Runcinated 5-simplexes
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In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.
 5-simplex    |
 Runcinated 5-simplex |
 Runcitruncated 5-simplex |
 Birectified 5-simplex |
 Runcicantellated 5-simplex |
 Runcicantitruncated 5-simplex |
| Orthogonal projections in A5 Coxeter plane | ||
|---|---|---|
There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations.
Runcinated 5-simplex
| Runcinated 5-simplex | ||
| Type | Uniform 5-polytope | |
| Schläfli symbol | t0,3{3,3,3,3} | |
| Coxeter-Dynkin diagram | | |
| 4-faces | 47 | 6 t0,3{3,3,3}  20 {3}×{3} 15 { }×r{3,3} 6 r{3,3,3}  |
| Cells | 255 | 45 {3,3}  180 { }×{3} 30 r{3,3}  |
| Faces | 420 | 240 {3}  180 {4} |
| Edges | 270 | |
| Vertices | 60 | |
| Vertex figure |  | |
| Coxeter group | A5 [3,3,3,3], order 720 | |
| Properties | convex | |
Alternate names
- Runcinated hexateron
- Small prismated hexateron (Acronym: spix) (Jonathan Bowers)[1]
Coordinates
The vertices of the runcinated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,1,1,1,2) or of (0,1,1,1,2,2), seen as facets of a runcinated 6-orthoplex, or a biruncinated 6-cube respectively.
Images
| Ak Coxeter plane |
A5 | A4 |
|---|---|---|
| Graph | |
 |
| Dihedral symmetry | [6] | [5] |
| Ak Coxeter plane |
A3 | A2 |
| Graph |  |
 |
| Dihedral symmetry | [4] | [3] |
Runcitruncated 5-simplex
| Runcitruncated 5-simplex | ||
| Type | Uniform 5-polytope | |
| Schläfli symbol | t0,1,3{3,3,3,3} | |
| Coxeter-Dynkin diagram | | |
| 4-faces | 47 | 6 t0,1,3{3,3,3} 20 {3}×{6} 15 { }×r{3,3} 6 rr{3,3,3} |
| Cells | 315 | |
| Faces | 720 | |
| Edges | 630 | |
| Vertices | 180 | |
| Vertex figure |  | |
| Coxeter group | A5 [3,3,3,3], order 720 | |
| Properties | convex, isogonal | |
Alternate names
- Runcitruncated hexateron
- Prismatotruncated hexateron (Acronym: pattix) (Jonathan Bowers)[2]
Coordinates
The coordinates can be made in 6-space, as 180 permutations of:
- (0,0,1,1,2,3)
This construction exists as one of 64 orthant facets of the runcitruncated 6-orthoplex.
Images
| Ak Coxeter plane |
A5 | A4 |
|---|---|---|
| Graph | |
 |
| Dihedral symmetry | [6] | [5] |
| Ak Coxeter plane |
A3 | A2 |
| Graph |  |
 |
| Dihedral symmetry | [4] | [3] |
Runcicantellated 5-simplex
| Runcicantellated 5-simplex | ||
| Type | Uniform 5-polytope | |
| Schläfli symbol | t0,2,3{3,3,3,3} | |
| Coxeter-Dynkin diagram | | |
| 4-faces | 47 | |
| Cells | 255 | |
| Faces | 570 | |
| Edges | 540 | |
| Vertices | 180 | |
| Vertex figure |  | |
| Coxeter group | A5 [3,3,3,3], order 720 | |
| Properties | convex, isogonal | |
Alternate names
- Runcicantellated hexateron
- Biruncitruncated 5-simplex/hexateron
- Prismatorhombated hexateron (Acronym: pirx) (Jonathan Bowers)[3]
Coordinates
The coordinates can be made in 6-space, as 180 permutations of:
- (0,0,1,2,2,3)
This construction exists as one of 64 orthant facets of the runcicantellated 6-orthoplex.
Images
| Ak Coxeter plane |
A5 | A4 |
|---|---|---|
| Graph | |
 |
| Dihedral symmetry | [6] | [5] |
| Ak Coxeter plane |
A3 | A2 |
| Graph |  |
 |
| Dihedral symmetry | [4] | [3] |
Runcicantitruncated 5-simplex
| Runcicantitruncated 5-simplex | ||
| Type | Uniform 5-polytope | |
| Schläfli symbol | t0,1,2,3{3,3,3,3} | |
| Coxeter-Dynkin diagram | | |
| 4-faces | 47 | 6 t0,1,2,3{3,3,3} 20 {3}×{6} 15 {}×t{3,3} 6 tr{3,3,3} |
| Cells | 315 | 45 t0,1,2{3,3} 120 { }×{3} 120 { }×{6} 30 t{3,3} |
| Faces | 810 | 120 {3} 450 {4} 240 {6} |
| Edges | 900 | |
| Vertices | 360 | |
| Vertex figure |  Irregular 5-cell | |
| Coxeter group | A5 [3,3,3,3], order 720 | |
| Properties | convex, isogonal | |
Alternate names
- Runcicantitruncated hexateron
- Great prismated hexateron (Acronym: gippix) (Jonathan Bowers)[4]
Coordinates
The coordinates can be made in 6-space, as 360 permutations of:
- (0,0,1,2,3,4)
This construction exists as one of 64 orthant facets of the runcicantitruncated 6-orthoplex.
Images
| Ak Coxeter plane |
A5 | A4 |
|---|---|---|
| Graph | |
 |
| Dihedral symmetry | [6] | [5] |
| Ak Coxeter plane |
A3 | A2 |
| Graph |  |
 |
| Dihedral symmetry | [4] | [3] |
Related uniform 5-polytopes
These polytopes are in a set of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
