Hexic 7-cubes
From Infogalactic: the planetary knowledge core
 7-demicube       |
120px Hexic 7-cube  |
 Hexicantic 7-cube |
 Hexiruncic 7-cube |
 Hexiruncicantic 7-cube |
 Hexisteric 7-cube |
 Hexistericantic 7-cube |
 Hexisteriruncic 7-cube |
 Hexisteriruncicantic 7-cube |
 Hexipentic 7-cube |
 Hexipenticantic 7-cube |
 Hexipentiruncic 7-cube |
 Hexipentiruncicantic 7-cube |
 Hexipentisteric 7-cube |
 Hexipentistericantic 7-cube |
 Hexipentisteriruncic 7-cube |
 Hexipentisteriruncicantic 7-cube |
|||
| 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.
Contents
- 1 Hexic 7-cube
- 2 Hexicantic 7-cube
- 3 Hexiruncic 7-cube
- 4 Hexisteric 7-cube
- 5 Hexipentic 7-cube
- 6 Hexiruncicantic 7-cube
- 7 Hexistericantic 7-cube
- 8 Hexipenticantic 7-cube
- 9 Hexisteriruncic 7-cube
- 10 Hexipentiruncic 7-cube
- 11 Hexipentisteric 7-cube
- 12 Hexisteriruncicantic 7-cube
- 13 Hexipentiruncicantic 7-cube
- 14 Hexipentisteriruncic 7-cube
- 15 Hexipentistericantic 7-cube
- 16 Hexipentisteriruncicantic 7-cube
- 17 Related polytopes
- 18 Notes
- 19 References
- 20 External links
Hexic 7-cube
| Hexic 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,5{3,34,1} h6{4,35} |
| Coxeter-Dynkin diagram | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 4704 |
| Vertices | 448 |
| Vertex figure | |
| Coxeter groups | D7, [34,1,1] |
| Properties | convex |
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
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexicantic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexiruncic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexisteric 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexipentic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexiruncicantic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexistericantic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexipenticantic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexisteriruncic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexipentiruncic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexipentisteric 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexisteriruncicantic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexipentiruncicantic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexipentisteriruncic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexipentistericantic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| Dihedral symmetry |
[6] | [4] |
Hexipentisteriruncicantic 7-cube
Images
| Coxeter plane |
B7 | D7 | D6 |
|---|---|---|---|
| Graph | 150px | |
150px |
| Dihedral symmetry |
[14/2] | [12] | [10] |
| Coxeter plane | D5 | D4 | D3 |
| Graph | 150px | 150px | 150px |
| Dihedral symmetry |
[8] | [6] | [4] |
| Coxeter plane |
A5 | A3 | |
| Graph | 150px | 150px | |
| 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 | ||||||||||||
