# Truncated 6-simplexes

(Redirected from Tritruncated 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
Class A6 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
( )v{3,3,3}
Coxeter group A6, , order 5040
Dual ?
Properties convex

### Alternate names

• Truncated heptapeton (Acronym: til) (Jonathan Bowers)

### 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.

## Bitruncated 6-simplex

Bitruncated 6-simplex
Type uniform 6-polytope
Class A6 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
{ }v{3,3}
Coxeter group A6, , order 5040
Properties convex

### Alternate names

• Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers)

### 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.

## Tritruncated 6-simplex

Tritruncated 6-simplex
Type uniform 6-polytope
Class A6 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
{3}v{3}
Coxeter group A6, [], order 10080
Properties convex, isotopic

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

The tritruncated 6-simplex is the intersection of two 6-simplexes in dual configuration:       and      .

### Alternate names

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

### 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. Alternately it can be centered on the origin as permutations of (-1,-1,-1,0,1,1,1).

### Images

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
Coxeter
Hexagon
=
t{3} = {6}
Octahedron
=
r{3,3} = {31,1} = {3,4}
$\left\{{\begin{array}{l}3\\3\end{array}}\right\}$
Decachoron

2t{33}
Dodecateron

2r{34} = {32,2}
$\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$

3t{35}

3r{36} = {33,3}
$\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$

4t{37}
Images
Vertex figure ( )v( )
{ }×{ }

{ }v{ }

{3}×{3}

{3}v{3}
{3,3}x{3,3}
{3,3}v{3,3}
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}
As
intersecting
dual
simplexes

## 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.