Steric 6-cubes
In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube.
6-demicube = |
Steric 6-cube = |
Stericantic 6-cube = |
Steriruncic 6-cube = |
Stericruncicantic 6-cube = | |
Orthogonal projections in D6 Coxeter plane |
---|
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
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
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
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
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:
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
- Klitzing, (x3o3o *b3o3x3o - sophax)
- Klitzing, (x3x3o *b3o3x3o - pithax)
- Klitzing, (x3o3o *b3x3x3o - prohax)
- 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
- (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
External links
- Weisstein, Eric W. "Hypercube". MathWorld.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary