# Truncated 6-simplexes

(Redirected from Truncated 6-simplex)
 Orthogonal projections in A7 Coxeter plane 6-simplex Truncated 6-simplex Bitruncated 6-simplex Tritruncated 6-simplex

In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.

There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the triangular faces of the 6-simplex. Vertices of the tritruncated 6-simplex are located inside the tetrahedral cells of the 6-simplex.

## Truncated 6-simplex

Truncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t{3,3,3,3,3}
Coxeter-Dynkin diagram
5-faces 14:
7 {3,3,3,3}
7 t{3,3,3,3}
4-faces 63:
42 {3,3,3}
21 t{3,3,3}
Cells 140:
105 {3,3}
35 t{3,3}
Faces 175:
140 {3}
35 {6}
Edges 126
Vertices 42
Vertex figure Elongated 5-cell pyramid
Coxeter group A6, [35], order 5040
Dual ?
Properties convex

### Alternate names

• Truncated heptapeton (Acronym: til) (Jonathan Bowers)[1]

### Coordinates

The vertices of the truncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,2). This construction is based on facets of the truncated 7-orthoplex.

### Images

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

## Bitruncated 6-simplex

Bitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol 2t{3,3,3,3,3}
Coxeter-Dynkin diagram
5-faces 14
4-faces 84
Cells 245
Faces 385
Edges 315
Vertices 105
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

### Alternate names

• Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers)[2]

### Coordinates

The vertices of the bitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 7-orthoplex.

### Images

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

## Tritruncated 6-simplex

Tritruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol 3t{3,3,3,3,3}
Coxeter-Dynkin diagram
or
5-faces 14 2t{3,3,3,3}
4-faces 84
Cells 280
Faces 490
Edges 420
Vertices 140
Vertex figure
Coxeter group A6, [[35]], order 10080
Properties convex, isotopic

The tritruncated 6-simplex is an isotopic uniform polytope, with 14 identical bitruncated 5-simplex facets.

### Alternate names

• Tetradecapeton (as a 14-facetted 6-polytope) (Acronym: fe) (Jonathan Bowers)[3]

### Coordinates

The vertices of the tritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,2). This construction is based on facets of the bitruncated 7-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Symmetry [[7]](*)=[14] [6] [[5]](*)=[10]
Ak Coxeter plane A3 A2
Graph
Symmetry [4] [[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

### Related polytopes

Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name t{3}
Hexagon
r{3,3}
Octahedron
2t{3,3,3}
Decachoron
2r{3,3,3,3}
Dodecateron
3t{3,3,3,3,3}
3r{3,3,3,3,3,3}
4t{3,3,3,3,3,3,3}
Coxeter
diagram
Images
Facets {3} t{3,3} r{3,3,3} 2t{3,3,3,3} 2r{3,3,3,3,3} 3t{3,3,3,3,3,3}

## Related uniform 6-polytopes

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

 t0 t1 t2 t0,1 t0,2 t1,2 t0,3 t1,3 t2,3 t0,4 t1,4 t0,5 t0,1,2 t0,1,3 t0,2,3 t1,2,3 t0,1,4 t0,2,4 t1,2,4 t0,3,4 t0,1,5 t0,2,5 t0,1,2,3 t0,1,2,4 t0,1,3,4 t0,2,3,4 t1,2,3,4 t0,1,2,5 t0,1,3,5 t0,2,3,5 t0,1,4,5 t0,1,2,3,4 t0,1,2,3,5 t0,1,2,4,5 t0,1,2,3,4,5

## Notes

1. ^ Klitzing, (o3x3o3o3o3o - til)
2. ^ Klitzing, (o3x3x3o3o3o - batal)
3. ^ Klitzing, (o3o3x3x3o3o - fe)

## 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, 6D, uniform polytopes (polypeta) o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe