Rectified 7-cubes


7-cube
Rectified 7-cube
Birectified 7-cube
Trirectified 7-cube

Birectified 7-orthoplex
Rectified 7-orthoplex
7-orthoplex
Orthogonal projections in B7 Coxeter plane

In seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube.

There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Vertices of the rectified 7-cube are located at the edge-centers of the 7-ocube. Vertices of the birectified 7-cube are located in the square face centers of the 7-cube. Vertices of the trirectified 7-cube are located in the cube cell centers of the 7-cube.

Rectified 7-cube

Rectified 7-cube
Type uniform 7-polytope
Schläfli symbol r{4,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces 128 + 14
5-faces 896 + 84
4-faces 2688 + 280
Cells 4480 + 560
Faces 4480 + 672
Edges 2688
Vertices 448
Vertex figure 5-simplex prism
Coxeter groups B7, [3,3,3,3,3,4]
Properties convex

Alternate names

  • rectified hepteract (Acronym rasa) (Jonathan Bowers)[1]

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 7-cube, centered at the origin, edge length are all permutations of:

(±1,±1,±1,±1,±1,±1,0)

Birectified 7-cube

Birectified 7-cube
Type uniform 7-polytope
Coxeter symbol 0411
Schläfli symbol 2r{4,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces 128 + 14
5-faces 448 + 896 + 84
4-faces 2688 + 2688 + 280
Cells 6720 + 4480 + 560
Faces 8960 + 4480
Edges 6720
Vertices 672
Vertex figure {3}x{3,3,3}
Coxeter groups B7, [3,3,3,3,3,4]
Properties convex

Alternate names

  • Birectified hepteract (Acronym bersa) (Jonathan Bowers)[2]

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 7-cube, centered at the origin, edge length are all permutations of:

(±1,±1,±1,±1,±1,0,0)

Trirectified 7-cube

Trirectified 7-cube
Type uniform 7-polytope
Schläfli symbol 3r{4,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces 128 + 14
5-faces 448 + 896 + 84
4-faces 672 + 2688 + 2688 + 280
Cells 3360 + 6720 + 4480
Faces 6720 + 8960
Edges 6720
Vertices 560
Vertex figure {3,3}x{3,3}
Coxeter groups B7, [3,3,3,3,3,4]
Properties convex

Alternate names

  • Trirectified hepteract
  • Trirectified 7-orthoplex
  • Trirectified heptacross (Acronym sez) (Jonathan Bowers)[3]

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Cartesian coordinates

Cartesian coordinates for the vertices of a trirectified 7-cube, centered at the origin, edge length are all permutations of:

(±1,±1,±1,±1,0,0,0)
2-isotopic hypercubes
Dim. 2 3 4 5 6 7 8 n
Name t{4} r{4,3} 2t{4,3,3} 2r{4,3,3,3} 3t{4,3,3,3,3} 3r{4,3,3,3,3,3} 4t{4,3,3,3,3,3,3} ...
Coxeter
diagram
Images
Facets {3}
{4} 		t{3,3}
t{3,4} 		r{3,3,3}
r{3,3,4} 		2t{3,3,3,3}
2t{3,3,3,4} 		2r{3,3,3,3,3}
2r{3,3,3,3,4} 		3t{3,3,3,3,3,3}
3t{3,3,3,3,3,4}
Vertex
figure 		( )v( )
{ }×{ }
{ }v{ }
{3}×{4}
{3}v{4} 		{3,3}×{3,4} {3,3}v{3,4}

Notes

  1. ^ Klitzing, (o3o3o3o3o3x4o - rasa)
  2. ^ Klitzing, (o3o3o3o3x3o4o - bersa)
  3. ^ Klitzing, (o3o3o3x3o3o4o - sez)

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. "7D uniform polytopes (polyexa)". o3o3o3x3o3o4o - sez, o3o3o3o3x3o4o - bersa, o3o3o3o3o3x4o - rasa
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