# Cantellated 6-demicube

 Orthogonal projections in D6Coxeter plane 6-cube Cantellated 6-demicube Cantitruncated 6-demicube

In six-dimensional geometry, a cantellated 6-demicube is a convex uniform 6-polytope, being a cantellation of the uniform 6-demicube. There are 2 unique cantellation for the 6-demicube including a truncation.

## Cantellated 6-demicube

Cantellated 6-demicube
Type uniform polypeton
Schläfli symbol t0,2{3,33,1}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges 3840
Vertices 640
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

### Alternate names

• Small rhombated hemihexeract (Acronym sirhax) (Jonathan Bowers)[1]

### Cartesian coordinates

The Cartesian coordinates for the vertices of a cantellated demihexeract centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±3)

with an odd number of plus signs.

### Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
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## Cantitruncated 6-demicube

Cantitruncated 6-demicube
Type uniform polypeton
Schläfli symbol t0,1,2{3,33,1}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges 5760
Vertices 1920
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

### Alternate names

• Great rhombated hemihexeract (Acronym girhax) (Jonathan Bowers)[2]

### Cartesian coordinates

The Cartesian coordinates for the vertices of a cantitruncated demihexeract centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5,±5)

with an odd number of plus signs.

### Images

orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
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## Related polytopes

This polytope is based on the 6-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:

 t0(131) t0,1(131) t0,2(131) t0,3(131) t0,4(131) t0,1,2(131) t0,1,3(131) t0,1,4(131) t0,2,3(131) t0,2,4(131) t0,3,4(131) t0,1,2,3(131) t0,1,2,4(131) t0,1,3,4(131) t0,2,3,4(131) t0,1,2,3,4(131)
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## Notes

1. ^ Klitzing, (x3o3o *b3x3o3o - sirhax)
2. ^ Klitzing, (x3x3o *b3x3o3o - girhax)
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## 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.
• Richard Klitzing, 6D, uniform polytopes (polypeta) x3o3o *b3x3o3o, x3x3o *b3x3o3o

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