# Rectified 7-cubes

(Redirected from Birectified 7-cube)
 Orthogonal projections in BC7 Coxeter plane 7-cube Rectified 7-cube Birectified 7-cube Trirectified 7-cube Birectified 7-orthoplex Rectified 7-orthoplex 7-orthoplex

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
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure 5-simplex prism
Coxeter groups BC7, [3,3,3,3,3,4]
Properties convex

### Alternate names

• rectified hepteract (Acronym resa) (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 $\sqrt{2}\$ are all permutations of:

(±1,±1,±1,±1,±1,±1,0)

## Birectified 7-cube

Birectified 7-cube
Type uniform 7-polytope
Schläfli symbol 2r{4,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3}x{3,3,3}
Coxeter groups BC7, [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 $\sqrt{2}\$ 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
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3}x{3,3}
Coxeter groups BC7, [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 $\sqrt{2}\$ are all permutations of:

(±1,±1,±1,±1,0,0,0)

## = Related polytopes

2-isotopic hypercubes
Dim. 2 3 4 5 6 7 8
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

Rectangle

Disphenoid

{3}×{4} duoprism
{3,3}×{3,4} duoprism

## 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.
• Richard Klitzing, 7D, uniform polytopes (polyexa) o3o3o3x3o3o4o - sez, o3o3o3o3x3o4o - bersa, o3o3o3o3o3x4o - rasa