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词条 Rectified 10-cubes
释义

  1. Rectified 10-cube

      Alternate names   Cartesian coordinates   Images 

  2. Birectified 10-cube

      Alternate names   Cartesian coordinates   Images 

  3. Trirectified 10-cube

      Alternate names   Cartesian coordinates   Images 

  4. Quadrirectified 10-cube

      Alternate names   Cartesian coordinates   Images 

  5. Notes

  6. References

  7. External links

10-orthoplex
{{CDD>node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}		Rectified 10-orthoplex
{{CDD>node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}		Birectified 10-orthoplex
{{CDD>node|4|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}		Trirectified 10-orthoplex
{{CDD>node|4|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}
Quadirectified 10-orthoplex
{{CDD>node|4|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}		Quadrirectified 10-cube
{{CDD>node|4|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}		Trirectified 10-cube
{{CDD>node|4|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}		Birectified 10-cube
{{CDD>node|4|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
Rectified 10-cube
{{CDD>node|4|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}		10-cube
{{CDD>node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
Orthogonal projections in BC10 Coxeter plane

In ten-dimensional geometry, a rectified 10-cube is a convex uniform 10-polytope, being a rectification of the regular 10-cube.

There are 10 rectifications of the 10-cube, with the zeroth being the 10-cube itself. Vertices of the rectified 10-cube are located at the edge-centers of the 10-cube. Vertices of the birectified 10-cube are located in the square face centers of the 10-cube. Vertices of the trirectified 10-cube are located in the cubic cell centers of the 10-cube. The others are more simply constructed relative to the 10-cube dual polytpoe, the 10-orthoplex.

These polytopes are part of a family 1023 uniform 10-polytopes with BC10 symmetry.

Rectified 10-cube

Rectified 10-orthoplex
Typeuniform 10-polytope
Schläfli symbol t1{38,4}
Coxeter-Dynkin diagramsnode|4|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
{{CDD|nodes_11|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges46080
Vertices5120
Vertex figure8-simplex prism
Coxeter groupsC10, [4,38]
D10, [37,1,1]
Propertiesconvex

Alternate names

  • Rectified dekeract (Acronym rade) (Jonathan Bowers)[1]

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 10-cube, centered at the origin, edge length are all permutations of:

(±1,±1,±1,±1,±1,±1,±1,±1,±1,0)

Images

{{B10 Coxeter plane graphs|t1|150|NoA9=true|NoA5=true|NoA7=true|NoA3=true}}

Birectified 10-cube

Birectified 10-orthoplex
Typeuniform 10-polytope
Coxeter symbol 0711
Schläfli symbol t2{38,4}
Coxeter-Dynkin diagramsnode|4|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
{{CDD|nodes|split2|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges184320
Vertices11520
Vertex figure{4}x{36}
Coxeter groupsC10, [4,38]
D10, [37,1,1]
Propertiesconvex

Alternate names

  • Birectified dekeract (Acronym brade) (Jonathan Bowers)[2]

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 10-cube, centered at the origin, edge length are all permutations of:

(±1,±1,±1,±1,±1,±1,±1,±1,0,0)

Images

{{B10 Coxeter plane graphs|t2|150|NoA9=true|NoA5=true|NoA7=true|NoA3=true}}

Trirectified 10-cube

Trirectified 10-orthoplex
Typeuniform 10-polytope
Schläfli symbol t3{38,4}
Coxeter-Dynkin diagramsnode|4|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
{{CDD|nodes|split2|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges322560
Vertices15360
Vertex figure{4,3}x{35}
Coxeter groupsC10, [4,38]
D10, [37,1,1]
Propertiesconvex

Alternate names

  • Tririrectified dekeract (Acronym trade) (Jonathan Bowers)[3]

Cartesian coordinates

Cartesian coordinates for the vertices of a triirectified 10-cube, centered at the origin, edge length are all permutations of:

(±1,±1,±1,±1,±1,±1,±1,0,0,0)

Images

{{B10 Coxeter plane graphs|t3|150|NoA9=true|NoA5=true|NoA7=true|NoA3=true}}

Quadrirectified 10-cube

Quadrirectified 10-orthoplex
Typeuniform 10-polytope
Schläfli symbol t4{38,4}
Coxeter-Dynkin diagramsnode|4|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
{{CDD|nodes|split2|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges322560
Vertices13440
Vertex figure{4,3,3}x{34}
Coxeter groupsC10, [4,38]
D10, [37,1,1]
Propertiesconvex

Alternate names

  • Quadrirectified dekeract
  • Quadrirectified decacross (Acronym terade) (Jonathan Bowers)[4]

Cartesian coordinates

Cartesian coordinates for the vertices of a quadrirectified 10-cube, centered at the origin, edge length are all permutations of:

(±1,±1,±1,±1,±1,±1,0,0,0,0)

Images

{{B10 Coxeter plane graphs|t4|150|NoA9=true|NoA5=true|NoA7=true|NoA3=true}}

Notes

1. ^Klitzing, (o3o3o3o3o3o3o3o3x4o - rade)
2. ^Klitzing, (o3o3o3o3o3o3o3x3o4o - brade)
3. ^Klitzing, (o3o3o3o3o3o3x3o3o4o - trade)
4. ^Klitzing, (o3o3o3o3o3x3o3o3o4o - terade)

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|polyxenna.htm|10D|uniform polytopes (polyxenna)}} x3o3o3o3o3o3o3o3o4o - ka, o3x3o3o3o3o3o3o3o4o - rake, o3o3x3o3o3o3o3o3o4o - brake, o3o3o3x3o3o3o3o3o4o - trake, o3o3o3o3x3o3o3o3o4o - terake, o3o3o3o3o3x3o3o3o4o - terade, o3o3o3o3o3o3x3o3o4o - trade, o3o3o3o3o3o3o3x3o4o - brade, o3o3o3o3o3o3o3o3x4o - rade, o3o3o3o3o3o3o3o3o4x - deker

External links

  • [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
  • Multi-dimensional Glossary
{{Polytopes}}

1 : 10-polytopes
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