Pentic 6-cubes

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

Orthogonal projections in D5 Coxeter plane

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

[edit]

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

[edit]

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

### Cartesian coordinates

[edit]

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±1,±3)

with an odd number of plus signs.

### Images

[edit]

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

[edit]

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

[edit]

* Steritruncated 6-demicube
* Steritruncated demihexeract
* Cellitruncated hemihexeract (Acronym: cathix) (Jonathan Bowers)[2]

### Cartesian coordinates

[edit]

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±3,±5)

with an odd number of plus signs.

### Images

[edit]

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

[edit]

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

[edit]

* Stericantellated 6-demicube
* Stericantellated demihexeract
* Cellirhombated hemihexeract (Acronym: crohax) (Jonathan Bowers)[3]

### Cartesian coordinates

[edit]

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3,±5)

with an odd number of plus signs.

### Images

[edit]

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

[edit]

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

[edit]

* Stericantitruncated demihexeract
* Stericantitruncated 6-demicube
* Great cellated hemihexeract (Acronym: cagrohax) (Jonathan Bowers)[4]

### Cartesian coordinates

[edit]

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±5,±7)

with an odd number of plus signs.

### Images

[edit]

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

[edit]

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

[edit]

* Steriruncinated 6-demicube
* Steriruncinated demihexeract
* Small celliprismated hemihexeract (Acronym: cophix) (Jonathan Bowers)[5]

### Cartesian coordinates

[edit]

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±5)

with an odd number of plus signs.

### Images

[edit]

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

[edit]

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

[edit]

* Steriruncitruncated demihexeract
* Steriruncitruncated 6-demicube
* Cellitruncated hemihexeract (Acronym: capthix) (Jonathan Bowers)[6]

### Cartesian coordinates

[edit]

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±5,±7)

with an odd number of plus signs.

### Images

[edit]

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

[edit]

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

[edit]

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

### Cartesian coordinates

[edit]

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5,±7)

with an odd number of plus signs.

### Images

[edit]

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

[edit]

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

[edit]

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

### Cartesian coordinates

[edit]

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±5,±7)

with an odd number of plus signs.

### Images

[edit]

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

[edit]

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

D6 polytopes

h{4,34} |
h2{4,34} |
h3{4,34} |
h4{4,34} |
h5{4,34} |
h2,3{4,34} |
h2,4{4,34} |
h2,5{4,34}

h3,4{4,34} |
h3,5{4,34} |
h4,5{4,34} |
h2,3,4{4,34} |
h2,3,5{4,34} |
h2,4,5{4,34} |
h3,4,5{4,34} |
h2,3,4,5{4,34}

## Notes

[edit]

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

[edit]

* 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, wiley.com, ISBN 978-0-471-01003-6
* (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) with acronyms". x3o3o *b3o3x3o3o - sochax, x3x3o *b3o3x3o3o - cathix, x3o3o *b3x3x3o3o - crohax, x3x3o *b3x3x3o3o - cagrohax, x3o3o *b3o3x3x3x - cophix, x3x3o *b3o3x3x3x - capthix, x3o3o *b3x3x3x3x - caprohax, x3x3o *b3x3x3x3o - gochax

## External links

[edit]

* Weisstein, Eric W. "Hypercube". MathWorld.
* Polytopes of Various Dimensions
* Multi-dimensional Glossary

* v
* t
* e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube |  | Dodecahedron • Icosahedron
Uniform polychoron | Pentachoron | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube |  |
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 |
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 |
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 |
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube |  |
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube |  |
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations
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