Top Qs
Timeline
Chat
Perspective
Runcinated 5-simplexes
From Wikipedia, the free encyclopedia
Remove ads
In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.
There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations.
Remove ads
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
Remove ads
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
Remove ads
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
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
Remove ads
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)
Remove ads
Notes
References
External links
Wikiwand - on
Seamless Wikipedia browsing. On steroids.
Remove ads