# Truncated 6-simplexes

Last updated
 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.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

## Contents

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.

In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.

## 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-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 figure
( )v{3,3,3}
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.

In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.

### Images

orthographic projections
Ak Coxeter plane A6A5A4
Graph
Dihedral symmetry [7][6][5]
Ak Coxeter planeA3A2
Graph
Dihedral symmetry[4][3]

## 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-faces14
4-faces84
Cells245
Faces385
Edges315
Vertices105
Vertex figure
{ }v{3,3}
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 A6A5A4
Graph
Dihedral symmetry [7][6][5]
Ak Coxeter planeA3A2
Graph
Dihedral symmetry[4][3]

## 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-faces14 2t{3,3,3,3}
4-faces84
Cells280
Faces490
Edges420
Vertices140
Vertex figure
{3}v{3}
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.

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

### Images

orthographic projections
Ak Coxeter plane A6A5A4
Graph
Symmetry[[7]](*)=[14][6][[5]](*)=[10]
Ak Coxeter planeA3A2
Graph
Symmetry[4][[3]](*)=[6]
Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.
Isotopic uniform truncated simplices
Dim.2345678
Name
Coxeter
Hexagon
=
t{3} = {6}
Octahedron
=
r{3,3} = {31,1} = {3,4}
${\displaystyle \left\{{\begin{array}{l}3\\3\end{array}}\right\}}$
Decachoron

2t{33}
Dodecateron

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

3t{35}

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

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

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)

## Related Research Articles

In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.

In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

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

In six-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.

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

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

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

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

In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.

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

In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination of the regular 6-simplex.

In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.

In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex.

In seven-dimensional geometry, a pentellated 7-simplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-simplex.

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

In eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex.

In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex.

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

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

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

## References

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds