# Rectified 9-orthoplexes

(Redirected from Rectified 9-orthoplex)
 Orthogonal projections in A9 Coxeter plane 9-orthoplex Rectified 9-orthoplex Birectified 9-orthoplex Trirectified 9-orthoplex Quadrirectified 9-cube Trirectified 9-cube Birectified 9-cube Rectified 9-cube 9-cube

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

There are 9 rectifications of the 9-orthoplex. Vertices of the rectified 9-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-orthoplex are located in the triangular face centers of the 9-orthoplex. Vertices of the trirectified 9-orthoplex are located in the tetrahedral cell centers of the 9-orthoplex.

These polytopes are part of a family 511 uniform 9-polytopes with BC9 symmetry.

## Rectified 9-orthoplex

Rectified 9-orthoplex
Type uniform 9-polytope
Schläfli symbol t1{37,4}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 2016
Vertices 144
Vertex figure 7-orthoplex prism
Petrie polygon octakaidecagon
Coxeter groups C9, [4,37]
D9, [36,1,1]
Properties convex

The rectified 9-orthoplex is the vertex figure for the demienneractic honeycomb.

or

## Rectified 9-orthoplex

### Alternate names

• rectified enneacross (Acronym riv) (Jonathan Bowers)[1]

### Construction

There are two Coxeter groups associated with the rectified 9-orthoplex, one with the C9 or [4,37] Coxeter group, and a lower symmetry with two copies of 8-orthoplex facets, alternating, with the D9 or [36,1,1] Coxeter group.

### Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 9-orthoplex, centered at the origin, edge length ${\displaystyle {\sqrt {2}}}$ are all permutations of:

(±1,±1,0,0,0,0,0,0,0)

#### Root vectors

Its 144 vertices represent the root vectors of the simple Lie group D9. The vertices can be seen in 3 hyperplanes, with the 36 vertices rectified 8-simplexs cells on opposite sides, and 72 vertices of an expanded 8-simplex passing through the center. When combined with the 18 vertices of the 9-orthoplex, these vertices represent the 162 root vectors of the B9 and C9 simple Lie groups.

### Images

orthographic projections
B9 B8 B7
[18] [16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]

## Birectified 9-orthoplex

### Alternate names

• Rectified 9-demicube
• Birectified enneacross (Acronym brav) (Jonathan Bowers)[2]

### Images

orthographic projections
B9 B8 B7
[18] [16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]

## Trirectified 9-orthoplex

### Alternate names

• trirectified enneacross (Acronym tarv) (Jonathan Bowers)[3]

### Images

orthographic projections
B9 B8 B7
[18] [16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]

## Notes

1. ^ Klitzing (o3x3o3o3o3o3o3o4o - riv)
2. ^ Klitzing (o3o3x3o3o3o3o3o4o - brav)
3. ^ Klitzing (o3o3o3x3o3o3o3o4o - tarv)

## 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)". x3o3o3o3o3o3o3o4o - vee, o3x3o3o3o3o3o3o4o - riv, o3o3x3o3o3o3o3o4o - brav, o3o3o3x3o3o3o3o4o - tarv, o3o3o3o3x3o3o3o4o - nav, o3o3o3o3o3x3o3o4o - tarn, o3o3o3o3o3o3x3o4o - barn, o3o3o3o3o3o3o3x4o - ren, o3o3o3o3o3o3o3o4x - enne