# Truncated 8-cubes

 Orthogonal projections in B8 Coxeter plane 8-cube Truncated 8-cube Bitruncated 8-cube Quadritruncated 8-cube Tritruncated 8-cube Tritruncated 8-orthoplex Bitruncated 8-orthoplex Truncated 8-orthoplex 8-orthoplex

In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.

There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edge of the 8-cube. Vertices of the bitruncated 8-cube are located on the square faces of the 8-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 8-cube. The final truncations are best expressed relative to the 8-orthoplex.

## Truncated 8-cube

Truncated 8-cube
Type uniform 8-polytope
Schläfli symbol t{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure Elongated 6-simplex pyramid
Coxeter groups B8, [3,3,3,3,3,3,4]
Properties convex

### Alternate names

• Truncated octeract (acronym tocto) (Jonathan Bowers)[1]

### Coordinates

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of

(±2,±2,±2,±2,±2,±2,±1,0)

### Images

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

### Related polytopes

The truncated 8-cube, is seventh in a sequence of truncated hypercubes:

 ... Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube

## Bitruncated 8-cube

Bitruncated 8-cube
Type uniform 8-polytope
Schläfli symbol 2t{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B8, [3,3,3,3,3,3,4]
Properties convex

### Alternate names

• Bitruncated octeract (acronym bato) (Jonathan Bowers)[2]

### Coordinates

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of

(±2,±2,±2,±2,±2,±1,0,0)

### Images

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

### Related polytopes

The bitruncated 8-cube is sixth in a sequence of bitruncated hypercubes:

 ... Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube

## Tritruncated 8-cube

Tritruncated 8-cube
Type uniform 8-polytope
Schläfli symbol 3t{4,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B8, [3,3,3,3,3,3,4]
Properties convex

### Alternate names

• Tritruncated octeract (acronym tato) (Jonathan Bowers)[3]

### Coordinates

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of

(±2,±2,±2,±2,±1,0,0,0)

### Images

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

Type uniform 8-polytope
Schläfli symbol 4t{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams

6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Properties convex

### Alternate names

• Quadritruncated octeract (acronym oke) (Jonathan Bowers)[4]

### Coordinates

Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±2,±2,±1,0,0,0)

### Images

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

### 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. ^ Klitizing, (o3o3o3o3o3o3x4x - tocto)
2. ^ Klitizing, (o3o3o3o3o3x3x4o - bato)
3. ^ Klitizing, (o3o3o3o3x3x3o4o - tato)
4. ^ Klitizing, (o3o3o3x3x3o3o4o - oke)

## 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)". o3o3o3o3o3o3x4x - tocto, o3o3o3o3o3x3x4o - bato, o3o3o3o3x3x3o4o - tato, o3o3o3x3x3o3o4o - oke