Runcinated 5-cubes

150px
5-cube
| 150px
Runcinated 5-cube
| 150px
Runcinated 5-orthoplex

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|

150px
Runcitruncated 5-cube
| 150px
Runcicantellated 5-cube
| 150px
Runcicantitruncated 5-cube

150px
Runcitruncated 5-orthoplex
| 150px
Runcicantellated 5-orthoplex
| 150px
Runcicantitruncated 5-orthoplex

Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube.

There are 8 unique degrees of runcinations of the 5-cube, along with permutations of truncations and cantellations. Four are more simply constructed relative to the 5-orthoplex.

## Contents

* 1 Runcinated 5-cube
* 1.1 Alternate names
* 1.2 Coordinates
* 1.3 Images
* 2 Runcitruncated 5-cube
* 2.1 Alternate names
* 2.2 Construction and coordinates
* 2.3 Images
* 3 Runcicantellated 5-cube
* 3.1 Alternate names
* 3.2 Coordinates
* 3.3 Images
* 4 Runcicantitruncated 5-cube
* 4.1 Alternate names
* 4.2 Coordinates
* 4.3 Images
* 5 Related polytopes
* 6 Notes
* 7 References
* 8 External links

## Runcinated 5-cube

Runcinated 5-cube

Type  | Uniform 5-polytope
Schläfli symbol | t0,3{4,3,3,3}
Coxeter diagram |
4-faces  | 202  | 10

Cells  | 1240  | 40
240
320
160
320
160
Faces  | 2160  | 240
960
640
320
Edges  | 1440  | 480+960
Vertices  | 320
Vertex figure | 80px
Coxeter group | B5 [4,3,3,3]
Properties  | convex

### Alternate names

* Small prismated penteract (Acronym: span) (Jonathan Bowers)[1]

### Coordinates

The Cartesian coordinates of the vertices of a runcinated 5-cube having edge length 2 are all permutations of:

(±1, ±1, ±1, ±(1+2), ±(1+2))

### Images

orthographic projections Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2

|
|
|

Graph  |  |  |
Dihedral symmetry | [10]  | [8]  | [6]
Coxeter plane  | B2 | A3
Graph  |  |
Dihedral symmetry  | [4]  | [4]

## Runcitruncated 5-cube

Runcitruncated 5-cube

Type  | Uniform 5-polytope
Schläfli symbol | t0,1,3{4,3,3,3}
Coxeter-Dynkin diagrams  |
4-faces  | 202  | 10

Cells  | 1560  | 40
240
320
320
160
320
160
Faces  | 3760  | 240
960
320
960
640
640

Edges  | 3360  | 480+960+1920
Vertices  | 960
Vertex figure | 80px
Coxeter group | B5, [3,3,3,4]
Properties  | convex

### Alternate names

* Runcitruncated penteract
* Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers)[2]

### Construction and coordinates

The Cartesian coordinates of the vertices of a runcitruncated 5-cube having edge length 2 are all permutations of:

(±1, ±(1+2), ±(1+2), ±(1+22), ±(1+22))

### Images

orthographic projections Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2

|
|
|

Graph  |  |  |
Dihedral symmetry | [10]  | [8]  | [6]
Coxeter plane  | B2 | A3
Graph  |  |
Dihedral symmetry  | [4]  | [4]

## Runcicantellated 5-cube

Runcicantellated 5-cube

Type  | Uniform 5-polytope
Schläfli symbol | t0,2,3{4,3,3,3}
Coxeter-Dynkin diagram  |
4-faces  | 202  | 10

Cells  | 1240  | 40
240
320
320
160
160
Faces  | 2960  | 240
480
960
320
640
320
Edges  | 2880  | 960+960+960
Vertices  | 960
Vertex figure | 80px
Coxeter group | B5 [4,3,3,3]
Properties  | convex

### Alternate names

* Runcicantellated penteract
* Prismatorhombated penteract (Acronym: prin) (Jonathan Bowers)[3]

### Coordinates

The Cartesian coordinates of the vertices of a runcicantellated 5-cube having edge length 2 are all permutations of:

(±1, ±1, ±(1+2), ±(1+22), ±(1+22))

### Images

orthographic projections Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2

|
|
|

Graph  |  |  |
Dihedral symmetry | [10]  | [8]  | [6]
Coxeter plane  | B2 | A3
Graph  |  |
Dihedral symmetry  | [4]  | [4]

## Runcicantitruncated 5-cube

Runcicantitruncated 5-cube

Type  | Uniform 5-polytope
Schläfli symbol | t0,1,2,3{4,3,3,3}
Coxeter-Dynkin
diagram  |
4-faces | 202
Cells | 1560
Faces | 4240
Edges | 4800
Vertices | 1920
Vertex figure | 80px
Irregular 5-cell
Coxeter group | B5 [4,3,3,3]
Properties  | convex, isogonal

### Alternate names

* Runcicantitruncated penteract
* Biruncicantitruncated pentacross
* Great prismated penteract (Acronym: ) (Jonathan Bowers)[4]

### Coordinates

The Cartesian coordinates of the vertices of a runcicantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:

(1, 1+2, 1+22, 1+32, 1+32)

### Images

orthographic projections Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2

|
|
|

Graph  |  |  |
Dihedral symmetry | [10]  | [8]  | [6]
Coxeter plane  | B2 | A3
Graph  |  |
Dihedral symmetry  | [4]  | [4]

## Related polytopes

These polytopes are a part of a set of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.

B5 polytopes

β5 |
t1β5 |
t2γ5 |
t1γ5 |
γ5 |
t0,1β5 |
t0,2β5 |
t1,2β5

t0,3β5 |
t1,3γ5 |
t1,2γ5 |
t0,4γ5 |
t0,3γ5 |
t0,2γ5 |
t0,1γ5 |
t0,1,2β5

t0,1,3β5 |
t0,2,3β5 |
t1,2,3γ5 |
t0,1,4β5 |
t0,2,4γ5 |
t0,2,3γ5 |
t0,1,4γ5 |
t0,1,3γ5

t0,1,2γ5 |
t0,1,2,3β5 |
t0,1,2,4β5 |
t0,1,3,4γ5 |
t0,1,2,4γ5 |
t0,1,2,3γ5 |
t0,1,2,3,4γ5

## Notes

1. ↑ Klitzing, (o3x3o3o4x - span).
2. ↑ Klitzing, (o3x3o3x4x - pattin).
3. ↑ Klitzing, (o3x3x3o4x - prin).
4. ↑ Klitzing, (o3x3x3x4x - gippin).

## 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, 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.
* Klitzing, Richard. "5D uniform polytopes (polytera) with acronyms". https://bendwavy.org/klitzing/dimensions/polytera.htm. o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin

## External links

* Glossary for hyperspace, George Olshevsky.
* Polytopes of Various Dimensions, Jonathan Bowers
* Runcinated uniform polytera (spid), Jonathan Bowers
* 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 4-polytope | 5-cell | 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

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