Hexic 7-cubes
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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
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 | File:7-demicube t05 B7.svg | File:7-demicube t05 D7.svg | File:7-demicube t05 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t05 D5.svg | File:7-demicube t05 D4.svg | File:7-demicube t05 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t05 A5.svg | File:7-demicube t05 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexicantic 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t015 B7.svg | File:7-demicube t015 D7.svg | File:7-demicube t015 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t015 D5.svg | File:7-demicube t015 D4.svg | File:7-demicube t015 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t015 A5.svg | File:7-demicube t015 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexiruncic 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t025 B7.svg | File:7-demicube t025 D7.svg | File:7-demicube t025 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t025 D5.svg | File:7-demicube t025 D4.svg | File:7-demicube t025 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t025 A5.svg | File:7-demicube t025 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexisteric 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t035 B7.svg | File:7-demicube t035 D7.svg | File:7-demicube t035 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t035 D5.svg | File:7-demicube t035 D4.svg | File:7-demicube t035 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t035 A5.svg | File:7-demicube t035 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexipentic 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t045 B7.svg | File:7-demicube t045 D7.svg | File:7-demicube t045 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t045 D5.svg | File:7-demicube t045 D4.svg | File:7-demicube t045 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t045 A5.svg | File:7-demicube t045 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexiruncicantic 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t0125 B7.svg | File:7-demicube t0125 D7.svg | File:7-demicube t0125 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t0125 D5.svg | File:7-demicube t0125 D4.svg | File:7-demicube t0125 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t0125 A5.svg | File:7-demicube t0125 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexistericantic 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t0135 B7.svg | File:7-demicube t0135 D7.svg | File:7-demicube t0135 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t0135 D5.svg | File:7-demicube t0135 D4.svg | File:7-demicube t0135 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t0135 A5.svg | File:7-demicube t0135 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexipenticantic 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t0145 B7.svg | File:7-demicube t0145 D7.svg | File:7-demicube t0145 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t0145 D5.svg | File:7-demicube t0145 D4.svg | File:7-demicube t0145 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t0145 A5.svg | File:7-demicube t0145 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexisteriruncic 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t0235 B7.svg | File:7-demicube t0235 D7.svg | File:7-demicube t0235 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t0235 D5.svg | File:7-demicube t0235 D4.svg | File:7-demicube t0235 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t0235 A5.svg | File:7-demicube t0235 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexipentiruncic 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t0245 B7.svg | File:7-demicube t0245 D7.svg | File:7-demicube t0245 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t0245 D5.svg | File:7-demicube t0245 D4.svg | File:7-demicube t0245 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t0245 A5.svg | File:7-demicube t0245 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexipentisteric 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t0345 B7.svg | File:7-demicube t0345 D7.svg | File:7-demicube t0345 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t0345 D5.svg | File:7-demicube t0345 D4.svg | File:7-demicube t0345 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t0345 A5.svg | File:7-demicube t0345 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexisteriruncicantic 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t01235 B7.svg | File:7-demicube t01235 D7.svg | File:7-demicube t01235 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t01235 D5.svg | File:7-demicube t01235 D4.svg | File:7-demicube t01235 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t01235 A5.svg | File:7-demicube t01235 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexipentiruncicantic 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t01245 B7.svg | File:7-demicube t01245 D7.svg | File:7-demicube t01245 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t01245 D5.svg | File:7-demicube t01245 D4.svg | File:7-demicube t01245 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t01245 A5.svg | File:7-demicube t01245 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexipentisteriruncic 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t02345 B7.svg | File:7-demicube t02345 D7.svg | File:7-demicube t02345 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t02345 D5.svg | File:7-demicube t02345 D4.svg | File:7-demicube t02345 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t02345 A5.svg | File:7-demicube t02345 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexipentistericantic 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t01345 B7.svg | File:7-demicube t01345 D7.svg | File:7-demicube t01345 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t01345 D5.svg | File:7-demicube t01345 D4.svg | File:7-demicube t01345 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t01345 A5.svg | File:7-demicube t01345 A3.svg | |
Dihedral symmetry |
[6] | [4] |
Hexipentisteriruncicantic 7-cube
Images
Coxeter plane |
B7 | D7 | D6 |
---|---|---|---|
Graph | File:7-demicube t012345 B7.svg | File:7-demicube t012345 D7.svg | File:7-demicube t012345 D6.svg |
Dihedral symmetry |
[14/2] | [12] | [10] |
Coxeter plane | D5 | D4 | D3 |
Graph | File:7-demicube t012345 D5.svg | File:7-demicube t012345 D4.svg | File:7-demicube t012345 D3.svg |
Dihedral symmetry |
[8] | [6] | [4] |
Coxeter plane |
A5 | A3 | |
Graph | File:7-demicube t012345 A5.svg | File:7-demicube t012345 A3.svg | |
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
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. "7D uniform polytopes (polyexa)".
External links
- Weisstein, Eric W. "Hypercube". MathWorld.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary