Rectified 8-cubes

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8-cube t0.svg
8-cube
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-cube t1.svg
Rectified 8-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-cube t2.svg
Birectified 8-cube
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-cube t3.svg
Trirectified 8-cube
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-cube t4.svg
Trirectified 8-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-cube t5.svg
Birectified 8-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-cube t6.svg
Rectified 8-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
8-cube t7.svg
8-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Orthogonal projections in BC8 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 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure 6-simplex prism
Coxeter groups C8, [36,4]
D8, [35,1,1]
Properties convex

Alternate names[edit]

  • rectified octeract

Images[edit]

orthographic projections
B8 B7
8-cube t1.svg 8-cube t1 B7.svg
[16] [14]
B6 B5
8-cube t1 B6.svg 8-cube t1 B5.svg
[12] [10]
B4 B3 B2
8-cube t1 B4.svg 8-cube t1 B3.svg 8-cube t1 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t1 A7.svg 8-cube t1 A5.svg 8-cube t1 A3.svg
[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 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3,3,3}x{4}
Coxeter groups C8, [36,4]
D8, [35,1,1]
Properties convex

Alternate names[edit]

  • birectified octeract

Images[edit]

orthographic projections
B8 B7
8-cube t2.svg 8-cube t2 B7.svg
[16] [14]
B6 B5
8-cube t2 B6.svg 8-cube t2 B5.svg
[12] [10]
B4 B3 B2
8-cube t2 B4.svg 8-cube t2 B3.svg 8-cube t2 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t2 A7.svg 8-cube t2 A5.svg 8-cube t2 A3.svg
[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-Dynkin diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3,3}x{3,4}
Coxeter groups C8, [36,4]
D8, [35,1,1]
Properties convex

Alternate names[edit]

  • trirectified octeract

Images[edit]

orthographic projections
B8 B7
8-cube t3.svg 8-cube t3 B7.svg
[16] [14]
B6 B5
8-cube t3 B6.svg 8-cube t3 B5.svg
[12] [10]
B4 B3 B2
8-cube t3 B4.svg 8-cube t3 B3.svg 8-cube t3 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t3 A7.svg 8-cube t3 A5.svg 8-cube t3 A3.svg
[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

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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