# Truncated 7-simplexes

(Redirected from Bitruncated 7-simplex)
 Orthogonal projections in A7 Coxeter plane 7-simplex Truncated 7-simplex Bitruncated 7-simplex Tritruncated 7-simplex

In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.

There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the triangular faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the tetrahedral cells of the 7-simplex.

## Truncated 7-simplex

Truncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces 16
5-faces
4-faces
Cells 350
Faces 336
Edges 196
Vertices 56
Vertex figure Elongated 5-simplex pyramid
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex, Vertex-transitive

In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.

### Alternate names

• Truncated octaexon (Acronym: toc) (Jonathan Bowers)[1]

### Coordinates

The vertices of the truncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 8-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Bitruncated 7-simplex

Bitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol 2t{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 588
Vertices 168
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex, Vertex-transitive

### Alternate names

• Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers)[2]

### Coordinates

The vertices of the bitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 8-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Tritruncated 7-simplex

Tritruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol 3t{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 980
Vertices 280
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex, Vertex-transitive

### Alternate names

• Tritruncated octaexon (acronym: tattoc) (Jonathan Bowers)[3]

### Coordinates

The vertices of the tritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 8-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Related polytopes

These three polytopes are from a set of 71 uniform 7-polytopes with A7 symmetry.

## Notes

1. ^ Klitizing, (x3x3o3o3o3o3o - toc)
2. ^ Klitizing, (o3x3x3o3o3o3o - roc)
3. ^ Klitizing, (o3o3x3x3o3o3o - tattoc)

## 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. "7D uniform polytopes (polyexa)". x3x3o3o3o3o3o - toc, o3x3x3o3o3o3o - roc, o3o3x3x3o3o3o - tattoc