# Rectified 8-cubes

(Redirected from Rectified 8-cube)
 Orthogonal projections in BC8 Coxeter plane 8-cube Rectified 8-cube Birectified 8-cube Trirectified 8-cube Trirectified 8-orthoplex Birectified 8-orthoplex Rectified 8-orthoplex 8-orthoplex

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

Rectified 8-cube
Type uniform 8-polytope
Schläfli symbol t1{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
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

• rectified octeract

### Images

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

## Birectified 8-cube

Birectified 8-cube
Type uniform 8-polytope
Coxeter symbol 0511
Schläfli symbol t2{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
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

• birectified octeract

### Images

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

## Trirectified 8-cube

Triectified 8-cube
Type uniform 8-polytope
Schläfli symbol t3{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
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

• trirectified octeract

### Images

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

## 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. "8D uniform polytopes (polyzetta)". o3o3o3o3o3o3x4o, o3o3o3o3o3x3o4o, o3o3o3o3x3o3o4o