# Truncated 6-orthoplexes

(Redirected from Truncated 6-orthoplex)
 Orthogonal projections in B6 Coxeter plane 6-orthoplex Truncated 6-orthoplex Bitruncated 6-orthoplex Tritruncated 6-cube 6-cube Truncated 6-cube Bitruncated 6-cube

In six-dimensional geometry, a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex.

There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex are located as pairs on the edge of the 6-orthoplex. Vertices of the bitruncated 6-orthoplex are located on the triangular faces of the 6-orthoplex. Vertices of the tritruncated 6-orthoplex are located inside the tetrahedral cells of the 6-orthoplex.

## Truncated 6-orthoplex

Truncated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol t{3,3,3,3,4}
Coxeter-Dynkin diagrams

5-faces 76
4-faces 576
Cells 1200
Faces 1120
Edges 540
Vertices 120
Vertex figure Elongated 16-cell pyramid
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

### Alternate names

• Truncated hexacross
• Truncated hexacontatetrapeton (Acronym: tag) (Jonathan Bowers)[1]

### Construction

There are two Coxeter groups associated with the truncated hexacross, one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.

### Coordinates

Cartesian coordinates for the vertices of a truncated 6-orthoplex, centered at the origin, are all 120 vertices are sign (4) and coordinate (30) permutations of

(±2,±1,0,0,0,0)

### Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

## Bitruncated 6-orthoplex

Bitruncated 6-orthoplex
Type uniform 6-polytope
Schläfli symbol 2t{3,3,3,3,4}
Coxeter-Dynkin diagrams

5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

### Alternate names

• Bitruncated hexacross
• Bitruncated hexacontatetrapeton (Acronym: botag) (Jonathan Bowers)[2]

### Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

## Related polytopes

Thes polytopes are a part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

## Notes

1. ^ Klitzing, (x3x3o3o3o4o - tag)
2. ^ Klitzing, (o3x3x3o3o4o - botag)

## 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. "6D uniform polytopes (polypeta)". x3x3o3o3o4o - tag, o3x3x3o3o4o - botag