Hexic 7-cubes
From HandWiki - Reading time: 4 min
| 120px 7-demicube      
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120px Hexic 7-cube 
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120px Hexicantic 7-cube
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120px Hexiruncic 7-cube
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120px Hexiruncicantic 7-cube
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| 120px Hexisteric 7-cube
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120px Hexistericantic 7-cube
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120px Hexisteriruncic 7-cube
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120px Hexisteriruncicantic 7-cube
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120px Hexipentic 7-cube
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| 120px Hexipenticantic 7-cube
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120px Hexipentiruncic 7-cube
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120px Hexipentiruncicantic 7-cube
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120px Hexipentisteric 7-cube
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120px Hexipentistericantic 7-cube
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| 120px Hexipentisteriruncic 7-cube
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120px Hexipentisteriruncicantic 7-cube
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| Orthogonal projections in D7 Coxeter plane | ||||
|---|---|---|---|---|
In seven-dimensional geometry, a hexic 7-cube is a convex uniform 7-polytope, constructed from the uniform 7-demicube. There are 16 unique forms.
Hexic 7-cube
| Hexic 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,5{3,34,1} h6{4,35} |
| Coxeter-Dynkin diagram |
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 4704 |
| Vertices | 448 |
| Vertex figure | |
| Coxeter groups | D7, [34,1,1] |
| Properties | convex |
Alternate names
- Small terated demihepteract (acronym: suthesa)[1]
Cartesian coordinates
The Cartesian coordinates for the vertices of a hexic 7-cube centered at the origin are coordinate permutations:
- (±1,±1,±1,±1,±1,±1,±3)
with an odd number of plus signs.
Images
Template:7-demicube Coxeter plane graphs
Hexicantic 7-cube
Alternate names
- Teritruncated demihepteract (acronym: tuthesa)[2]
Images
Template:7-demicube Coxeter plane graphs
Hexiruncic 7-cube
Alternate names
- Terirhombated demihepteract (acronym: turhesa)[3]
Images
Template:7-demicube Coxeter plane graphs
Hexisteric 7-cube
Alternate names
- Teriprismated demihepteract (acronym: tuphesa)[4]
Images
Template:7-demicube Coxeter plane graphs
Hexipentic 7-cube
Alternate names
- Tericellated demihepteract (acronym: tuchesa)[5]
Images
Template:7-demicube Coxeter plane graphs
Hexiruncicantic 7-cube
Alternate names
- Terigreatorhombated demihepteract (acronym: tugrohesa)[6]
Images
Template:7-demicube Coxeter plane graphs
Hexistericantic 7-cube
Alternate names
- Teriprismatotruncated demihepteract (acronym: tupthesa)[7]
Images
Template:7-demicube Coxeter plane graphs
Hexipenticantic 7-cube
Alternate names
- Tericellitruncated demihepteract (acronym: tucothesa)[8]
Images
Template:7-demicube Coxeter plane graphs
Hexisteriruncic 7-cube
Alternate names
- Teriprismatorhombated demihepteract (acronym: tuprohesa)[9]
Images
Template:7-demicube Coxeter plane graphs
Hexipentiruncic 7-cube
Alternate names
- Tericellirhombated demihepteract (acronym: tucrohesa)[10]
Images
Template:7-demicube Coxeter plane graphs
Hexipentisteric 7-cube
Alternate names
- Tericelliprismated demihepteract (acronym: tucophesa)[11]
Images
Template:7-demicube Coxeter plane graphs
Hexisteriruncicantic 7-cube
Alternate names
- Terigreatoprismated demihepteract (acronym: tugphesa)[12]
Images
Template:7-demicube Coxeter plane graphs
Hexipentiruncicantic 7-cube
Alternate names
- Tericelligreatorhombated demihepteract (acronym: tucagrohesa)[13]
Images
Template:7-demicube Coxeter plane graphs
Hexipentisteriruncic 7-cube
Alternate names
- Tericelliprismatorhombated demihepteract (acronym: tucprohesa)[14]
Images
Template:7-demicube Coxeter plane graphs
Hexipentistericantic 7-cube
Alternate names
- Tericelliprismatotruncated demihepteract (acronym: tucpathesa)[15]
Images
Template:7-demicube Coxeter plane graphs
Hexipentisteriruncicantic 7-cube
Alternate names
- Great terated demihepteract (acronym: guthesa)[16]
Images
Template:7-demicube Coxeter plane graphs
Related polytopes
These polytopes are based on the 7-demicube, a member of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 95 uniform polytopes with D7 symmetry, 63 are shared by the BC7 symmetry, and 32 are unique: Template:Demihepteract family
Notes
- ↑ Klitzing, (x3o3o *b3o3o3o3x - suthesa)
- ↑ Klitzing, (x3x3o *b3o3o3o3x - tuthesa)
- ↑ Klitzing, (x3o3o *b3x3o3o3x - turhesa)
- ↑ Klitzing, (x3o3o *b3o3x3o3x - tuphesa)
- ↑ Klitzing, (x3o3o *b3o3o3x3x - tuchesa)
- ↑ Klitzing, (x3x3o *b3x3o3o3x - tugrohesa)
- ↑ Klitzing, (x3x3o *b3o3x3o3x - tupthesa)
- ↑ Klitzing, (x3x3o *b3o3o3x3x - tucothesa)
- ↑ Klitzing, (x3o3o *b3x3x3o3x - tuprohesa)
- ↑ Klitzing, (x3o3o *b3x3o3x3x - tucrohesa)
- ↑ Klitzing, (x3o3o *b3o3x3x3x - tucophesa)
- ↑ Klitzing, (x3x3o *b3x3x3o3x - tugphesa)
- ↑ Klitzing, (x3x3o *b3x3o3x3x - tucagrohesa)
- ↑ Klitzing, (x3o3o *b3x3x3x3x - tucprohesa)
- ↑ Klitzing, (x3x3o *b3o3x3x3x - tucpathesa)
- ↑ Klitzing, (x3x3o *b3x3x3x3x - guthesa)
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 Ivić 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. "7D uniform polytopes (polyexa) with acronyms". https://bendwavy.org/klitzing/dimensions/polyexa.htm.
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
| ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 | ||||||||||||
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