# Rectified 9-simplexes

 Orthogonal projections in A9 Coxeter plane 9-simplex Rectified 9-simplex Birectified 9-simplex Trirectified 9-simplex Quadrirectified 9-simplex

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex.

These polytopes are part of a family of 271 uniform 9-polytopes with A9 symmetry.

There are unique 4 degrees of rectifications. Vertices of the rectified 9-simplex are located at the edge-centers of the 9-simplex. Vertices of the birectified 9-simplex are located in the triangular face centers of the 9-simplex. Vertices of the trirectified 9-simplex are located in the tetrahedral cell centers of the 9-simplex. Vertices of the quadrirectified 9-simplex are located in the 5-cell centers of the 9-simplex.

## Rectified 9-simplex

Rectified 9-simplex
Type uniform 9-polytope
Schläfli symbol t1{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces 20
7-faces 135
6-faces 480
5-faces 1050
4-faces 1512
Cells 1470
Faces 960
Edges 360
Vertices 45
Vertex figure 8-simplex prism
Petrie polygon decagon
Coxeter groups A9, [3,3,3,3,3,3,3,3]
Properties convex

The rectified 9-simplex is the vertex figure of the 10-demicube.

### Alternate names

• Rectified decayotton (reday) (Jonathan Bowers)[1]

### Coordinates

The Cartesian coordinates of the vertices of the rectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 10-orthoplex.

### Images

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

## Birectified 9-simplex

Birectified 9-simplex
Type uniform 9-polytope
Schläfli symbol t2{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 1260
Vertices 120
Vertex figure {3}x{3,3,3,3,3}
Coxeter groups A9, [3,3,3,3,3,3,3,3]
Properties convex

This polytope is the vertex figure for the 162 honeycomb. Its 120 vertices represent the kissing number of the related hyperbolic 10-dimensional sphere packing.

### Alternate names

• Birectified decayotton (breday) (Jonathan Bowers)[2]

### Coordinates

The Cartesian coordinates of the vertices of the birectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 10-orthoplex.

### Images

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

## Trirectified 9-simplex

Trirectified 9-simplex
Type uniform 9-polytope
Schläfli symbol t3{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3}x{3,3,3,3}
Coxeter groups A9, [3,3,3,3,3,3,3,3]
Properties convex

### Alternate names

• Trirectified decayotton (treday) (Jonathan Bowers)[3]

### Coordinates

The Cartesian coordinates of the vertices of the trirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 10-orthoplex.

### Images

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

Type uniform 9-polytope
Schläfli symbol t4{3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3,3}x{3,3,3}
Coxeter groups A9, [3,3,3,3,3,3,3,3]
Properties convex

### Alternate names

• Icosayotton (icoy) (Jonathan Bowers)[4]

### Coordinates

The Cartesian coordinates of the vertices of the quadrirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 10-orthoplex.

### Images

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

## Notes

1. ^ Klitzing, (o3x3o3o3o3o3o3o3o - reday)
2. ^ Klitzing, (o3o3x3o3o3o3o3o3o - breday)
3. ^ Klitzing, (o3o3o3x3o3o3o3o3o - treday)
4. ^ Klitzing, (o3o3o3o3x3o3o3o3o - icoy)

## 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)
• Klitzing, Richard. "9D uniform polytopes (polyyotta)". o3x3o3o3o3o3o3o3o - reday, o3o3x3o3o3o3o3o3o - breday, o3o3o3x3o3o3o3o3o - treday, o3o3o3o3x3o3o3o3o - icoy