# Rectified 10-simplexes

(Redirected from Rectified 10-simplex)
 Orthogonal projections in A9 Coxeter plane 10-simplex Rectified 10-simplex Birectified 10-simplex Trirectified 10-simplex Quadrirectified 10-simplex

In ten-dimensional geometry, a rectified 10-simplex is a convex uniform 10-polytope, being a rectification of the regular 10-simplex.

These polytopes are part of a family of 527 uniform 10-polytopes with A10 symmetry.

There are unique 5 degrees of rectifications including the zeroth, the 10-simplex itself. Vertices of the rectified 10-simplex are located at the edge-centers of the 10-simplex. Vertices of the birectified 10-simplex are located in the triangular face centers of the 10-simplex. Vertices of the trirectified 10-simplex are located in the tetrahedral cell centers of the 10-simplex. Vertices of the quadrirectified 10-simplex are located in the 5-cell centers of the 10-simplex.

## Rectified 10-simplex

Rectified 10-simplex
Type uniform polyxennon
Schläfli symbol t1{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 495
Vertices 55
Vertex figure 9-simplex prism
Petrie polygon decagon
Coxeter groups A10, [3,3,3,3,3,3,3,3,3]
Properties convex

The rectified 10-simplex is the vertex figure of the 11-demicube.

### Alternate names

• Rectified hendecaxennon (Acronym ru) (Jonathan Bowers)[1]

### Coordinates

The Cartesian coordinates of the vertices of the rectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 11-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A10 A9 A8
Graph
Dihedral symmetry [11] [10] [9]
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 10-simplex

Birectified 10-simplex
Type uniform polyyotton
Schläfli symbol t2{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 1980
Vertices 165
Vertex figure {3}x{3,3,3,3,3,3}
Coxeter groups A10, [3,3,3,3,3,3,3,3,3]
Properties convex

### Alternate names

• Birectified hendecaxennon (Acronym bru) (Jonathan Bowers)[2]

### Coordinates

The Cartesian coordinates of the vertices of the birectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 11-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A10 A9 A8
Graph
Dihedral symmetry [11] [10] [9]
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 10-simplex

Trirectified 10-simplex
Type uniform polyxennon
Schläfli symbol t3{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 4620
Vertices 330
Vertex figure {3,3}x{3,3,3,3,3}
Coxeter groups A10, [3,3,3,3,3,3,3,3,3]
Properties convex

### Alternate names

• Trirectified hendecaxennon (Jonathan Bowers)[3]

### Coordinates

The Cartesian coordinates of the vertices of the trirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the triirectified 11-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A10 A9 A8
Graph
Dihedral symmetry [11] [10] [9]
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Type uniform polyxennon
Schläfli symbol t4{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 6930
Vertices 462
Vertex figure {3,3,3}x{3,3,3,3}
Coxeter groups A10, [3,3,3,3,3,3,3,3,3]
Properties convex

### Alternate names

• Quadrirectified hendecaxennon (Acronym teru) (Jonathan Bowers)[4]

### Coordinates

The Cartesian coordinates of the vertices of the quadrirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 11-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A10 A9 A8
Graph
Dihedral symmetry [11] [10] [9]
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Notes

1. ^ Klitzing, (o3x3o3o3o3o3o3o3o3o - ru)
2. ^ Klitzing, (o3o3x3o3o3o3o3o3o3o - bru)
3. ^ Klitzing, (o3o3o3x3o3o3o3o3o3o - tru)
4. ^ Klitzing, (o3o3o3o3x3o3o3o3o3o - teru)

## References

• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Richard Klitzing, 10D, uniform polytopes (polyxenna) x3o3o3o3o3o3o3o3o3o - ux, o3x3o3o3o3o3o3o3o3o - ru, o3o3x3o3o3o3o3o3o3o - bru, o3o3o3x3o3o3o3o3o3o - tru, o3o3o3o3x3o3o3o3o3o - teru