# Cantellated 5-cubes

(Redirected from Cantitruncated 5-cube)
 Orthogonal projections in B5 Coxeter plane 5-cube Cantellated 5-cube Bicantellated 5-cube Cantellated 5-orthoplex 5-orthoplex Cantitruncated 5-cube Bicantitruncated 5-cube Cantitruncated 5-orthoplex

In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.

There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex

## Cantellated 5-cube

 Cantellated 5-cube Type Uniform 5-polytope Schläfli symbol rr{4,3,3,3} = ${\displaystyle r\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}}$ Coxeter-Dynkin diagram = 4-faces 122 Cells 680 Faces 1520 Edges 1280 Vertices 320 Vertex figure Coxeter group B5 [4,3,3,3] Properties convex

### Alternate names

• Small rhombated penteract (Acronym: sirn) (Jonathan Bowers)

### Coordinates

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

${\displaystyle \left(\pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}})\right)}$

### 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]

## Bicantellated 5-cube

 Bicantellated 5-cube Type Uniform 5-polytope Schläfli symbols 2rr{4,3,3,3} = ${\displaystyle r\left\{{\begin{array}{l}3,4\\3,3\end{array}}\right\}}$ r{32,1,1} = ${\displaystyle r\left\{{\begin{array}{l}3,3\\3\\3\end{array}}\right\}}$ Coxeter-Dynkin diagrams = 4-faces 122 Cells 840 Faces 2160 Edges 1920 Vertices 480 Vertex figure Coxeter group B5 [4,3,3,3] Properties convex

In five-dimensional geometry, a bicantellated 5-cube is a uniform 5-polytope.

### Alternate names

• Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross
• Small birhombated penteractitriacontiditeron (Acronym: sibrant) (Jonathan Bowers)

### Coordinates

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

(0,1,1,2,2)

### 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]

## Cantitruncated 5-cube

 Cantitruncated 5-cube Type Uniform 5-polytope Schläfli symbol tr{4,3,3,3} = ${\displaystyle t\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}}$ Coxeter-Dynkindiagram = 4-faces 122 Cells 680 Faces 1520 Edges 1600 Vertices 640 Vertex figure Irr. 5-cell Coxeter group B5 [4,3,3,3] Properties convex, isogonal

### Alternate names

• Tricantitruncated 5-orthoplex / tricantitruncated pentacross
• Great rhombated penteract (girn) (Jonathan Bowers)

### Coordinates

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

${\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}}\right)}$

### 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]

## Bicantitruncated 5-cube

Bicantitruncated 5-cube
Type uniform 5-polytope
Schläfli symbol 2tr{3,3,3,4} = ${\displaystyle t\left\{{\begin{array}{l}3,4\\3,3\end{array}}\right\}}$
t{32,1,1} = ${\displaystyle t\left\{{\begin{array}{l}3,3\\3\\3\end{array}}\right\}}$
Coxeter-Dynkin diagrams           =

4-faces 122
Cells 840
Faces 2160
Edges 2400
Vertices 960
Vertex figure
Coxeter groups B5, [3,3,3,4]
D5, [32,1,1]
Properties convex

### Alternate names

• Bicantitruncated penteract
• Bicantitruncated pentacross
• Great birhombated penteractitriacontiditeron (Acronym: gibrant) (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a bicantitruncated 5-cube, centered at the origin, are all sign and coordinate permutations of

(±3,±3,±2,±1,0)

### 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 from a set of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

It is third in a series of cantitruncated hypercubes:

## 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, editied 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.
• Klitzing, Richard. "5D uniform polytopes (polytera)". o3o3x3o4x - sirn, o3x3o3x4o - sibrant, o3o3x3x4x - girn, o3x3x3x4o - gibrant