Rectified 8-cubes

In eight-dimensional geometry, a rectified 8-cube is a convex uniform 8-polytope, being a rectification of the regular 8-cube.


8-cube
Rectified 8-cube
Birectified 8-cube
Trirectified 8-cube

Trirectified 8-orthoplex
Birectified 8-orthoplex
Rectified 8-orthoplex
8-orthoplex
Orthogonal projections in B8 Coxeter plane

There are unique 8 degrees of rectifications, the zeroth being the 8-cube, and the 7th and last being the 8-orthoplex. Vertices of the rectified 8-cube are located at the edge-centers of the 8-cube. Vertices of the birectified 8-cube are located in the square face centers of the 8-cube. Vertices of the trirectified 8-cube are located in the 7-cube cell centers of the 8-cube.

Rectified 8-cube
Rectified 8-cube
Typeuniform 8-polytope
Schläfli symbolt1{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces256 + 16
6-faces2048 + 112
5-faces7168 + 448
4-faces14336 + 1120
Cells17920 +* 1792
Faces4336 + 1792
Edges7168
Vertices1024
Vertex figure6-simplex prism
{3,3,3,3,3}×{}
Coxeter groupsB8, [36,4]
D8, [35,1,1]
Propertiesconvex

Alternate names
  • rectified octeract

Images
orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

Birectified 8-cube
Birectified 8-cube
Typeuniform 8-polytope
Coxeter symbol0511
Schläfli symbolt2{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces256 + 16
6-faces1024 + 2048 + 112
5-faces7168 + 7168 + 448
4-faces21504 + 14336 + 1120
Cells35840 + 17920 + 1792
Faces35840 + 14336
Edges21504
Vertices1792
Vertex figure{3,3,3,3}x{4}
Coxeter groupsB8, [36,4]
D8, [35,1,1]
Propertiesconvex

Alternate names
  • Birectified octeract
  • Rectified 8-demicube

Images
orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

Trirectified 8-cube
Triectified 8-cube
Typeuniform 8-polytope
Schläfli symbolt3{4,3,3,3,3,3,3}
Coxeter diagrams
7-faces16+256
6-faces1024 + 2048 + 112
5-faces1792 + 7168 + 7168 + 448
4-faces1792 + 10752 + 21504 +14336
Cells8960 + 26880 + 35840
Faces17920+35840
Edges17920
Vertices1152
Vertex figure{3,3,3}x{3,4}
Coxeter groupsB8, [36,4]
D8, [35,1,1]
Propertiesconvex

Alternate names
  • trirectified octeract

Images
orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

Notes

    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.
    • Klitzing, Richard. "8D uniform polytopes (polyzetta)". o3o3o3o3o3o3x4o, o3o3o3o3o3x3o4o, o3o3o3o3x3o3o4o
    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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