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.
 7-demicube       |
 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 | ||||
|---|---|---|---|---|
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
| 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] |
Hexicantic 7-cube
Alternate names
- Teritruncated demihepteract (acronym: tuthesa)[2]
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] |
Hexiruncic 7-cube
Alternate names
- Terirhombated demihepteract (acronym: turhesa)[3]
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] |
Hexisteric 7-cube
Alternate names
- Teriprismated demihepteract (acronym: tuphesa)[4]
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] |
Hexipentic 7-cube
Alternate names
- Tericellated demihepteract (acronym: tuchesa)[5]
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] |
Hexiruncicantic 7-cube
Alternate names
- Terigreatorhombated demihepteract (acronym: tugrohesa)[6]
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] |
Hexistericantic 7-cube
Alternate names
- Teriprismatotruncated demihepteract (acronym: tupthesa)[7]
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] |
Hexipenticantic 7-cube
Alternate names
- Tericellitruncated demihepteract (acronym: tucothesa)[8]
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] |
Hexisteriruncic 7-cube
Alternate names
- Teriprismatorhombated demihepteract (acronym: tuprohesa)[9]
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] |
Hexipentiruncic 7-cube
Alternate names
- Tericellirhombated demihepteract (acronym: tucrohesa)[10]
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] |
Hexipentisteric 7-cube
Alternate names
- Tericelliprismated demihepteract (acronym: tucophesa)[11]
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] |
Hexisteriruncicantic 7-cube
Alternate names
- Terigreatoprismated demihepteract (acronym: tugphesa)[12]
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] |
Hexipentiruncicantic 7-cube
Alternate names
- Tericelligreatorhombated demihepteract (acronym: tucagrohesa)[13]
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] |
Hexipentisteriruncic 7-cube
Alternate names
- Tericelliprismatorhombated demihepteract (acronym: tucprohesa)[14]
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] |
Hexipentistericantic 7-cube
Alternate names
- Tericelliprismatotruncated demihepteract (acronym: tucpathesa)[15]
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] |
Hexipentisteriruncicantic 7-cube
Alternate names
- Great terated demihepteract (acronym: guthesa)[16]
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
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:
