Jump to content

Rectified 8-cubes

From Wikipedia, the free encyclopedia
(Redirected from Rectified 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

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

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

[edit]
Rectified 8-cube
Type uniform 8-polytope
Schläfli symbol t1{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces 256 + 16
6-faces 2048 + 112
5-faces 7168 + 448
4-faces 14336 + 1120
Cells 17920 +* 1792
Faces 4336 + 1792
Edges 7168
Vertices 1024
Vertex figure 6-simplex prism
{3,3,3,3,3}×{}
Coxeter groups B8, [36,4]
D8, [35,1,1]
Properties convex

Alternate names

[edit]
  • rectified octeract

Images

[edit]
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

[edit]
Birectified 8-cube
Type uniform 8-polytope
Coxeter symbol 0511
Schläfli symbol t2{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
7-faces 256 + 16
6-faces 1024 + 2048 + 112
5-faces 7168 + 7168 + 448
4-faces 21504 + 14336 + 1120
Cells 35840 + 17920 + 1792
Faces 35840 + 14336
Edges 21504
Vertices 1792
Vertex figure {3,3,3,3}x{4}
Coxeter groups B8, [36,4]
D8, [35,1,1]
Properties convex

Alternate names

[edit]
  • Birectified octeract
  • Rectified 8-demicube

Images

[edit]
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

[edit]
Triectified 8-cube
Type uniform 8-polytope
Schläfli symbol t3{4,3,3,3,3,3,3}
Coxeter diagrams
7-faces 16+256
6-faces 1024 + 2048 + 112
5-faces 1792 + 7168 + 7168 + 448
4-faces 1792 + 10752 + 21504 +14336
Cells 8960 + 26880 + 35840
Faces 17920+35840
Edges 17920
Vertices 1152
Vertex figure {3,3,3}x{3,4}
Coxeter groups B8, [36,4]
D8, [35,1,1]
Properties convex

Alternate names

[edit]
  • trirectified octeract

Images

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

Notes

[edit]

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.
  • Klitzing, Richard. "8D uniform polytopes (polyzetta)". o3o3o3o3o3o3x4o, o3o3o3o3o3x3o4o, o3o3o3o3x3o3o4o
[edit]
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 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