# Rectified 6-orthoplexes

| 6-orthoplex | Rectified 6-orthoplex | Birectified 6-orthoplex |
| Birectified 6-cube | Rectified 6-cube | 6-cube |
| Orthogonal projections in B$_6$ Coxeter plane |||
In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.
There are unique 6 degrees of rectifications, the zeroth being the 6-orthoplex, and the 6th and last being the 6-cube. Vertices of the rectified 6-orthoplex are located at the edge-centers of the 6-orthoplex. Vertices of the birectified 6-orthoplex are located in the triangular face centers of the 6-orthoplex.

## Rectified 6-orthoplex

Rectified hexacross :
| Type | uniform 6-polytope |
| Schläfli symbols | t$_1${3$^4$,4} or r{3$^4$,4} $\left\{{\begin{array}{l}3,3,3,4\\3\end{array}}\right\}$ r{3,3,3,3$^{1,1}$} |
| Coxeter-Dynkin diagrams | = = |
| 5-faces | 76 total: 64 rectified 5-simplex 12 5-orthoplex |
| 4-faces | 576 total: 192 rectified 5-cell 384 5-cell |
| Cells | 1200 total: 240 octahedron 960 tetrahedron |
| Faces | 1120 total: 160 and 960 triangles |
| Edges | 480 |
| Vertices | 60 |
| Vertex figure | 16-cell prism |
| Petrie polygon | Dodecagon |
| Coxeter groups | B$_6$, [3,3,3,3,4] D$_6$, [3$^{3,1,1}$] |
| Properties | convex |
The rectified 6-orthoplex is the vertex figure for the demihexeractic honeycomb.
    or

### Alternate names

* rectified hexacross
* rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers)

### Construction

There are two Coxeter groups associated with the rectified hexacross, one with the C$_6$ or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D$_6$ or [3$^{3,1,1}$] Coxeter group.

### Cartesian coordinates

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

### Images

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

### Root vectors

The 60 vertices represent the root vectors of the simple Lie group D$_6$. The vertices can be seen in 3 hyperplanes, with the 15 vertices rectified 5-simplices cells on opposite sides, and 30 vertices of an expanded 5-simplex passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B$_6$ and C$_6$ simple Lie groups.
The 60 roots of D$_6$ can be geometrically folded into H$_3$ (Icosahedral symmetry), as to , creating 2 copies of 30-vertex icosidodecahedra, with the Golden ratio between their radii:
| Rectified 6-orthoplex | Rectified 6-orthoplex | 2 icosidodecahedra |
| 3D (H3 projection) | A$_4$/B$_5$/D$_6$ Coxeter plane | H$_2$ Coxeter plane |

## Birectified 6-orthoplex

Birectified 6-orthoplex :
| Type | uniform 6-polytope |
| Schläfli symbols | t$_2${3$^4$,4} or 2r{3$^4$,4} $\left\{{\begin{array}{l}3,3,4\\3,3\end{array}}\right\}$ t$_2${3,3,3,3$^{1,1}$} |
| Coxeter-Dynkin diagrams | = = |
| 5-faces | 76 |
| 4-faces | 636 |
| Cells | 2160 |
| Faces | 2880 |
| Edges | 1440 |
| Vertices | 160 |
| Vertex figure | {3}×{3,4} duoprism |
| Petrie polygon | Dodecagon |
| Coxeter groups | B$_6$, [3,3,3,3,4] D$_6$, [3$^{3,1,1}$] |
| Properties | convex |
The birectified 6-orthoplex can tessellation space in the trirectified 6-cubic honeycomb.

### Alternate names

* birectified hexacross
* birectified hexacontitetrapeton (acronym: brag) (Jonathan Bowers)

### Cartesian coordinates

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

### Images

orthographic projections :
| Coxeter plane | B$_6$ | B$_5$ | B$_4$ |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B$_3$ | B$_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
It can also be projected into 3D-dimensions as → , a dodecahedron envelope.

## Related polytopes

These polytopes are a part a family of 63 Uniform 6-polytopes generated from the B$_6$ Coxeter plane, including the regular 6-cube or 6-orthoplex.
B6 polytopes :
| β$_6$ | t$_1$β$_6$ | t$_2$β$_6$ | t$_2$γ$_6$ | t$_1$γ$_6$ | γ$_6$ | t$_{0,1}$β$_6$ | t$_{0,2}$β$_6$ |
| t$_{1,2}$β$_6$ | t$_{0,3}$β$_6$ | t$_{1,3}$β$_6$ | t$_{2,3}$γ$_6$ | t$_{0,4}$β$_6$ | t$_{1,4}$γ$_6$ | t$_{1,3}$γ$_6$ | t$_{1,2}$γ$_6$ |
| t$_{0,5}$γ$_6$ | t$_{0,4}$γ$_6$ | t$_{0,3}$γ$_6$ | t$_{0,2}$γ$_6$ | t$_{0,1}$γ$_6$ | t$_{0,1,2}$β$_6$ | t$_{0,1,3}$β$_6$ | t$_{0,2,3}$β$_6$ |
| t$_{1,2,3}$β$_6$ | t$_{0,1,4}$β$_6$ | t$_{0,2,4}$β$_6$ | t$_{1,2,4}$β$_6$ | t$_{0,3,4}$β$_6$ | t$_{1,2,4}$γ$_6$ | t$_{1,2,3}$γ$_6$ | t$_{0,1,5}$β$_6$ |
| t$_{0,2,5}$β$_6$ | t$_{0,3,4}$γ$_6$ | t$_{0,2,5}$γ$_6$ | t$_{0,2,4}$γ$_6$ | t$_{0,2,3}$γ$_6$ | t$_{0,1,5}$γ$_6$ | t$_{0,1,4}$γ$_6$ | t$_{0,1,3}$γ$_6$ |
| t$_{0,1,2}$γ$_6$ | t$_{0,1,2,3}$β$_6$ | t$_{0,1,2,4}$β$_6$ | t$_{0,1,3,4}$β$_6$ | t$_{0,2,3,4}$β$_6$ | t$_{1,2,3,4}$γ$_6$ | t$_{0,1,2,5}$β$_6$ | t$_{0,1,3,5}$β$_6$ |
| t$_{0,2,3,5}$γ$_6$ | t$_{0,2,3,4}$γ$_6$ | t$_{0,1,4,5}$γ$_6$ | t$_{0,1,3,5}$γ$_6$ | t$_{0,1,3,4}$γ$_6$ | t$_{0,1,2,5}$γ$_6$ | t$_{0,1,2,4}$γ$_6$ | t$_{0,1,2,3}$γ$_6$ |
| t$_{0,1,2,3,4}$β$_6$ | t$_{0,1,2,3,5}$β$_6$ | t$_{0,1,2,4,5}$β$_6$ | t$_{0,1,2,4,5}$γ$_6$ | t$_{0,1,2,3,5}$γ$_6$ | t$_{0,1,2,3,4}$γ$_6$ | t$_{0,1,2,3,4,5}$γ$_6$ |