Truncated 5-cubes

120px
5-cube
| 120px
Truncated 5-cube
| 120px
Bitruncated 5-cube

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|

120px
5-orthoplex
| 120px
Truncated 5-orthoplex
| 120px
Bitruncated 5-orthoplex

Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube.

There are four unique truncations of the 5-cube. Vertices of the truncated 5-cube are located as pairs on the edge of the 5-cube. Vertices of the bitruncated 5-cube are located on the square faces of the 5-cube. The third and fourth truncations are more easily constructed as second and first truncations of the 5-orthoplex.

## Contents

* 1 Truncated 5-cube
* 1.1 Alternate names
* 1.2 Construction and coordinates
* 1.3 Images
* 1.4 Related polytopes
* 2 Bitruncated 5-cube
* 2.1 Alternate names
* 2.2 Construction and coordinates
* 2.3 Images
* 2.4 Related polytopes
* 3 Related polytopes
* 4 Notes
* 5 References
* 6 External links

## Truncated 5-cube

Truncated 5-cube

Type | uniform 5-polytope
Schläfli symbol | t{4,3,3,3}
Coxeter-Dynkin diagram |
4-faces | 42 | 10

Cells | 200 | 40
160
Faces | 400 | 80
320
Edges | 400 | 80
320
Vertices | 160
Vertex figure | 60px
( )v{3,3}
Coxeter group | B5, [3,3,3,4], order 3840
Properties | convex

### Alternate names

* Truncated penteract (Acronym: tan) (Jonathan Bowers)[1]

### Construction and coordinates

The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at 1/(2+2) of the edge length. A regular 5-cell is formed at each truncated vertex.

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

(±1, ±(1+2), ±(1+2), ±(1+2), ±(1+2))

### Images

The truncated 5-cube is constructed by a truncation applied to the 5-cube. All edges are shortened, and two new vertices are added on each original edge.

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

The truncated 5-cube, is fourth in a sequence of truncated hypercubes: Template:Truncated hypercube polytopes

## Bitruncated 5-cube

Bitruncated 5-cube

Type | uniform 5-polytope
Schläfli symbol | 2t{4,3,3,3}
Coxeter-Dynkin diagrams |

4-faces | 42 | 10

Cells | 280 | 40
160

Faces | 720 | 80
320
320
Edges | 800 | 320
480
Vertices | 320
Vertex figure | 80px
{ }v{3}
Coxeter groups | B5, [3,3,3,4], order 3840
Properties | convex

### Alternate names

* Bitruncated penteract (Acronym: bittin) (Jonathan Bowers)[2]

### Construction and coordinates

The bitruncated 5-cube may be constructed by bitruncating the vertices of the 5-cube at 2 of the edge length.

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

(0, ±1, ±2, ±2, ±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]

### Related polytopes

The bitruncated 5-cube is third in a sequence of bitruncated hypercubes: Template:Bitruncated hypercube polytopes

## Related polytopes

The truncated 5-cube and bitruncated 5-cube are from the family of 31 uniform 5-polytopes 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, (o3o3o3x4x - tan).
2. ↑ Klitzing, (o3o3x3x4o - bittin).

## 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. o3o3o3x4x - tan, o3o3x3x4o - bittin

## External links

* 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 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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