthoplex
Typeuniform 8-polytope
Schläfli symbol t0,1{3,3,3,3,3,3,4}
Coxeter-Dynkin diagramsnode_1|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node|split1|nodes}}
6-faces
5-faces
4-faces
Cells
Faces
Edges1456
Vertices224
Vertex figure( )v{3,3,3,4}
Coxeter groupsB8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Propertiesconvex

Alternate names

  • Truncated octacross (acronym tek) (Jonthan Bowers)[1]

Construction

There are two Coxeter groups associated with the truncated 8-orthoplex, one with the C8 or [4,3,3,3,3,3,3] Coxeter group, and a lower symmetry with the D8 or [35,1,1] Coxeter group.

Coordinates

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

(±2,±1,0,0,0,0,0,0)

Images

{{8-cube Coxeter plane graphs|t67|200}}

Bitruncated 8-orthoplex

Bitruncated 8-orthoplex
Typeuniform 8-polytope
Schläfli symbol t1,2{3,3,3,3,3,3,4}
Coxeter-Dynkin diagramsnode|3|node_1|3|node_1|3|node|3|node|3|node|3|node|3|node}}
{{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node|split1|nodes}}
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure{ }v{3,3,3,4}
Coxeter groupsB8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Propertiesconvex

Alternate names

  • Bitruncated octacross (acronym batek) (Jonthan Bowers)[2]

Coordinates

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

(±2,±2,±1,0,0,0,0,0)

Images

{{8-cube Coxeter plane graphs|t56|200}}

Tritruncated 8-orthoplex

Tritruncated 8-orthoplex
Typeuniform 8-polytope
Schläfli symbol t2,3{3,3,3,3,3,3,4}
Coxeter-Dynkin diagramsnode|3|node|3|node_1|3|node_1|3|node|3|node|3|node|4|node}}
{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node|split1|nodes}}
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure{3}v{3,3,4}
Coxeter groupsB8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Propertiesconvex

Alternate names

  • Tritruncated octacross (acronym tatek) (Jonthan Bowers)[3]

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,±1,0,0,0,0)

Images

{{8-cube Coxeter plane graphs|t45|200}}

Notes

1. ^Klitizing, (x3x3o3o3o3o3o4o - tek)
2. ^Klitizing, (o3x3x3o3o3o3o4o - batek)
3. ^Klitizing, (o3o3x3x3o3o3o4o - tatek)

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}}  
    • (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|polyzetta.htm|8D|uniform polytopes (polyzetta)}} x3x3o3o3o3o3o4o - tek, o3x3x3o3o3o3o4o - batek, o3o3x3x3o3o3o4o - tatek

External links

  • Polytopes of Various Dimensions
  • Multi-dimensional Glossary
{{Polytopes}}

1 : 8-polytopes

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