Steric 6-cubes
![]() 6-demicube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Steric 6-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Stericantic 6-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Steriruncic 6-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Stericruncicantic 6-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
Orthogonal projections in D6 Coxeter plane |
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In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube.
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
Related polytopes
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
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
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
Related polytopes
There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:
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, 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)". https://bendwavy.org/klitzing/dimensions/polypeta.htm. x3o3o *b3o3x3o - sophax, x3x3o *b3o3x3o - pithax, x3o3o *b3x3x3o - prohax, x3x3o *b3x3x3o - gophax
External links
- Weisstein, Eric W.. "Hypercube". http://mathworld.wolfram.com/Hypercube.html.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
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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 4-polytope | 5-cell | 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 |
![]() | Original source: https://en.wikipedia.org/wiki/Steric 6-cubes.
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