# Stericated 5-simplexes

| 5-simplex | Stericated 5-simplex |
| Steritruncated 5-simplex | Stericantellated 5-simplex |
| Stericantitruncated 5-simplex | Steriruncitruncated 5-simplex |
| Steriruncicantitruncated 5-simplex (Omnitruncated 5-simplex) ||
| Orthogonal projections in A$_5$ and A$_4$ Coxeter planes ||
In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) 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

Stericated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | 2r2r{3,3,3,3} 2r{3$^{2,2}$} = $2r\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$ | 2r2r{3,3,3,3} 2r{3$^{2,2}$} = $2r\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagram | or | or |
| 4-faces | 62 | 6+6 {3,3,3} 15+15×{3,3} 20 {3}×{3} |
| Cells | 180 | 60 {3,3} 120×{3} |
| Faces | 210 | 120 {3} 90 {4} |
| Edges | 120 | 120 |
| Vertices | 30 | 30 |
| Vertex figure | Tetrahedral antiprism | Tetrahedral antiprism |
| Coxeter group | A$_5$×2, [[3,3,3,3]], order 1440 | A$_5$×2, [[3,3,3,3]], order 1440 |
| Properties | convex, isogonal, isotoxal | convex, isogonal, isotoxal |
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 (120 triangles and 90 squares), 180 cells (60 tetrahedra and 120 triangular prisms) and 62 4-faces (12 5-cells, 30 tetrahedral prisms and 20 3-3 duoprisms).

### Alternate names

* Expanded 5-simplex
* Stericated hexateron
* Small cellated dodecateron (Acronym: scad) (Jonathan Bowers)

### Cross-sections

The maximal cross-section of the stericated hexateron with a 4-dimensional hyperplane is a runcinated 5-cell. This cross-section divides the stericated hexateron into two pentachoral hypercupolas consisting of 6 5-cells, 15 tetrahedral prisms and 10 3-3 duoprisms each.

### Coordinates

The vertices of the stericated 5-simplex can be constructed on a hyperplane in 6-space as permutations of (0,1,1,1,1,2). 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:
    (1,-1,0,0,0,0)
The Cartesian coordinates in 5-space for the normalized vertices of an origin-centered stericated hexateron are:
    $\left(\pm 1,\ 0,\ 0,\ 0,\ 0\right)$
    $\left(0,\ \pm 1,\ 0,\ 0,\ 0\right)$
    $\left(0,\ 0,\ \pm 1,\ 0,\ 0\right)$
    $\left(\pm 1/2,\ 0,\ \pm 1/2,\ -{\sqrt {1/8}},\ -{\sqrt {3/8}}\right)$
    $\left(\pm 1/2,\ 0,\ \pm 1/2,\ {\sqrt {1/8}},\ {\sqrt {3/8}}\right)$
    $\left(0,\ \pm 1/2,\ \pm 1/2,\ -{\sqrt {1/8}},\ {\sqrt {3/8}}\right)$
    $\left(0,\ \pm 1/2,\ \pm 1/2,\ {\sqrt {1/8}},\ -{\sqrt {3/8}}\right)$
    $\left(\pm 1/2,\ \pm 1/2,\ 0,\ \pm {\sqrt {1/2}},\ 0\right)$

### Root system

Its 30 vertices represent the root vectors of the simple Lie group A$_5$. It is also the vertex figure of the 5-simplex honeycomb.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [[5]]=[10] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [[3]]=[6] |
* orthogonal projection with [6] symmetry

## Steritruncated 5-simplex

Steritruncated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,1,4}${3,3,3,3} | t$_{0,1,4}${3,3,3,3} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 62 | 6 t{3,3,3} 15×t{3,3} 20 {3}×{6} 15×{3,3} 6 t$_{0,3}${3,3,3} |
| Cells | 330 | |
| Faces | 570 | |
| Edges | 420 | 420 |
| Vertices | 120 | 120 |
| Vertex figure | | |
| Coxeter group | A$_5$ [3,3,3,3], order 720 | A$_5$ [3,3,3,3], order 720 |
| Properties | convex, isogonal | convex, isogonal |

### Alternate names

* Steritruncated hexateron
* Celliprismated hexateron (Acronym: cappix) (Jonathan Bowers)

### Coordinates

The coordinates can be made in 6-space, as 180 permutations of:
    (0,1,1,1,2,3)
This construction exists as one of 64 orthant facets of the steritruncated 6-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [5] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [3] |

## Stericantellated 5-simplex

Stericantellated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,2,4}${3,3,3,3} | t$_{0,2,4}${3,3,3,3} |
| Coxeter-Dynkin diagram | or | or |
| 4-faces | 62 | 12 rr{3,3,3} 30 rr{3,3}x 20 {3}×{3} |
| Cells | 420 | 60 rr{3,3} 240×{3} 90×× 30 r{3,3} |
| Faces | 900 | 360 {3} 540 {4} |
| Edges | 720 | 720 |
| Vertices | 180 | 180 |
| Vertex figure | | |
| Coxeter group | A$_5$×2, [[3,3,3,3]], order 1440 | A$_5$×2, [[3,3,3,3]], order 1440 |
| Properties | convex, isogonal | convex, isogonal |

### Alternate names

* Stericantellated hexateron
* Cellirhombated dodecateron (Acronym: card) (Jonathan Bowers)

### Coordinates

The coordinates can be made in 6-space, as permutations of:
    (0,1,1,2,2,3)
This construction exists as one of 64 orthant facets of the stericantellated 6-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [[5]]=[10] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [[3]]=[6] |

