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Pentic 6-cubes
| 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
| Pentic 6-cube |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagram |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
The pentic 6-cube, , has half of the vertices of a pentellated 6-cube, .
Alternate names
- Stericated 6-demicube
- Stericated demihexeract
- Small cellated hemihexeract (Acronym: sochax) (Jonathan Bowers)
Cartesian coordinates
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
Penticantic 6-cube
| Penticantic 6-cube |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagram |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
The penticantic 6-cube, , has half of the vertices of a penticantellated 6-cube, .
Alternate names
- Steritruncated 6-demicube
- Steritruncated demihexeract
- Cellitruncated hemihexeract (Acronym: cathix) (Jonathan Bowers)
Cartesian coordinates
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
Pentiruncic 6-cube
| Pentiruncic 6-cube |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagram |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
The pentiruncic 6-cube, , has half of the vertices of a pentiruncinated 6-cube (penticantellated 6-orthoplex), .
Alternate names
- Stericantellated 6-demicube
- Stericantellated demihexeract
- Cellirhombated hemihexeract (Acronym: crohax) (Jonathan Bowers)
Cartesian coordinates
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
Pentiruncicantic 6-cube
| Pentiruncicantic 6-cube |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagram |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
The pentiruncicantic 6-cube, , has half of the vertices of a pentiruncicantellated 6-cube or (pentiruncicantellated 6-orthoplex),
Alternate names
- Stericantitruncated demihexeract
- Stericantitruncated 6-demicube
- Great cellated hemihexeract (Acronym: cagrohax) (Jonathan Bowers)
Cartesian coordinates
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
Pentisteric 6-cube
| Pentisteric 6-cube |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagram |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
The pentisteric 6-cube, , has half of the vertices of a pentistericated 6-cube (pentitruncated 6-orthoplex),
Alternate names
- Steriruncinated 6-demicube
- Steriruncinated demihexeract
- Small celliprismated hemihexeract (Acronym: cophix) (Jonathan Bowers)
Cartesian coordinates
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
Pentistericantic 6-cube
| Pentistericantic 6-cube |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagram |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
The pentistericantic 6-cube, , has half of the vertices of a pentistericantellated 6-cube (pentiruncitruncated 6-orthoplex), .
Alternate names
- Steriruncitruncated demihexeract
- Steriruncitruncated 6-demicube
- Cellitruncated hemihexeract (Acronym: capthix) (Jonathan Bowers)
Cartesian coordinates
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
Pentisteriruncic 6-cube
| Pentisteriruncic 6-cube |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagram |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
The pentisteriruncic 6-cube, , has half of the vertices of a pentisteriruncinated 6-cube (penticantitruncated 6-orthoplex), .
Alternate names
- Steriruncicantellated 6-demicube
- Steriruncicantellated demihexeract
- Celliprismatorhombated hemihexeract (Acronym: caprohax) (Jonathan Bowers)
Cartesian coordinates
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
Pentisteriruncicantic 6-cube
| Pentisteriruncicantic 6-cube |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagram |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
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)
Cartesian coordinates
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
Notes
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, wiley.com,
- (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.
- x3o3o *b3o3x3o3o - sochax, x3x3o *b3o3x3o3o - cathix, x3o3o *b3x3x3o3o - crohax, x3x3o *b3x3x3o3o - cagrohax, x3o3o *b3o3x3x3x - cophix, x3x3o *b3o3x3x3x - capthix, x3o3o *b3x3x3x3x - caprohax, x3x3o *b3x3x3x3o - gochax
References
- Klitzing, (x3x3o *b3x3x3o3o - cagrohax)
- Klitzing, (x3x3o *b3x3x3x3o - gochax)
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