# Rectified 8-orthoplexes

| 8-orthoplex | Rectified 8-orthoplex | Birectified 8-orthoplex | Trirectified 8-orthoplex |
| Trirectified 8-cube | Birectified 8-cube | Rectified 8-cube | 8-cube |
| Orthogonal projections in A$_8$ Coxeter plane ||||
In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.
There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being the 8-cube. Vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the tetrahedral cell centers of the 8-orthoplex.

## Rectified 8-orthoplex

Rectified 8-orthoplex :
| Type | uniform 8-polytope |
| Schläfli symbol | t$_1${3,3,3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 7-faces | 272 |
| 6-faces | 3072 |
| 5-faces | 8960 |
| 4-faces | 12544 |
| Cells | 10080 |
| Faces | 4928 |
| Edges | 1344 |
| Vertices | 112 |
| Vertex figure | 6-orthoplex prism |
| Petrie polygon | hexakaidecagon |
| Coxeter groups | C$_8$, [4,3$^6$] D$_8$, [3$^{5,1,1}$] |
| Properties | convex |
The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the simple Lie group D$_8$. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, and 56 vertices of an expanded 7-simplex passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B$_8$ and C$_8$ simple Lie groups.

### Related polytopes

The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb.
    or

### Alternate names

* rectified octacross
* rectified diacosipentacontahexazetton (Acronym: rek) (Jonathan Bowers)

### Construction

There are two Coxeter groups associated with the rectified 8-orthoplex, one with the C$_8$ or [4,3$^6$] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D$_8$ or [3$^{5,1,1}$] Coxeter group.

### Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 8-orthoplex, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:
    (±1,±1,0,0,0,0,0,0)

### Images

orthographic projections :
| B$_8$ | B$_8$ | B$_8$ | B$_7$ | B$_7$ | B$_7$ |
| [16] | [16] | [16] | [14] | [14] | [14] |
| B$_6$ | B$_6$ | B$_6$ | B$_5$ | B$_5$ | B$_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| B$_4$ | B$_4$ | B$_3$ | B$_3$ | B$_2$ | B$_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| A$_7$ | A$_7$ | A$_5$ | A$_5$ | A$_3$ | A$_3$ |
| [8] | [8] | [6] | [6] | [4] | [4] |

## Birectified 8-orthoplex

Birectified 8-orthoplex :
| Type | uniform 8-polytope |
| Schläfli symbol | t$_2${3,3,3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 7-faces | 272 |
| 6-faces | 3184 |
| 5-faces | 16128 |
| 4-faces | 34048 |
| Cells | 36960 |
| Faces | 22400 |
| Edges | 6720 |
| Vertices | 448 |
| Vertex figure | {3,3,3,4}x{3} |
| Coxeter groups | C$_8$, [3,3,3,3,3,3,4] D$_8$, [3$^{5,1,1}$] |
| Properties | convex |

### Alternate names

* birectified octacross
* birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)

### Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 8-orthoplex, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:
    (±1,±1,±1,0,0,0,0,0)

### Images

orthographic projections :
| B$_8$ | B$_8$ | B$_8$ | B$_7$ | B$_7$ | B$_7$ |
| [16] | [16] | [16] | [14] | [14] | [14] |
| B$_6$ | B$_6$ | B$_6$ | B$_5$ | B$_5$ | B$_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| B$_4$ | B$_4$ | B$_3$ | B$_3$ | B$_2$ | B$_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| A$_7$ | A$_7$ | A$_5$ | A$_5$ | A$_3$ | A$_3$ |
| [8] | [8] | [6] | [6] | [4] | [4] |

## Trirectified 8-orthoplex

Trirectified 8-orthoplex :
| Type | uniform 8-polytope |
| Schläfli symbol | t$_3${3,3,3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 7-faces | 16+256 |
| 6-faces | 1024 + 2048 + 112 |
| 5-faces | 1792 + 7168 + 7168 + 448 |
| 4-faces | 1792 + 10752 + 21504 + 14336 |
| Cells | 8960 + 126880 + 35840 |
| Faces | 17920 + 35840 |
| Edges | 17920 |
| Vertices | 1120 |
| Vertex figure | {3,3,4}x{3,3} |
| Coxeter groups | C$_8$, [3,3,3,3,3,3,4] D$_8$, [3$^{5,1,1}$] |
| Properties | convex |
The trirectified 8-orthoplex can tessellate space in the quadrirectified 8-cubic honeycomb.

### Alternate names

* trirectified octacross
* trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)

### Cartesian coordinates

Cartesian coordinates for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:
    (±1,±1,±1,±1,0,0,0,0)

### Images

orthographic projections :
| B$_8$ | B$_8$ | B$_8$ | B$_7$ | B$_7$ | B$_7$ |
| [16] | [16] | [16] | [14] | [14] | [14] |
| B$_6$ | B$_6$ | B$_6$ | B$_5$ | B$_5$ | B$_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| B$_4$ | B$_4$ | B$_3$ | B$_3$ | B$_2$ | B$_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| A$_7$ | A$_7$ | A$_5$ | A$_5$ | A$_3$ | A$_3$ |
| [8] | [8] | [6] | [6] | [4] | [4] |