## Stericantitruncated 5-simplex

Stericantitruncated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,1,2,4}${3,3,3,3} | t$_{0,1,2,4}${3,3,3,3} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 62 | |
| Cells | 480 | |
| Faces | 1140 | |
| Edges | 1080 | 1080 |
| Vertices | 360 | 360 |
| Vertex figure | | |
| Coxeter group | A$_5$ [3,3,3,3], order 720 | A$_5$ [3,3,3,3], order 720 |
| Properties | convex, isogonal | convex, isogonal |

### Alternate names

* Stericantitruncated hexateron
* Celligreatorhombated hexateron (Acronym: cograx) (Jonathan Bowers)

### Coordinates

The coordinates can be made in 6-space, as 360 permutations of:
    (0,1,1,2,3,4)
This construction exists as one of 64 orthant facets of the stericantitruncated 6-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [5] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [3] |

## Steriruncitruncated 5-simplex

Steriruncitruncated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,1,3,4}${3,3,3,3} 2t{3$^{2,2}$} | t$_{0,1,3,4}${3,3,3,3} 2t{3$^{2,2}$} |
| Coxeter-Dynkin diagram | or | or |
| 4-faces | 62 | 12 t$_{0,1,3}${3,3,3} 30×t{3,3} 20 {6}×{6} |
| Cells | 450 | |
| Faces | 1110 | |
| Edges | 1080 | 1080 |
| Vertices | 360 | 360 |
| Vertex figure | | |
| Coxeter group | A$_5$×2, [[3,3,3,3]], order 1440 | A$_5$×2, [[3,3,3,3]], order 1440 |
| Properties | convex, isogonal | convex, isogonal |

### Alternate names

* Steriruncitruncated hexateron
* Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)

### Coordinates

The coordinates can be made in 6-space, as 360 permutations of:
    (0,1,2,2,3,4)
This construction exists as one of 64 orthant facets of the steriruncitruncated 6-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [[5]]=[10] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [[3]]=[6] |

## Omnitruncated 5-simplex

Omnitruncated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,1,2,3,4}${3,3,3,3} 2tr{3$^{2,2}$} | t$_{0,1,2,3,4}${3,3,3,3} 2tr{3$^{2,2}$} |
| Coxeter-Dynkin diagram | or | or |
| 4-faces | 62 | 12 t$_{0,1,2,3}${3,3,3} 30×tr{3,3} 20 {6}×{6} |
| Cells | 540 | 360 t{3,4} 90 {4,3} 90×{6} |
| Faces | 1560 | 480 {6} 1080 {4} |
| Edges | 1800 | 1800 |
| Vertices | 720 | 720 |
| Vertex figure | Irregular 5-cell | Irregular 5-cell |
| Coxeter group | A$_5$×2, [[3,3,3,3]], order 1440 | A$_5$×2, [[3,3,3,3]], order 1440 |
| Properties | convex, isogonal, zonotope | convex, isogonal, zonotope |
The omnitruncated 5-simplex has 720 vertices, 1800 edges, 1560 faces (480 hexagons and 1080 squares), 540 cells (360 truncated octahedra, 90 cubes, and 90 hexagonal prisms), and 62 4-faces (12 omnitruncated 5-cells, 30 truncated octahedral prisms, and 20 6-6 duoprisms).

### Alternate names

* Steriruncicantitruncated 5-simplex (Full description of omnitruncation for 5-polytopes by Johnson)
* Omnitruncated hexateron
* Great cellated dodecateron (Acronym: gocad) (Jonathan Bowers)

### Coordinates

The vertices of the omnitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,1,2,3,4,5). These coordinates come from the positive orthant facet of the steriruncicantitruncated 6-orthoplex, t$_{0,1,2,3,4}${3$^4$,4}, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [[5]]=[10] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [[3]]=[6] |

### 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 | ${\tilde {I}}_{1}$ | ${\tilde {A}}_{2}$ | ${\tilde {A}}_{3}$ | ${\tilde {A}}_{4}$ | ${\tilde {A}}_{5}$ |
| Coxeter-Dynkin | | | | | |
| Picture | | | | | |
| Name | Apeirogon | Hextille | Omnitruncated 3-simplex honeycomb | Omnitruncated 4-simplex honeycomb | Omnitruncated 5-simplex honeycomb |
| Facets | | | | | |

### Full snub 5-simplex

The full snub 5-simplex or omnisnub 5-simplex, defined as an alternation of the omnitruncated 5-simplex is not uniform, but it can be given Coxeter diagram and symmetry [[3,3,3,3]]$^+$, and constructed from 12 snub 5-cells, 30 snub tetrahedral antiprisms, 20 3-3 duoantiprisms, and 360 irregular 5-cells filling the gaps at the deleted vertices.

## Related uniform polytopes

These polytopes are a part of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A$_5$ Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
A5 polytopes :
| t$_0$ | t$_1$ | t$_2$ | t$_{0,1}$ | t$_{0,2}$ | t$_{1,2}$ | t$_{0,3}$ |
| t$_{1,3}$ | t$_{0,4}$ | t$_{0,1,2}$ | t$_{0,1,3}$ | t$_{0,2,3}$ | t$_{1,2,3}$ | t$_{0,1,4}$ |
| t$_{0,2,4}$ | t$_{0,1,2,3}$ | t$_{0,1,2,4}$ | t$_{0,1,3,4}$ | t$_{0,1,2,3,4}$ |