Runcic 5-cubes
Concept in geometry
From Wikipedia, the free encyclopedia
In six-dimensional geometry, a runcic 5-cube or (runcic 5-demicube, runcihalf 5-cube) is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.
 5-cube     |
 Runcic 5-cube   =  | ||
 5-demicube = |
 Runcicantic 5-cube = | ||
| Orthogonal projections in B5 Coxeter plane | |||
|---|---|---|---|
Runcic 5-cube
| Runcic 5-cube | |
|---|---|
| Type | uniform 5-polytope |
| Schläfli symbol | h3{4,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | 42 |
| Cells | 360 |
| Faces | 880 |
| Edges | 720 |
| Vertices | 160 |
| Vertex figure | |
| Coxeter groups | D5, [32,1,1] |
| Properties | convex |
Alternate names
- Cantellated 5-demicube/demipenteract
- Small rhombated hemipenteract (sirhin) (Jonathan Bowers)[1]
Cartesian coordinates
The Cartesian coordinates for the 960 vertices of a runcic 5-cubes centered at the origin are coordinate permutations:
- (±1,±1,±1,±3,±3)
with an odd number of plus signs.
Images
| Coxeter plane | B5 | |
|---|---|---|
| Graph | | |
| Dihedral symmetry | [10/2] | |
| Coxeter plane | D5 | D4 |
| Graph |  |
 |
| Dihedral symmetry | [8] | [6] |
| Coxeter plane | D3 | A3 |
| Graph |  |
 |
| Dihedral symmetry | [4] | [4] |
Related polytopes
It has half the vertices of the runcinated 5-cube, as compared here in the B5 Coxeter plane projections:
Runcic 5-cube |
 Runcinated 5-cube |
Runcicantic 5-cube
| Runcicantic 5-cube | |
|---|---|
| Type | uniform 5-polytope |
| Schläfli symbol | t0,1,2{3,32,1} h3{4,33} |
| Coxeter-Dynkin diagram | |
| 4-faces | 42 |
| Cells | 360 |
| Faces | 1040 |
| Edges | 1200 |
| Vertices | 480 |
| Vertex figure | |
| Coxeter groups | D5, [32,1,1] |
| Properties | convex |
Alternate names
- Cantitruncated 5-demicube/demipenteract
- Great rhombated hemipenteract (girhin) (Jonathan Bowers)[2]
Cartesian coordinates
The Cartesian coordinates for the 480 vertices of a runcicantic 5-cube centered at the origin are coordinate permutations:
- (±1,±1,±3,±5,±5)
with an odd number of plus signs.
Images
| Coxeter plane | B5 | |
|---|---|---|
| Graph | | |
| Dihedral symmetry | [10/2] | |
| Coxeter plane | D5 | D4 |
| Graph |  |
 |
| Dihedral symmetry | [8] | [6] |
| Coxeter plane | D3 | A3 |
| Graph |  |
 |
| Dihedral symmetry | [4] | [4] |
Related polytopes
It has half the vertices of the runcicantellated 5-cube, as compared here in the B5 Coxeter plane projections:
Runcicantic 5-cube |
 Runcicantellated 5-cube |
Related polytopes
This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 23 uniform 5-polytopes that can be constructed from the D5 symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.
