# Rectified 7-simplexes

(Redirected from Trirectified 7-simplex)
 Orthogonal projections in A7 Coxeter plane 7-simplex Rectified 7-simplex Birectified 7-simplex Trirectified 7-simplex

In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.

There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex. Vertices of the birectified 7-simplex are located in the triangular face centers of the 7-simplex. Vertices of the trirectified 7-simplex are located in the tetrahedral cell centers of the 7-simplex.

## Rectified 7-simplex

Rectified 7-simplex
Type uniform 7-polytope
Coxeter symbol 051
Schläfli symbol r{3,3,3,3,3,3}
Coxeter diagrams
Or
6-faces 16
5-faces 84
4-faces 224
Cells 350
Faces 336
Edges 168
Vertices 28
Vertex figure 6-simplex prism
Petrie polygon Octagon
Coxeter group A7, [36], order 40320
Properties convex

The rectified 7-simplex is the edge figure of the 251 honeycomb.

### Alternate names

• Rectified octaexon (Acronym: roc) (Jonathan Bowers)

### Coordinates

The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 8-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Birectified 7-simplex

Birectified 7-simplex
Type uniform 7-polytope
Coxeter symbol 042
Schläfli symbol 2r{3,3,3,3,3,3}
Coxeter diagrams
Or
6-faces 16:
8 r{35}
8 2r{35}
5-faces 112:
28 {34}
56 r{34}
28 2r{34}
4-faces 392:
168 {33}
(56+168) r{33}
Cells 770:
(420+70) {3,3}
280 {3,4}
Faces 840:
(280+560) {3}
Edges 420
Vertices 56
Vertex figure {3}x{3,3,3}
Coxeter group A7, [36], order 40320
Properties convex

### Alternate names

• Birectified octaexon (Acronym: broc) (Jonathan Bowers)

### Coordinates

The vertices of the birectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 8-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Trirectified 7-simplex

Trirectified 7-simplex
Type uniform 7-polytope
Coxeter symbol 033
Schläfli symbol 3r{36}
Coxeter diagrams
Or
6-faces 16 2r{35}
5-faces 112
4-faces 448
Cells 980
Faces 1120
Edges 560
Vertices 70
Vertex figure {3,3}x{3,3}
Coxeter group|A7×2, [[36]], order 80640
Properties convex, isotopic

This polytope is the vertex figure of the 133 honeycomb.

### Alternate names

• Hexadecaexon (Acronym: he) (Jonathan Bowers)

### Coordinates

The vertices of the trirectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 8-orthoplex.

The trirectified 7-simplex is the intersection of two regular 7-simplices in dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space: the 70 distinct permutations of (1,1,1,1,−1,−1,−1,-1).

### Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

### Related polytopes

Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name t{3}
Hexagon
r{3,3}
Octahedron
2t{3,3,3}
Decachoron
2r{3,3,3,3}
Dodecateron
3t{3,3,3,3,3}
3r{3,3,3,3,3,3}
4t{3,3,3,3,3,3,3}
Coxeter
diagram
Images
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}

## Related polytopes

These polytopes are three of 71 uniform 7-polytopes with A7 symmetry.

 t0 t1 t2 t3 t0,1 t0,2 t1,2 t0,3 t1,3 t2,3 t0,4 t1,4 t2,4 t0,5 t1,5 t0,6 t0,1,2 t0,1,3 t0,2,3 t1,2,3 t0,1,4 t0,2,4 t1,2,4 t0,3,4 t1,3,4 t2,3,4 t0,1,5 t0,2,5 t1,2,5 t0,3,5 t1,3,5 t0,4,5 t0,1,6 t0,2,6 t0,3,6 t0,1,2,3 t0,1,2,4 t0,1,3,4 t0,2,3,4 t1,2,3,4 t0,1,2,5 t0,1,3,5 t0,2,3,5 t1,2,3,5 t0,1,4,5 t0,2,4,5 t1,2,4,5 t0,3,4,5 t0,1,2,6 t0,1,3,6 t0,2,3,6 t0,1,4,6 t0,2,4,6 t0,1,5,6 t0,1,2,3,4 t0,1,2,3,5 t0,1,2,4,5 t0,1,3,4,5 t0,2,3,4,5 t1,2,3,4,5 t0,1,2,3,6 t0,1,2,4,6 t0,1,3,4,6 t0,2,3,4,6 t0,1,2,5,6 t0,1,3,5,6 t0,1,2,3,4,5 t0,1,2,3,4,6 t0,1,2,3,5,6 t0,1,2,4,5,6 t0,1,2,3,4,5,6