Rectified 10-simplexes

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


10-simplex
Rectified 10-simplex
Birectified 10-simplex

Trirectified 10-simplex
Quadrirectified 10-simplex
Orthogonal projections in A9 Coxeter plane

These polytopes are part of a family of 527 uniform 10-polytopes with A10 symmetry.

There are unique 5 degrees of rectifications including the zeroth, the 10-simplex itself. Vertices of the rectified 10-simplex are located at the edge-centers of the 10-simplex. Vertices of the birectified 10-simplex are located in the triangular face centers of the 10-simplex. Vertices of the trirectified 10-simplex are located in the tetrahedral cell centers of the 10-simplex. Vertices of the quadrirectified 10-simplex are located in the 5-cell centers of the 10-simplex.

Rectified 10-simplex
Rectified 10-simplex
Typeuniform polyxennon
Schläfli symbolt1{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
9-faces22
8-faces165
7-faces660
6-faces1650
5-faces2772
4-faces3234
Cells2640
Faces1485
Edges495
Vertices55
Vertex figure9-simplex prism
Petrie polygondecagon
Coxeter groupsA10, [3,3,3,3,3,3,3,3,3]
Propertiesconvex

The rectified 10-simplex is the vertex figure of the 11-demicube.

Alternate names
  • Rectified hendecaxennon (Acronym ru) (Jonathan Bowers)[1]

Coordinates

The Cartesian coordinates of the vertices of the rectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 11-orthoplex.

Images
orthographic projections
Ak Coxeter plane A10 A9 A8
Graph
Dihedral symmetry [11] [10] [9]
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Birectified 10-simplex
Birectified 10-simplex
Typeuniform 9-polytope
Schläfli symbolt2{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges1980
Vertices165
Vertex figure{3}x{3,3,3,3,3,3}
Coxeter groupsA10, [3,3,3,3,3,3,3,3,3]
Propertiesconvex

Alternate names
  • Birectified hendecaxennon (Acronym bru) (Jonathan Bowers)[2]

Coordinates

The Cartesian coordinates of the vertices of the birectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 11-orthoplex.

Images
orthographic projections
Ak Coxeter plane A10 A9 A8
Graph
Dihedral symmetry [11] [10] [9]
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Trirectified 10-simplex
Trirectified 10-simplex
Typeuniform polyxennon
Schläfli symbolt3{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges4620
Vertices330
Vertex figure{3,3}x{3,3,3,3,3}
Coxeter groupsA10, [3,3,3,3,3,3,3,3,3]
Propertiesconvex

Alternate names
  • Trirectified hendecaxennon (Jonathan Bowers)[3]

Coordinates

The Cartesian coordinates of the vertices of the trirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 11-orthoplex.

Images
orthographic projections
Ak Coxeter plane A10 A9 A8
Graph
Dihedral symmetry [11] [10] [9]
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Quadrirectified 10-simplex
Quadrirectified 10-simplex
Typeuniform polyxennon
Schläfli symbolt4{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges6930
Vertices462
Vertex figure{3,3,3}x{3,3,3,3}
Coxeter groupsA10, [3,3,3,3,3,3,3,3,3]
Propertiesconvex

Alternate names
  • Quadrirectified hendecaxennon (Acronym teru) (Jonathan Bowers)[4]

Coordinates

The Cartesian coordinates of the vertices of the quadrirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 11-orthoplex.

Images
orthographic projections
Ak Coxeter plane A10 A9 A8
Graph
Dihedral symmetry [11] [10] [9]
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Notes
  1. Klitzing, (o3x3o3o3o3o3o3o3o3o - ru)
  2. Klitzing, (o3o3x3o3o3o3o3o3o3o - bru)
  3. Klitzing, (o3o3o3x3o3o3o3o3o3o - tru)
  4. Klitzing, (o3o3o3o3x3o3o3o3o3o - teru)

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)
  • Klitzing, Richard. "10D uniform polytopes (polyxenna)". x3o3o3o3o3o3o3o3o3o - ux, o3x3o3o3o3o3o3o3o3o - ru, o3o3x3o3o3o3o3o3o3o - bru, o3o3o3x3o3o3o3o3o3o - tru, o3o3o3o3x3o3o3o3o3o - teru
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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