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Rectified 8-cubes

(Redirected from Birectified 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

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
B8B7
[16][14]
B6B5
[12][10]
B4B3B2
[8][6][4]
A7A5A3
[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
B8B7
[16][14]
B6B5
[12][10]
B4B3B2
[8][6][4]
A7A5A3
[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
B8B7
[16][14]
B6B5
[12][10]
B4B3B2
[8][6][4]
A7A5A3
[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 [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
Fundamental convex regular and uniform polytopes in dimensions 2–10
FamilyAnBnI2(p) / DnE6 / E7 / E8 / F4 / G2Hn
Regular polygonTriangleSquarep-gonHexagonPentagon
Uniform polyhedronTetrahedronOctahedron • CubeDemicubeDodecahedron • Icosahedron
Uniform polychoronPentachoron16-cell • TesseractDemitesseract24-cell120-cell • 600-cell
Uniform 5-polytope5-simplex5-orthoplex • 5-cube5-demicube
Uniform 6-polytope6-simplex6-orthoplex • 6-cube6-demicube122 • 221
Uniform 7-polytope7-simplex7-orthoplex • 7-cube7-demicube132 • 231 • 321
Uniform 8-polytope8-simplex8-orthoplex • 8-cube8-demicube142 • 241 • 421
Uniform 9-polytope9-simplex9-orthoplex • 9-cube9-demicube
Uniform 10-polytope10-simplex10-orthoplex • 10-cube10-demicube
Uniform n-polytopen-simplexn-orthoplex • n-cuben-demicube1k2 • 2k1 • k21n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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