Rectified 10-cubes

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


10-orthoplex
Rectified 10-orthoplex
Birectified 10-orthoplex
Trirectified 10-orthoplex

Quadirectified 10-orthoplex
Quadrirectified 10-cube
Trirectified 10-cube
Birectified 10-cube

Rectified 10-cube
10-cube
Orthogonal projections in BC10 Coxeter plane

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 polytope, 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 symbolt1{38,4}
Coxeter-Dynkin diagrams
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
Orthographic projections
B10 B9 B8
[20] [18] [16]
B7 B6 B5
[14] [12] [10]
B4 B3 B2
[8] [6] [4]
A9 A5
— —
[10] [6]
A7 A3
— —
[8] [4]

Birectified 10-cube
Birectified 10-orthoplex
Typeuniform 10-polytope
Coxeter symbol0711
Schläfli symbolt2{38,4}
Coxeter-Dynkin diagrams
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
Orthographic projections
B10 B9 B8
[20] [18] [16]
B7 B6 B5
[14] [12] [10]
B4 B3 B2
[8] [6] [4]
A9 A5
— —
[10] [6]
A7 A3
— —
[8] [4]

Trirectified 10-cube
Trirectified 10-orthoplex
Typeuniform 10-polytope
Schläfli symbolt3{38,4}
Coxeter-Dynkin diagrams
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
Orthographic projections
B10 B9 B8
[20] [18] [16]
B7 B6 B5
[14] [12] [10]
B4 B3 B2
[8] [6] [4]
A9 A5
— —
[10] [6]
A7 A3
— —
[8] [4]

Quadrirectified 10-cube
Quadrirectified 10-orthoplex
Typeuniform 10-polytope
Schläfli symbolt4{38,4}
Coxeter-Dynkin diagrams
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
Orthographic projections
B10 B9 B8
[20] [18] [16]
B7 B6 B5
[14] [12] [10]
B4 B3 B2
[8] [6] [4]
A9 A5
— —
[10] [6]
A7 A3
— —
[8] [4]

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