Pentic 7-cubes
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| 130px 7-demicube (half 7-cube, h{4,35})      
|
130px Pentic 7-cube h5{4,35} 
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130px Penticantic 7-cube h2,5{4,35}
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| 130px Pentiruncic 7-cube h3,5{4,35}
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130px Pentiruncicantic 7-cube h2,3,5{4,35}
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130px Pentisteric 7-cube h4,5{4,35}
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| 130px Pentistericantic 7-cube h2,4,5{4,35}
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130px Pentisteriruncic 7-cube h3,4,5{4,35}
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130px Penticsteriruncicantic 7-cube h2,3,4,5{4,35}
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| Orthogonal projections in D7 Coxeter plane | ||
|---|---|---|
In seven-dimensional geometry, a pentic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 8 unique forms.
Pentic 7-cube
| Pentic 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,4{3,34,1} h5{4,35} |
| Coxeter-Dynkin diagram | 
|
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 13440 |
| Vertices | 1344 |
| Vertex figure | |
| Coxeter groups | D7, [34,1,1] |
| Properties | convex |
Alternate names
- Small cellated demihepteract (acronym: sochesa)[1]
Cartesian coordinates
The Cartesian coordinates for the vertices of a pentic 7-cube centered at the origin are coordinate permutations:
- (±1,±1,±1,±1,±1,±3,±3)
with an odd number of plus signs.
Images
Template:7-demicube Coxeter plane graphs
Related polytopes
Penticantic 7-cube
Alternate names
- Cellitruncated demihepteract (acronym: cothesa)[2]
Images
Template:7-demicube Coxeter plane graphs
Pentiruncic 7-cube
Alternate names
- Cellirhombated demihepteract (acronym: crohesa)[3]
Images
Template:7-demicube Coxeter plane graphs
Pentiruncicantic 7-cube
Alternate names
- Celligreatorhombated demihepteract (acronym: cagrohesa)[4]
Images
Template:7-demicube Coxeter plane graphs
Pentisteric 7-cube
Alternate names
- Celliprismated demihepteract (acronym: caphesa)[5]
Images
Template:7-demicube Coxeter plane graphs
Pentistericantic 7-cube
Alternate names
- Celliprismatotruncated demihepteract (acronym: capthesa)[6]
Images
Template:7-demicube Coxeter plane graphs
Pentisteriruncic 7-cube
Alternate names
- Celliprismatorhombated demihepteract (acronym: coprahesa)[7]
Images
Template:7-demicube Coxeter plane graphs
Pentisteriruncicantic 7-cube
Alternate names
- Great cellated demihepteract (acronym: gochesa)[8]
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 *b3o3o3x3o - sochesa)
- ↑ Klitzing, (x3x3o *b3o3o3x3o - cothesa)
- ↑ Klitzing, (x3o3o *b3x3o3x3o - crohesa)
- ↑ Klitzing, (x3x3o *b3x3o3x3o - cagrohesa)
- ↑ Klitzing, (x3o3o *b3o3x3x3o - caphesa)
- ↑ Klitzing, (x3x3o *b3o3x3x3o - capthesa)
- ↑ Klitzing, (x3o3o *b3x3x3x3o - coprahesa)
- ↑ Klitzing, (x3x3o *b3x3x3x3o - gochesa)
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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