Rectified 7-cubes explained

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

bgcolor=#e7dcc3 colspan=2	Rectified 7-cube
Type	uniform 7-polytope
Schläfli symbol	r
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]

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

bgcolor=#e7dcc3 colspan=2	Birectified 7-cube
Type	uniform 7-polytope
Coxeter symbol	0411
Schläfli symbol	2r
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	x
Coxeter groups	B7, [3,3,3,3,3,4]
Properties	convex

Alternate names

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

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

bgcolor=#e7dcc3 colspan=2	Trirectified 7-cube
Type	uniform 7-polytope
Schläfli symbol	3r
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	x
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]

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)

Related polytopes

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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
    • (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

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

Notes and References

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