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Rectified 7-cubes

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
Schläfli symbol
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • rectified hepteract (Acronym rasa) (Jonathan Bowers)

Images

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 7-cube, centered at the origin, edge length \sqrt{2}\ are all permutations of: : (±1,±1,±1,±1,±1,±1,0)

Birectified 7-cube

Birectified 7-cube
Type
Coxeter symbol
Schläfli symbol
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Birectified hepteract (Acronym bersa) (Jonathan Bowers)

Images

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 7-cube, centered at the origin, edge length \sqrt{2}\ are all permutations of: : (±1,±1,±1,±1,±1,0,0)

Trirectified 7-cube

Trirectified 7-cube
Type
Schläfli symbol
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

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

Images

Cartesian coordinates

Cartesian coordinates for the vertices of a trirectified 7-cube, centered at the origin, edge length \sqrt{2}\ are all permutations of: : (±1,±1,±1,±1,0,0,0)

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, wiley.com,
      • (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.
  • o3o3o3x3o3o4o - sez, o3o3o3o3x3o4o - bersa, o3o3o3o3o3x4o - rasa

References

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