Rectified 9-simplexes

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9-simplex t0.svg
9-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
9-simplex t1.svg
Rectified 9-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
9-simplex t2.svg
Birectified 9-simplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
9-simplex t3.svg
Trirectified 9-simplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
9-simplex t4.svg
Quadrirectified 9-simplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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[edit]

Rectified 9-simplex
Type uniform 9-polytope
Schläfli symbol t1{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A9 A8 A7 A6
Graph 9-simplex t1.svg 9-simplex t1 A8.svg 9-simplex t1 A7.svg 9-simplex t1 A6.svg
Dihedral symmetry [10] [9] [8] [7]
Ak Coxeter plane A5 A4 A3 A2
Graph 9-simplex t1 A5.svg 9-simplex t1 A4.svg 9-simplex t1 A3.svg 9-simplex t1 A2.svg
Dihedral symmetry [6] [5] [4] [3]

Birectified 9-simplex[edit]

Birectified 9-simplex
Type uniform 9-polytope
Schläfli symbol t2{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 1260
Vertices 120
Vertex figure {3}x{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 10-dimensional sphere packing.

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A9 A8 A7 A6
Graph 9-simplex t2.svg 9-simplex t2 A8.svg 9-simplex t2 A7.svg 9-simplex t2 A6.svg
Dihedral symmetry [10] [9] [8] [7]
Ak Coxeter plane A5 A4 A3 A2
Graph 9-simplex t2 A5.svg 9-simplex t2 A4.svg 9-simplex t2 A3.svg 9-simplex t2 A2.svg
Dihedral symmetry [6] [5] [4] [3]

Trirectified 9-simplex[edit]

Trirectified 9-simplex
Type uniform 9-polytope
Schläfli symbol t3{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3}x{3,3,3,3}
Coxeter groups A9, [3,3,3,3,3,3,3,3]
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A9 A8 A7 A6
Graph 9-simplex t3.svg 9-simplex t3 A8.svg 9-simplex t3 A7.svg 9-simplex t3 A6.svg
Dihedral symmetry [10] [9] [8] [7]
Ak Coxeter plane A5 A4 A3 A2
Graph 9-simplex t3 A5.svg 9-simplex t3 A4.svg 9-simplex t3 A3.svg 9-simplex t3 A2.svg
Dihedral symmetry [6] [5] [4] [3]

Quadrirectified 9-simplex[edit]

Quadrirectified 9-simplex
Type uniform 9-polytope
Schläfli symbol t4{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3,3}x{3,3,3}
Coxeter groups A9, [3,3,3,3,3,3,3,3]
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A9 A8 A7 A6
Graph 9-simplex t4.svg 9-simplex t4 A8.svg 9-simplex t4 A7.svg 9-simplex t4 A6.svg
Dihedral symmetry [10] [9] [8] [7]
Ak Coxeter plane A5 A4 A3 A2
Graph 9-simplex t4 A5.svg 9-simplex t4 A4.svg 9-simplex t4 A3.svg 9-simplex t4 A2.svg
Dihedral symmetry [6] [5] [4] [3]

Notes[edit]

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

References[edit]

  • 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

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds