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Rectified 9-simplexes

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9-simplex
Rectified 9-simplex

Birectified 9-simplex
Trirectified 9-simplex
Quadrirectified 9-simplex
Orthogonal projections in A9 Coxeter plane

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex.

These polytopes are part of a family of 271 uniform 9-polytopes with A9 symmetry.

There are unique 4 degrees of rectifications. Vertices of the rectified 9-simplex are located at the edge-centers of the 9-simplex. Vertices of the birectified 9-simplex are located in the triangular face centers of the 9-simplex. Vertices of the trirectified 9-simplex are located in the tetrahedral cell centers of the 9-simplex. Vertices of the quadrirectified 9-simplex are located in the 5-cell centers of the 9-simplex.

Rectified 9-simplex

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Rectified 9-simplex
Type uniform 9-polytope
Schläfli symbol t1{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams                  
8-faces 20
7-faces 135
6-faces 480
5-faces 1050
4-faces 1512
Cells 1470
Faces 960
Edges 360
Vertices 45
Vertex figure 8-simplex prism
Petrie polygon decagon
Coxeter groups A9, [3,3,3,3,3,3,3,3]
Properties convex

The rectified 9-simplex is the vertex figure of the 10-demicube.

Alternate names

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  • Rectified decayotton (reday) (Jonathan Bowers)[1]

Coordinates

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The Cartesian coordinates of the vertices of the rectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 10-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A9 A8 A7 A6
Graph        
Dihedral symmetry [10] [9] [8] [7]
Ak Coxeter plane A5 A4 A3 A2
Graph        
Dihedral symmetry [6] [5] [4] [3]

Birectified 9-simplex

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Birectified 9-simplex
Type uniform 9-polytope
Schläfli symbol t2{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams                  
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 1260
Vertices 120
Vertex figure {3}×{3,3,3,3,3}
Coxeter groups A9, [3,3,3,3,3,3,3,3]
Properties convex

This polytope is the vertex figure for the 162 honeycomb. Its 120 vertices represent the kissing number of the related hyperbolic 9-dimensional sphere packing.

Alternate names

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  • Birectified decayotton (breday) (Jonathan Bowers)[2]

Coordinates

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The Cartesian coordinates of the vertices of the birectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 10-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A9 A8 A7 A6
Graph        
Dihedral symmetry [10] [9] [8] [7]
Ak Coxeter plane A5 A4 A3 A2
Graph        
Dihedral symmetry [6] [5] [4] [3]

Trirectified 9-simplex

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Trirectified 9-simplex
Type uniform 9-polytope
Schläfli symbol t3{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams                  
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3}×{3,3,3,3}
Coxeter groups A9, [3,3,3,3,3,3,3,3]
Properties convex

Alternate names

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  • Trirectified decayotton (treday) (Jonathan Bowers)[3]

Coordinates

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The Cartesian coordinates of the vertices of the trirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 10-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A9 A8 A7 A6
Graph        
Dihedral symmetry [10] [9] [8] [7]
Ak Coxeter plane A5 A4 A3 A2
Graph        
Dihedral symmetry [6] [5] [4] [3]

Quadrirectified 9-simplex

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Quadrirectified 9-simplex
Type uniform 9-polytope
Schläfli symbol t4{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams                  
or          
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3,3}×{3,3,3}
Coxeter groups A9×2, [[38]]
Properties convex

Alternate names

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  • Quadrirectified decayotton
  • Icosayotton (icoy) (Jonathan Bowers)[4]

Coordinates

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The Cartesian coordinates of the vertices of the quadrirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 10-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A9 A8 A7 A6
Graph        
Dihedral symmetry [10] [9] [8] [7]
Ak Coxeter plane A5 A4 A3 A2
Graph        
Dihedral symmetry [6] [5] [4] [3]

Notes

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  1. ^ Klitzing, (o3x3o3o3o3o3o3o3o - reday)
  2. ^ Klitzing, (o3o3x3o3o3o3o3o3o - breday)
  3. ^ Klitzing, (o3o3o3x3o3o3o3o3o - treday)
  4. ^ Klitzing, (o3o3o3o3x3o3o3o3o - icoy)

References

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  • 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 [1]
      • (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)
  • Klitzing, Richard. "9D uniform polytopes (polyyotta)". o3x3o3o3o3o3o3o3o - reday, o3o3x3o3o3o3o3o3o - breday, o3o3o3x3o3o3o3o3o - treday, o3o3o3o3x3o3o3o3o - icoy
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Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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