Runcic 7-cubes
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In seven-dimensional geometry, a runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms.
 7-demicube       |
 Runcic 7-cube  |
 Runcicantic 7-cube | |
| Orthogonal projections in D7 Coxeter plane | |||
|---|---|---|---|
Runcic 7-cube
| Runcic 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,2{3,34,1} h3{4,35} |
| Coxeter-Dynkin diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 16800 |
| Vertices | 2240 |
| Vertex figure | |
| Coxeter groups | D7, [34,1,1] |
| Properties | convex |
A runcic 7-cube, h3{4,35}, has half the vertices of a runcinated 7-cube, t0,3{4,35}.
Alternate names
- Small rhombated hemihepteract (acronym: sirhesa) (Jonathan Bowers)[1]
Cartesian coordinates
The Cartesian coordinates for the vertices of a cantellated demihepteract centered at the origin are coordinate permutations:
- (±1,±1,±1,±3,±3,±3,±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] |
Runcicantic 7-cube
| Runcicantic 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2{3,34,1} h2,3{4,35} |
| Coxeter-Dynkin diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23520 |
| Vertices | 6720 |
| Vertex figure | |
| Coxeter groups | D6, [33,1,1] |
| Properties | convex |
A runcicantic 7-cube, h2,3{4,35}, has half the vertices of a runcicantellated 7-cube, t0,1,3{4,35}.
Alternate names
- Great rhombated hemihepteract (acronym: girhesa) (Jonathan Bowers)[2]
Cartesian coordinates
The Cartesian coordinates for the vertices of a runcicantic 7-cube centered at the origin are coordinate permutations:
- (±1,±1,±1,±1,±3,±5,±5)
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] |
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 BC6 symmetry, and 32 are unique:
