Truncated 8-simplexes
# Truncated 8-simplexes

| 8-simplex | Truncated 8-simplex | Rectified 8-simplex |
| - | - | - |
| Quadritruncated 8-simplex | Tritruncated 8-simplex | Bitruncated 8-simplex |
| Orthogonal projections in $A_8$ Coxeter plane |||

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

There are four unique degrees of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated 8-simplex are located on the triangular faces of the 8-simplex. Vertices of the tritruncated 8-simplex are located inside the tetrahedral cells of the 8-simplex.

## Truncated 8-simplex

Truncated 8-simplex :

| Type | uniform 8-polytope |
| - | - |
| Schläfli symbol | $t{3^7}$ |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 288 |
| Vertices | 72 |
| Vertex figure | ( )v{3,3,3,3,3} |
| Coxeter group | $A_8,$ $[3^7],$ order 362880 |
| Properties | convex |

### Alternate names

* Truncated enneazetton (Acronym: tene) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the truncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 9-orthoplex.

### Images

orthographic projections :

| $A_k$ Coxeter plane | $A_8$ | $A_7$ | $A_6$ | $A_5$ |
| - | - | - | - | - |
| Graph | | | | |
| Dihedral symmetry | [9] | [8] | [7] | [6] |
| $A_k$ Coxeter plane | $A_4$ | $A_3$ | $A_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Bitruncated 8-simplex

Bitruncated 8-simplex :

| Type | uniform 8-polytope |
| - | - |
| Schläfli symbol | $2t{3^7}$ |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1008 |
| Vertices | 252 |
| Vertex figure | { }v{3,3,3,3} |
| Coxeter group | $A_8,$ $[3^7],$ order 362880 |
| Properties | convex |

### Alternate names

* Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the bitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 9-orthoplex.

### Images

orthographic projections :

| $A_k$ Coxeter plane | $A_8$ | $A_7$ | $A_6$ | $A_5$ |
| - | - | - | - | - |
| Graph | | | | |
| Dihedral symmetry | [9] | [8] | [7] | [6] |
| $A_k$ Coxeter plane | $A_4$ | $A_3$ | $A_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Tritruncated 8-simplex

tritruncated 8-simplex :

| Type | uniform 8-polytope |
| - | - |
| Schläfli symbol | $3t{3^7}$ |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 2016 |
| Vertices | 504 |
| Vertex figure | {3}v{3,3,3} |
| Coxeter group | $A_8,$ $[3^7],$ order 362880 |
| Properties | convex |

### Alternate names

* Tritruncated enneazetton (Acronym: tatene) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the tritruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 9-orthoplex.

### Images

orthographic projections :

| $A_k$ Coxeter plane | $A_8$ | $A_7$ | $A_6$ | $A_5$ |
| - | - | - | - | - |
| Graph | | | | |
| Dihedral symmetry | [9] | [8] | [7] | [6] |
| $A_k$ Coxeter plane | $A_4$ | $A_3$ | $A_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Quadritruncated 8-simplex

Quadritruncated 8-simplex :

| Type | uniform 8-polytope |
| - | - |
| Schläfli symbol | $4t{3^7}$ |
| Coxeter-Dynkin diagrams | or |
| 6-faces | 18 3t{3,3,3,3,3,3} |
| 7-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 2520 |
| Vertices | 630 |
| Vertex figure | {3,3}v{3,3} |
| Coxeter group | $A_8,$ $[[3^7]],$ order 725760 |
| Properties | convex, isotopic |

The quadritruncated 8-simplex an isotopic polytope, constructed from 18 tritruncated 7-simplex facets.

### Alternate names

* Octadecazetton (18-facetted 8-polytope) (Acronym: be) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the quadritruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,2,2,2). This construction is based on facets of the quadritruncated 9-orthoplex.

