# Rectified 7-orthoplexes

| 7-orthoplex | Rectified 7-orthoplex | Birectified 7-orthoplex | Trirectified 7-orthoplex |
| Birectified 7-cube | Rectified 7-cube | 7-cube |
| Orthogonal projections in B$_7$ Coxeter plane ||||
In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.
There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and the 6th and last being the 7-cube. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex are located in the tetrahedral cell centers of the 7-orthoplex.

## Rectified 7-orthoplex

Rectified 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | r{3,3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 142 |
| 5-faces | 1344 |
| 4-faces | 3360 |
| Cells | 3920 |
| Faces | 2520 |
| Edges | 840 |
| Vertices | 84 |
| Vertex figure | 5-orthoplex prism |
| Coxeter groups | B$_7$, [3,3,3,3,3,4] D$_7$, [3$^{4,1,1}$] |
| Properties | convex |
The rectified 7-orthoplex is the vertex figure for the demihepteractic honeycomb. The rectified 7-orthoplex's 84 vertices represent the kissing number of a sphere-packing constructed from this honeycomb.
    or

### Alternate names

* rectified heptacross
* rectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - rectified 128-faceted polyexon

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

### Construction

There are two Coxeter groups associated with the rectified heptacross, one with the C$_7$ or [4,3,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D$_7$ or [3$^{4,1,1}$] Coxeter group.

### Cartesian coordinates

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

#### Root vectors

Its 84 vertices represent the root vectors of the simple Lie group D$_7$. The vertices can be seen in 3 hyperplanes, with the 21 vertices rectified 6-simplexs cells on opposite sides, and 42 vertices of an expanded 6-simplex passing through the center. When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B$_7$ and C$_7$ simple Lie groups.

## Birectified 7-orthoplex

Birectified 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | 2r{3,3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 142 |
| 5-faces | 1428 |
| 4-faces | 6048 |
| Cells | 10640 |
| Faces | 8960 |
| Edges | 3360 |
| Vertices | 280 |
| Vertex figure | {3}×{3,3,4} |
| Coxeter groups | B$_7$, [3,3,3,3,3,4] D$_7$, [3$^{4,1,1}$] |
| Properties | convex |

### Alternate names

* Birectified heptacross
* Birectified hecatonicosoctaexon (Acronym barz) (Jonathan Bowers) - birectified 128-faceted polyexon

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

### Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 7-orthoplex, centered at the origin, edge length ${\sqrt {2}}\$ are all permutations of:
    (±1,±1,±1,0,0,0,0)

## Trirectified 7-orthoplex

A trirectified 7-orthoplex is the same as a trirectified 7-cube.