Stericated 5-simplexes
In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations of the regular 5-simplex.
There are six unique sterications of the 5-simplex, including permutations of truncations, cantellations, and runcinations. The simplest stericated 5-simplex is also called an expanded 5-simplex, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-simplex. The highest form, the steriruncicantitruncated 5-simplex is more simply called an omnitruncated 5-simplex with all of the nodes ringed.
Stericated 5-simplex
A stericated 5-simplex can be constructed by an expansion operation applied to the regular 5-simplex, and thus is also sometimes called an expanded 5-simplex. It has 30 vertices, 120 edges, 210 faces, 180 cells and 62 4-faces.Alternate names
- Expanded 5-simplex
- Stericated hexateron
- Small cellated dodecateron
Cross-sections
Coordinates
The vertices of the stericated 5-simplex can be constructed on a hyperplane in 6-space as permutations of. This represents the positive orthant facet of the stericated 6-orthoplex.A second construction in 6-space, from the center of a rectified 6-orthoplex is given by coordinate permutations of:
The Cartesian coordinates in 5-space for the normalized vertices of an origin-centered stericated hexateron are:
Root system
Its 30 vertices represent the root vectors of the simple Lie group A5. It is also the vertex figure of the 5-simplex honeycomb.Images
orthogonal projection with symmetry |
Steritruncated 5-simplex
Alternate names
- Steritruncated hexateron
- Celliprismated hexateron
Coordinates
This construction exists as one of 64 orthant facets of the steritruncated 6-orthoplex.
Images
Stericantellated 5-simplex
Alternate names
- Stericantellated hexateron
- Cellirhombated dodecateron
Coordinates
This construction exists as one of 64 orthant facets of the stericantellated 6-orthoplex.
Images
Stericantitruncated 5-simplex
Alternate names
- Stericantitruncated hexateron
- Celligreatorhombated hexateron
Coordinates
This construction exists as one of 64 orthant facets of the stericantitruncated 6-orthoplex.
Images
Steriruncitruncated 5-simplex
Alternate names
- Steriruncitruncated hexateron
- Celliprismatotruncated dodecateron
Coordinates
This construction exists as one of 64 orthant facets of the steriruncitruncated 6-orthoplex.
Images
Omnitruncated 5-simplex
The omnitruncated 5-simplex has 720 vertices, 1800 edges, 1560 faces, 540 cells, and 62 4-faces.Alternate names
- Steriruncicantitruncated 5-simplex
- Omnitruncated hexateron
- Great cellated dodecateron
Coordinates
Images
Permutohedron
The omnitruncated 5-simplex is the permutohedron of order 6. It is also a zonotope, the Minkowski sum of six line segments parallel to the six lines through the origin and the six vertices of the 5-simplex.Orthogonal projection, vertices labeled as a permutohedron. |
Related honeycomb
The omnitruncated 5-simplex honeycomb is constructed by omnitruncated 5-simplex facets with 3 facets around each ridge. It has Coxeter-Dynkin diagram of.| Coxeter group | |||||
| Coxeter-Dynkin | |||||
| Picture | |||||
| Name | Apeirogon | Hextille | Omnitruncated 3-simplex honeycomb | Omnitruncated 4-simplex honeycomb | Omnitruncated 5-simplex honeycomb |
| Facets |