### Images

orthographic projections :

| $A_k$ Coxeter plane | $A_8$ | $A_7$ | $A_6$ | $A_5$ |
| - | - | - | - | - |
| Graph | | | | |
| Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
| $A_k$ Coxeter plane | $A_4$ | $A_3$ | $A_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |

### Related polytopes

Isotopic uniform truncated simplices :

| Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| - | - | - | - | - | - | - | - |
| Name Coxeter | Hexagon = t{3} = {6} | Octahedron = r{3,3} = ${3^{1,1}}$ = {3,4} $\left\{{\begin{array}{l}3\\3\end{array}}\right\}$ | Decachoron $2t{3^3}$ | Dodecateron $2r{3^4}$ = ${3^{2,2}}$ $\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$ | Tetradecapeton $3t{3^5}$ | Hexadecaexon $3r{3^6}$ = ${3^{3,3}}$ $\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$ | Octadecazetton $4t{3^7}$ |
| Images | | | | | | | |
| Vertex figure | ( )∨( ) | { }×{ } | { }∨{ } | {3}×{3} | {3}∨{3} | {3,3}×{3,3} | {3,3}∨{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 polytopes

This polytope is one of 135 uniform 8-polytopes with $A_8$ symmetry.

A8 polytopes :

| $t_0$ | $t_1$ | $t_2$ | $t_3$ | $t_{01}$ | $t_{02}$ | $t_{12}$ | $t_{03}$ | $t_{13}$ | $t_{23}$ | $t_{04}$ | $t_{14}$ | $t_{24}$ | $t_{34}$ | $t_{05}$ |
| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |
| $t_{15}$ | $t_{25}$ | $t_{06}$ | $t_{16}$ | $t_{07}$ | $t_{012}$ | $t_{013}$ | $t_{023}$ | $t_{123}$ | $t_{014}$ | $t_{024}$ | $t_{124}$ | $t_{034}$ | $t_{134}$ | $t_{234}$ |
| $t_{015}$ | $t_{025}$ | $t_{125}$ | $t_{035}$ | $t_{135}$ | $t_{235}$ | $t_{045}$ | $t_{145}$ | $t_{016}$ | $t_{026}$ | $t_{126}$ | $t_{036}$ | $t_{136}$ | $t_{046}$ | $t_{056}$ |
| $t_{017}$ | $t_{027}$ | $t_{037}$ | $t_{0123}$ | $t_{0124}$ | $t_{0134}$ | $t_{0234}$ | $t_{1234}$ | $t_{0125}$ | $t_{0135}$ | $t_{0235}$ | $t_{1235}$ | $t_{0145}$ | $t_{0245}$ | $t_{1245}$ |
| $t_{0345}$ | $t_{1345}$ | $t_{2345}$ | $t_{0126}$ | $t_{0136}$ | $t_{0236}$ | $t_{1236}$ | $t_{0146}$ | $t_{0246}$ | $t_{1246}$ | $t_{0346}$ | $t_{1346}$ | $t_{0156}$ | $t_{0256}$ | $t_{1256}$ |
| $t_{0356}$ | $t_{0456}$ | $t_{0127}$ | $t_{0137}$ | $t_{0237}$ | $t_{0147}$ | $t_{0247}$ | $t_{0347}$ | $t_{0157}$ | $t_{0257}$ | $t_{0167}$ | $t_{01234}$ | $t_{01235}$ | $t_{01245}$ | $t_{01345}$ |
| $t_{02345}$ | $t_{12345}$ | $t_{01236}$ | $t_{01246}$ | $t_{01346}$ | $t_{02346}$ | $t_{12346}$ | $t_{01256}$ | $t_{01356}$ | $t_{02356}$ | $t_{12356}$ | $t_{01456}$ | $t_{02456}$ | $t_{03456}$ | $t_{01237}$ |
| $t_{01247}$ | $t_{01347}$ | $t_{02347}$ | $t_{01257}$ | $t_{01357}$ | $t_{02357}$ | $t_{01457}$ | $t_{01267}$ | $t_{01367}$ | $t_{012345}$ | $t_{012346}$ | $t_{012356}$ | $t_{012456}$ | $t_{013456}$ | $t_{023456}$ |
| $t_{123456}$ | $t_{012347}$ | $t_{012357}$ | $t_{012457}$ | $t_{013457}$ | $t_{023457}$ | $t_{012367}$ | $t_{012467}$ | $t_{013467}$ | $t_{012567}$ | $t_{0123456}$ | $t_{0123457}$ | $t_{0123467}$ | $t_{0123567}$ | $t_{01234567}$ |
