# Rectified 7-simplexes

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

 Orthogonal projections in A7 Coxeter plane 7-simplex Rectified 7-simplex Birectified 7-simplex Trirectified 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
Typeuniform 7-polytope
Coxeter symbol051
Schläfli symbolr{36} = {35,1}
or $\left\{{\begin{array}{l}3,3,3,3,3\\3\end{array}}\right\}$ Coxeter diagrams
Or
6-faces16
5-faces84
4-faces224
Cells350
Faces336
Edges168
Vertices28
Vertex figure6-simplex prism
Petrie polygonOctagon
Coxeter groupA7, , order 40320
Propertiesconvex

The rectified 7-simplex is the edge figure of the 251 honeycomb. It is called 05,1 for its branching Coxeter-Dynkin diagram, shown as .

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
7
.

### 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   
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry   

## Birectified 7-simplex

Birectified 7-simplex
Typeuniform 7-polytope
Coxeter symbol042
Schläfli symbol2r{3,3,3,3,3,3} = {34,2}
or $\left\{{\begin{array}{l}3,3,3,3\\3,3\end{array}}\right\}$ Coxeter diagrams
Or
6-faces16:
8 r{35}
8 2r{35}
5-faces112:
28 {34}
56 r{34}
28 2r{34}
4-faces392:
168 {33}
(56+168) r{33}
Cells770:
(420+70) {3,3}
280 {3,4}
Faces840:
(280+560) {3}
Edges420
Vertices56
Vertex figure{3}x{3,3,3}
Coxeter groupA7, , order 40320
Propertiesconvex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2
7
. It is also called 04,2 for its branching Coxeter-Dynkin diagram, shown as .

### 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   
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry   

## Trirectified 7-simplex

Trirectified 7-simplex
Typeuniform 7-polytope
Coxeter symbol033
Schläfli symbol3r{36} = {33,3}
or $\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$ Coxeter diagrams
Or
6-faces16 2r{35}
5-faces112
4-faces448
Cells980
Faces1120
Edges560
Vertices70
Vertex figure{3,3}x{3,3}
Coxeter groupA7×2, [], order 80640
Propertiesconvex, isotopic

The trirectified 7-simplex is the intersection of two regular 7-simplexes in dual configuration.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S3
7
.

This polytope is the vertex figure of the 133 honeycomb. It is called 03,3 for its branching Coxeter-Dynkin diagram, shown as .

### 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  [] 
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry []  []
Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name
Coxeter
Hexagon
=
t{3} = {6}
Octahedron
=
r{3,3} = {31,1} = {3,4}
$\left\{{\begin{array}{l}3\\3\end{array}}\right\}$ Decachoron

2t{33}
Dodecateron

2r{34} = {32,2}
$\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$ Tetradecapeton

3t{35}
Hexadecaexon

3r{36} = {33,3}
$\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$ Octadecazetton

4t{37}
Images
Vertex figure ( )v( )
{ }×{ }

{ }v{ }

{3}×{3}

{3}v{3}
{3,3}x{3,3}
{3,3}v{3,3}
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}
As
intersecting
dual
simplexes

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

## See also

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