# Stericated 8-simplexes

| 8-simplex | Stericated 8-simplex | Bistericated 8-simplex |
| Steri-truncated 8-simplex | Bisteri-truncated 8-simplex | Steri-cantellated 8-simplex | Bisteri-cantellated 8-simplex |
| Stericanti-truncated 8-simplex | Bistericanti-truncated 8-simplex | Steri-runcinated 8-simplex | Bisteri-runcinated 8-simplex |
| Steriruncitruncated 8-simplex | Bisterirun-citruncated 8-simplex | Sterirunci-cantellated 8-simplex | Bisterirunci-cantellated 8-simplex |
| Steriruncicanti-truncated 8-simplex | Bisteriruncicanti-truncated 8-simplex |
| Orthogonal projections in A$_8$ Coxeter plane |||
In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications for the 8-simplex including permutations of truncation, cantellation, and runcination.

## Stericated 8-simplex

Stericated 8-simplex :
| Type | uniform 8-polytope |
| Schläfli symbol | t$_{0,4}${3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 6300 |
| Vertices | 630 |
| Vertex figure | |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |

### Coordinates

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

## Bistericated 8-simplex

bistericated 8-simplex :
| Type | uniform 8-polytope |
| Schläfli symbol | t$_{1,5}${3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 12600 |
| Vertices | 1260 |
| Vertex figure | |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |

### Coordinates

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

## Steritruncated 8-simplex

Steritruncated 8-simplex :
| Type | uniform 8-polytope |
| Schläfli symbol | t$_{0,1,4}${3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |

### 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] |

## Bisteritruncated 8-simplex

Bisteritruncated 8-simplex :
| Type | uniform 8-polytope |
| Schläfli symbol | t$_{1,2,5}${3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |

### 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] |

## Stericantellated 8-simplex
### 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] |

## Bistericantellated 8-simplex
### 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] |

## Stericantitruncated 8-simplex
### 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] |

## Bistericantitruncated 8-simplex
### 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] |

## Steriruncinated 8-simplex
### 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] |

## Bisteriruncinated 8-simplex
### 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] |

## Steriruncitruncated 8-simplex
### 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] |

## Bisteriruncitruncated 8-simplex
### 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] |

## Steriruncicantellated 8-simplex
### 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] |

## Bisteriruncicantellated 8-simplex
### 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] |

## Steriruncicantitruncated 8-simplex
### 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] |

## Bisteriruncicantitruncated 8-simplex
### 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] |

## 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$_{01234567}$ |