Pentic 7-cubes
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(Redirected from Steriruncicantitruncated 7-demicube)
 7-demicube (half 7-cube, h{4,35})       |
120px Pentic 7-cube h5{4,35}  |
 Penticantic 7-cube h2,5{4,35} |
 Pentiruncic 7-cube h3,5{4,35} |
 Pentiruncicantic 7-cube h2,3,5{4,35} |
 Pentisteric 7-cube h4,5{4,35} |
 Pentistericantic 7-cube h2,4,5{4,35} |
 Pentisteriruncic 7-cube h3,4,5{4,35} |
 Penticsteriruncicantic 7-cube h2,3,4,5{4,35} |
| 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 |  |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 13440 |
| Vertices | 1344 |
| Vertex figure | |
| Coxeter groups | D7, [34,1,1] |
| Properties | convex |
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
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph |  |
120px |  |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph |  |
 |
 |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph |  |
 |
|
| Dihedral symmetry |
[6] | [4] |
Related polytopes
| Dimensional family of pentic n-cubes | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| n | 6 | 7 | 8 | ||||||||
| [1+,4,3n-2] = [3,3n-3,1] |
[1+,4,34] = [3,33,1] |
[1+,4,35] = [3,34,1] |
[1+,4,36] = [3,35,1] |
||||||||
| Cantic figure |
 |
120px |  |
||||||||
| Coxeter | = |
= |
= |
||||||||
| Schläfli | h5{4,34} | h5{4,35} | h5{4,36} | ||||||||
Penticantic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph |  |
|
 |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph |  |
 |
 |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph |  |
 |
|
| Dihedral symmetry |
[6] | [4] |
Pentiruncic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph |  |
|
 |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph |  |
 |
 |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph |  |
 |
|
| Dihedral symmetry |
[6] | [4] |
Pentiruncicantic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph |  |
|
 |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph |  |
 |
 |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph |  |
 |
|
| Dihedral symmetry |
[6] | [4] |
Pentisteric 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph |  |
|
 |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph |  |
 |
 |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph |  |
 |
|
| Dihedral symmetry |
[6] | [4] |
Pentistericantic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph |  |
|
 |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph |  |
 |
 |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph |  |
 |
|
| Dihedral symmetry |
[6] | [4] |
Pentisterirunic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph |  |
|
 |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph |  |
 |
 |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph |  |
 |
|
| Dihedral symmetry |
[6] | [4] |
Pentisteriruncicantic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph |  |
|
 |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph |  |
 |
 |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph |  |
 |
|
| Dihedral symmetry |
[6] | [4] |
Related polytopes
This polytope is based on the 7-demicube, a part 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:
Notes
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- 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.
- Richard Klitzing, 7D, uniform polytopes (polyexa)
External links
- Weisstein, Eric W., "Hypercube", MathWorld.
- Olshevsky, George, Measure polytope at Glossary for Hyperspace.
- 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 | ||||||||||||
