Rectified 9-cubes

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

Orthogonal projections in BC9 Coxeter plane

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

There are 9 rectifications of the 9-cube. The zeroth is the 9-cube itself, and the 8th is the dual 9-orthoplex. Vertices of the rectified 9-cube are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-cube are located in the square face centers of the 9-cube. Vertices of the trirectified 9-orthoplex are located in the cube cell centers of the 9-cube. Vertices of the quadrirectified 9-cube are located in the tesseract centers of the 9-cube.

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

## Rectified 9-cube

[edit]

### Alternate names

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* Rectified enneract (Acronym ren) (Jonathan Bowers)[1]

### Images

[edit]

orthographic projections B9 | B8 | B7

|
|

|  |  |  |  |
|  |
[18]  | [16]  | [14]
B6 | B5

[12]  | [10]
B4 | B3 | B2
|  |
[8]  | [6]  | [4]
A7 | A5 | A3
—  | —  | —
[8]  | [6]  | [4]

## Birectified 9-cube

[edit]

### Alternate names

[edit]

* Birectified enneract (Acronym barn) (Jonathan Bowers)[2]

### Images

[edit]

orthographic projections B9 | B8 | B7

|
|

|  |  |  |  |
|  |
[18]  | [16]  | [14]
B6 | B5

[12]  | [10]
B4 | B3 | B2
|  |
[8]  | [6]  | [4]
A7 | A5 | A3
—  | —  | —
[8]  | [6]  | [4]

## Trirectified 9-cube

[edit]

### Alternate names

[edit]

* Trirectified enneract (Acronym tarn) (Jonathan Bowers)[3]

### Images

[edit]

orthographic projections B9 | B8 | B7

|
|

|  |  |  |  |
|  |
[18]  | [16]  | [14]
B6 | B5

[12]  | [10]
B4 | B3 | B2
|  |
[8]  | [6]  | [4]
A7 | A5 | A3
—  | —  | —
[8]  | [6]  | [4]

## Quadrirectified 9-cube

[edit]

### Alternate names

[edit]

* Quadrirectified enneract (Acronym nav) (Jonathan Bowers)[4]

### Images

[edit]

orthographic projections B9 | B8 | B7

|
|

|  |  |  |  |
|  |
[18]  | [16]  | [14]
B6 | B5

[12]  | [10]
B4 | B3 | B2
|  |
[8]  | [6]  | [4]
A7 | A5 | A3
—  | —  | —
[8]  | [6]  | [4]

## Notes

[edit]

1. ^ Klitzing (o3o3o3o3o3o3o3x4o - ren)
2. ^ Klitzing (o3o3o3o3o3o3x3o4o - barn)
3. ^ Klitzing (o3o3o3o3o3x3o3o4o - tarn)
4. ^ Klitzing (o3o3o3o3x3o3o3o4o - nav)

## References

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* 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

## External links

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* Polytopes of Various Dimensions
* Multi-dimensional Glossary

* v
* t
* e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube |  | Dodecahedron • Icosahedron
Uniform polychoron | Pentachoron | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube |  |
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 |
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 |
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 |
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube |  |
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube |  |
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations
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