Rectified 10-cubes

10-orthoplex

Rectified 10-orthoplex

Birectified 10-orthoplex

Trirectified 10-orthoplex

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|
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Quadrirectified 10-orthoplex

Quadrirectified 10-cube

Trirectified 10-cube

Birectified 10-cube

Rectified 10-cube

10-cube

Orthogonal projections in BC10 Coxeter plane

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

There are 10 rectifications of the 10-cube, with the zeroth being the 10-cube itself. Vertices of the rectified 10-cube are located at the edge-centers of the 10-cube. Vertices of the birectified 10-cube are located in the square face centers of the 10-cube. Vertices of the trirectified 10-cube are located in the cubic cell centers of the 10-cube. The others are more simply constructed relative to the 10-cube dual polytope, the 10-orthoplex.

These polytopes are part of a family of 1023 uniform 10-polytopes with BC10 symmetry.

## Rectified 10-cube

[edit]

Rectified 10-cube

Type | uniform 10-polytope
Schläfli symbol | t1{38,4}
Coxeter-Dynkin diagrams |

9-faces |
8-faces |
7-faces |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges | 46080
Vertices | 5120
Vertex figure | 8-simplex prism
Coxeter groups | C10, [4,38]
D10, [37,1,1]
Properties | convex

### Alternate names

[edit]

* Rectified dekeract (Acronym: rade) (Jonathan Bowers)[1]

### Cartesian coordinates

[edit]

Cartesian coordinates for the vertices of a rectified 10-cube, centered at the origin, edge length  2 {\displaystyle {\sqrt {2}}} are all permutations of:

(±1,±1,±1,±1,±1,±1,±1,±1,±1,0)

### Images

[edit]

Orthographic projections B10 | B9 | B8

|
|

|  |
[20]  | [18]  | [16]
B7 | B6 | B5
|  |
[14]  | [12]  | [10]
B4 | B3 | B2
|  |
[8]  | [6]  | [4]
A9 | A5
—  | —
[10]  | [6]
A7 | A3
—  | —
[8]  | [4]

## Birectified 10-cube

[edit]

Birectified 10-cube

Type | uniform 10-polytope
Coxeter symbol | 0711
Schläfli symbol | t2{38,4}
Coxeter-Dynkin diagrams |

9-faces |
8-faces |
7-faces |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges | 184320
Vertices | 11520
Vertex figure | {4}x{36}
Coxeter groups | C10, [4,38]
D10, [37,1,1]
Properties | convex

### Alternate names

[edit]

* Birectified dekeract (Acronym: brade) (Jonathan Bowers)[2]

### Cartesian coordinates

[edit]

Cartesian coordinates for the vertices of a birectified 10-cube, centered at the origin, edge length  2 {\displaystyle {\sqrt {2}}} are all permutations of:

(±1,±1,±1,±1,±1,±1,±1,±1,0,0)

### Images

[edit]

Orthographic projections B10 | B9 | B8

|
|

|  |
[20]  | [18]  | [16]
B7 | B6 | B5
|  |
[14]  | [12]  | [10]
B4 | B3 | B2
|  |
[8]  | [6]  | [4]
A9 | A5
—  | —
[10]  | [6]
A7 | A3
—  | —
[8]  | [4]

## Trirectified 10-cube

[edit]

Trirectified 10-cube

Type | uniform 10-polytope
Schläfli symbol | t3{38,4}
Coxeter-Dynkin diagrams |

9-faces |
8-faces |
7-faces |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges | 322560
Vertices | 15360
Vertex figure | {4,3}x{35}
Coxeter groups | C10, [4,38]
D10, [37,1,1]
Properties | convex

### Alternate names

[edit]

* Trirectified dekeract (Acronym: trade) (Jonathan Bowers)[3]

### Cartesian coordinates

[edit]

Cartesian coordinates for the vertices of a triirectified 10-cube, centered at the origin, edge length  2 {\displaystyle {\sqrt {2}}} are all permutations of:

(±1,±1,±1,±1,±1,±1,±1,0,0,0)

### Images

[edit]

Orthographic projections B10 | B9 | B8

|
|

|  |
[20]  | [18]  | [16]
B7 | B6 | B5
|  |
[14]  | [12]  | [10]
B4 | B3 | B2
|  |
[8]  | [6]  | [4]
A9 | A5
—  | —
[10]  | [6]
A7 | A3
—  | —
[8]  | [4]

## Quadrirectified 10-cube

[edit]

Quadrirectified 10-cube

Type | uniform 10-polytope
Schläfli symbol | t4{38,4}
Coxeter-Dynkin diagrams |

9-faces |
8-faces |
7-faces |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges | 322560
Vertices | 13440
Vertex figure | {4,3,3}x{34}
Coxeter groups | C10, [4,38]
D10, [37,1,1]
Properties | convex

### Alternate names

[edit]

* Quadrirectified dekeract
* Quadrirectified decacross (Acronym: terade) (Jonathan Bowers)[4]

### Cartesian coordinates

[edit]

Cartesian coordinates for the vertices of a quadrirectified 10-cube, centered at the origin, edge length  2 {\displaystyle {\sqrt {2}}} are all permutations of:

(±1,±1,±1,±1,±1,±1,0,0,0,0)

### Images

[edit]

Orthographic projections B10 | B9 | B8

|
|

|  |
[20]  | [18]  | [16]
B7 | B6 | B5
|  |
[14]  | [12]  | [10]
B4 | B3 | B2
|  |
[8]  | [6]  | [4]
A9 | A5
—  | —
[10]  | [6]
A7 | A3
—  | —
[8]  | [4]

## Notes

[edit]

1. ^ Klitzing, (o3o3o3o3o3o3o3o3x4o - rade).
2. ^ Klitzing, (o3o3o3o3o3o3o3x3o4o - brade).
3. ^ Klitzing, (o3o3o3o3o3o3x3o3o4o - trade)
4. ^ Klitzing, (o3o3o3o3o3x3o3o3o4o - terade)

## References

[edit]

* 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, wiley.com, ISBN 978-0-471-01003-6
* (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. "10D uniform polytopes (polyxenna) with acronyms". x3o3o3o3o3o3o3o3o4o - ka, o3x3o3o3o3o3o3o3o4o - rake, o3o3x3o3o3o3o3o3o4o - brake, o3o3o3x3o3o3o3o3o4o - trake, o3o3o3o3x3o3o3o3o4o - terake, o3o3o3o3o3x3o3o3o4o - terade, o3o3o3o3o3o3x3o3o4o - trade, o3o3o3o3o3o3o3x3o4o - brade, o3o3o3o3o3o3o3o3x4o - rade, o3o3o3o3o3o3o3o3o4x - deker

## External links

[edit]

* Polytopes of Various Dimensions
* Multi-dimensional Glossary

* v
* t
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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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