Truncated 6-simplexes


6-simplex

Truncated 6-simplex

Bitruncated 6-simplex

Tritruncated 6-simplex
Orthogonal projections in A7 Coxeter plane

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
Typeuniform 6-polytope
Schläfli symbol t{3,3,3,3,3}
Coxeter-Dynkin diagram
5-faces14:
7 {3,3,3,3}
7 t{3,3,3,3}
4-faces63:
42 {3,3,3}
21 t{3,3,3}
Cells140:
105 {3,3}
35 t{3,3}
Faces175:
140 {3}
35 {6}
Edges126
Vertices42
Vertex figureElongated 5-cell pyramid
Coxeter groupA6, [35], order 5040
Dual?
Propertiesconvex

Alternate names

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
Typeuniform 6-polytope
Schläfli symbol 2t{3,3,3,3,3}
Coxeter-Dynkin diagram
5-faces14
4-faces84
Cells245
Faces385
Edges315
Vertices105
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

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
Typeuniform 6-polytope
Schläfli symbol 3t{3,3,3,3,3}
Coxeter-Dynkin diagram
or
5-faces14 2t{3,3,3,3}
4-faces84
Cells280
Faces490
Edges420
Vertices140
Vertex figure
Coxeter groupA6, [[35]], order 10080
Propertiesconvex, 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

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

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
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\}
Tetradecapeton

3t{35}
Hexadecaexon

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

4t{37}
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}
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.

Notes

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

References

External links

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