# Pentic 6-cubes

(Redirected from Stericated 6-demicube)
 Orthogonal projections in D6 Coxeter plane 6-demicube (half 6-cube) = Pentic 6-cube = Penticantic 6-cube = Pentiruncic 6-cube = Pentiruncicantic 6-cube = Pentisteric 6-cube = Pentistericantic 6-cube = Pentisteriruncic 6-cube = Pentisteriruncicantic 6-cube =

In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope.

There are 8 pentic forms of the 6-cube.

## Pentic 6-cube

Pentic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,4{3,34,1}
h5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 1440
Vertices 192
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

The pentic 6-cube, , has half of the vertices of a pentellated 6-cube, .

### Alternate names

• Stericated 6-demicube/demihexeract
• Small cellated hemihexeract (Acronym: sochax) (Jonathan Bowers)[1]

### Cartesian coordinates

The Cartesian coordinates for the vertices of a pentic 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]

## Penticantic 6-cube

Penticantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,4{3,34,1}
h2,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 9600
Vertices 1920
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

The penticantic 6-cube, , has half of the vertices of a penticantellated 6-cube, .

### Alternate names

• Steritruncated 6-demicube/demihexeract
• cellitruncated hemihexeract (Acronym: cathix) (Jonathan Bowers)[2]

### Cartesian coordinates

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

(±1,±1,±3,±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]

## Pentiruncic 6-cube

Pentiruncic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,2,4{3,34,1}
h3,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 10560
Vertices 1920
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

The pentiruncic 6-cube, , has half of the vertices of a pentiruncinated 6-cube (penticantellated 6-orthoplex), .

### Alternate names

• Stericantellated 6-demicube/demihexeract
• cellirhombated hemihexeract (Acronym: crohax) (Jonathan Bowers)[3]

### Cartesian coordinates

The Cartesian coordinates for the vertices of a pentiruncic 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]

## Pentiruncicantic 6-cube

Pentiruncicantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2,4{3,32,1}
h2,3,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 20160
Vertices 5760
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

The pentiruncicantic 6-cube, , has half of the vertices of a pentiruncicantellated 6-cube or (pentiruncicantellated 6-orthoplex),

### Alternate names

• Stericantitruncated demihexeract, stericantitruncated 7-demicube
• Great cellated hemihexeract (Acronym: cagrohax) (Jonathan Bowers)[4]

### Cartesian coordinates

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

(±1,±1,±3,±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]

## Pentisteric 6-cube

Pentisteric 6-cube
Type uniform 6-polytope
Schläfli symbol t0,3,4{3,34,1}
h4,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 5280
Vertices 960
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

The pentisteric 6-cube, , has half of the vertices of a pentistericated 6-cube (pentitruncated 6-orthoplex),

### Alternate names

• Steriruncinated 6-demicube/demihexeract
• Small cellipriamated hemihexeract (Acronym: cophix) (Jonathan Bowers)[5]

### Cartesian coordinates

The Cartesian coordinates for the vertices of a pentisteric 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]

## Pentistericantic 6-cube

Pentistericantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,3,4{3,34,1}
h2,4,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 23040
Vertices 5760
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

The pentistericantic 6-cube, , has half of the vertices of a pentistericantellated 6-cube (pentiruncitruncated 6-orthoplex), .

### Alternate names

• Steriruncicantitruncated demihexeract/7-demicube
• cellitruncated hemihexeract (Acronym: capthix) (Jonathan Bowers)[6]

### Cartesian coordinates

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

(±1,±1,±3,±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]

## Pentisteriruncic 6-cube

Pentisteriruncic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,2,3,4{3,34,1}
h3,4,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 15360
Vertices 3840
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

The pentisteriruncic 6-cube, , has half of the vertices of a pentisteriruncinated 6-cube (penticantitruncated 6-orthoplex), .

### Alternate names

• Steriruncicantellated 6-demicube/demihexeract
• Celliprismatorhombated hemihexeract (Acronym: caprohax) (Jonathan Bowers)[7]

### Cartesian coordinates

The Cartesian coordinates for the vertices of a pentisteriruncic 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]

## Pentisteriruncicantic 6-cube

Pentisteriruncicantic 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2,3,4{3,32,1}
h2,3,4,5{4,34}
Coxeter-Dynkin diagram =
5-faces
4-faces
Cells
Faces
Edges 34560
Vertices 11520
Vertex figure
Coxeter groups D6, [33,1,1]
Properties convex

The pentisteriruncicantic 6-cube, , has half of the vertices of a pentisteriruncicantellated 6-cube (pentisteriruncicantitruncated 6-orthoplex), .

### Alternate names

• Steriruncicantitruncated 6-demicube/demihexeract
• Great cellated hemihexeract (Acronym: gochax) ((Jonathan Bowers)[8]

### Cartesian coordinates

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

(±1,±1,±3,±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 BC6 symmetry, and 16 are unique:

## Notes

1. ^ Klitzing, (x3o3o *b3o3x3o3o - sochax)
2. ^ Klitzing, (x3x3o *b3o3x3o3o - cathix)
3. ^ Klitzing, (x3o3o *b3x3x3o3o - crohax)
4. ^ Klitzing, (x3x3o *b3x3x3o3o - cagrohax)
5. ^ Klitzing, (x3o3o *b3o3x3x3x - cophix)
6. ^ Klitzing, (x3x3o *b3o3x3x3x - capthix)
7. ^ Klitzing, (x3o3o *b3x3x3x3x - caprohax)
8. ^ Klitzing, (x3x3o *b3x3x3x3o - gochax)

## 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 *b3o3x3o3o - sochax, x3x3o *b3o3x3o3o - cathix, x3o3o *b3x3x3o3o - crohax, x3x3o *b3x3x3o3o - cagrohax, x3o3o *b3o3x3x3x - cophix, x3x3o *b3o3x3x3x - capthix, x3o3o *b3x3x3x3x - caprohax, x3x3o *b3x3x3x3o - gochax