# Truncated 5-cubes

(Redirected from Bitruncated 5-cube)
 Orthogonal projections in B5 Coxeter plane 5-cube Truncated 5-cube Bitruncated 5-cube 5-orthoplex Truncated 5-orthoplex Bitruncated 5-orthoplex

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.

## Truncated 5-cube

Truncated 5-cube
Type uniform 5-polytope
Schläfli symbol t{4,3,3,3}
Coxeter-Dynkin diagrams
4-faces 42
Cells 200
Faces 400
Edges 400
Vertices 160
Vertex figure
( )v{3,3}
Coxeter groups B5, [3,3,3,4]
Properties convex

### Alternate names

• Truncated penteract (Acronym: tan) (Jonathan Bowers)

### Construction and coordinates

The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at ${\displaystyle 1/({\sqrt {2}}+2)}$  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:

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

### 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:

 Image Name Coxeter diagram Vertex figure ... Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube ( )v( ) ( )v{ } ( )v{3} ( )v{3,3} ( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}

## Bitruncated 5-cube

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

4-faces 42
Cells 280
Faces 720
Edges 800
Vertices 320
Vertex figure
{ }v{3}
Coxeter groups B5, [3,3,3,4]
Properties convex

### Alternate names

• Bitruncated penteract (Acronym: bittin) (Jonathan Bowers)

### Construction and coordinates

The bitruncated 5-cube may be constructed by bitruncating the vertices of the 5-cube at ${\displaystyle {\sqrt {2}}}$  of the edge length.

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

${\displaystyle \left(0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2\right)}$

### 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]

### Related polytopes

The bitruncated 5-cube is third in a sequence of bitruncated hypercubes:

 Image Name Coxeter Vertex figure ... Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube ( )v{ } { }v{ } { }v{3} { }v{3,3} { }v{3,3,3} { }v{3,3,3,3}

## Related polytopes

This polytope is one of 31 uniform 5-polytope generated from the regular 5-cube or 5-orthoplex.

## 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, 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)". o3o3o3x4x - tan, o3o3x3x4o - bittin