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 D6 Coxeter plane |
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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/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]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/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]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/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]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 7-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]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/demihexeract
- Small cellipriamated 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]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/7-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]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/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]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]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]- ^ Klitzing, (x3o3o *b3o3x3o3o - sochax)
- ^ Klitzing, (x3x3o *b3o3x3o3o - cathix)
- ^ Klitzing, (x3o3o *b3x3x3o3o - crohax)
- ^ Klitzing, (x3x3o *b3x3x3o3o - cagrohax)
- ^ Klitzing, (x3o3o *b3o3x3x3x - cophix)
- ^ Klitzing, (x3x3o *b3o3x3x3x - capthix)
- ^ Klitzing, (x3o3o *b3x3x3x3x - caprohax)
- ^ 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, 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
External links
[edit]- Weisstein, Eric W. "Hypercube". MathWorld.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary