# Steric 6-cubes

(Redirected from Runcicantitruncated 6-demicube)
 Orthogonal projections in D6 Coxeter plane 6-demicube = Steric 6-cube = Stericantic 6-cube = Steriruncic 6-cube = Stericruncicantic 6-cube =

In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube.

## Steric 6-cube

Steric 6-cube
Type uniform 6-polytope
Schläfli symbol t0,3{3,33,1}
h4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 3360
Vertices 480
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

### Alternate names

• Runcinated demihexeract/6-demicube
• Small prismated hemihexeract (Acronym sophax) (Jonathan Bowers)[1]

### Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a steric 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±1,±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]

## Stericantic 6-cube

Stericantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,3{3,33,1}
h2,4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 12960
Vertices 2880
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

### Alternate names

• Runcitruncated demihexeract/6-demicube
• Prismatotruncated hemihexeract (Acronym pithax) (Jonathan Bowers)[2]

### Cartesian coordinates

The Cartesian coordinates for the 2880 vertices of a stericantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3,±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]

## Steriruncic 6-cube

Steriruncic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,2,3{3,33,1}
h3,4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 7680
Vertices 1920
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

### Alternate names

• Runcicantellated demihexeract/6-demicube
• Prismatorhombated hemihexeract (Acronym prohax) (Jonathan Bowers)[3]

### Cartesian coordinates

The Cartesian coordinates for the 1920 vertices of a steriruncic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±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]

## Steriruncicantic 6-cube

Steriruncicantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2,3{3,32,1}
h2,3,4{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 17280
Vertices 5760
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

### Alternate names

• Runcicantitruncated demihexeract/6-demicube
• Great prismated hemihexeract (Acronym gophax) (Jonathan Bowers)[4]

### Cartesian coordinates

The Cartesian coordinates for the 5760 vertices of a steriruncicantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5,±7)

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]

## Related polytopes

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

## Notes

1. ^ Klitzing, (x3o3o *b3o3x3o - sophax)
2. ^ Klitzing, (x3x3o *b3o3x3o - pithax)
3. ^ Klitzing, (x3o3o *b3x3x3o - prohax)
4. ^ Klitzing, (x3x3o *b3x3x3o - gophax)

## 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)". x3o3o *b3o3x3o - sophax, x3x3o *b3o3x3o - pithax, x3o3o *b3x3x3o - prohax, x3x3o *b3x3x3o - gophax