# Cantellated 5-cell

| 5-cell | Cantellated 5-cell | Cantitruncated 5-cell |
| Orthogonal projections in A$_4$ Coxeter plane |||
In four-dimensional geometry, a cantellated 5-cell is a convex uniform 4-polytope, being a cantellation (a 2nd order truncation, up to edge-planing) of the regular 5-cell.

## Cantellated 5-cell

Cantellated 5-cell :
| Schlegel diagram with octahedral cells shown |||
| Type | Uniform 4-polytope | Uniform 4-polytope |
| Schläfli symbol | t$_{0,2}${3,3,3} rr{3,3,3} | t$_{0,2}${3,3,3} rr{3,3,3} |
| Coxeter diagram | | |
| Cells | 20 | 5 (3.4.3.4) 5 (3.3.3.3) 10 (3.4.4) |
| Faces | 80 | 50{3} 30{4} |
| Edges | 90 | 90 |
| Vertices | 30 | 30 |
| Vertex figure | Square wedge | Square wedge |
| Symmetry group | A$_4$, [3,3,3], order 120 | A$_4$, [3,3,3], order 120 |
| Properties | convex, isogonal | convex, isogonal |
| Uniform index | 3 4 5 | 3 4 5 |
The cantellated 5-cell or small rhombated pentachoron is a uniform 4-polytope. It has 30 vertices, 90 edges, 80 faces, and 20 cells. The cells are 5 cuboctahedra, 5 octahedra, and 10 triangular prisms. Each vertex is surrounded by 2 cuboctahedra, 2 triangular prisms, and 1 octahedron; the vertex figure is a nonuniform triangular prism.

### Alternate names

* Cantellated pentachoron
* Cantellated 4-simplex
* (small) prismatodispentachoron
* Rectified dispentachoron
* Small rhombated pentachoron (Acronym: Srip) (Jonathan Bowers)

### Configuration

Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
| f$_k$ | f$_0$ | f$_1$ | f$_1$ | f$_2$ | f$_2$ | f$_2$ | f$_2$ | f$_3$ | f$_3$ | f$_3$ |
| f$_0$ | 30 | 2 | 4 | 1 | 4 | 2 | 2 | 2 | 2 | 1 |
| f$_1$ | 2 | 30 | * | 1 | 2 | 0 | 0 | 2 | 1 | 0 |
| f$_1$ | 2 | * | 60 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| f$_2$ | 3 | 3 | 0 | 10 | * | * | * | 2 | 0 | 0 |
| f$_2$ | 4 | 2 | 2 | * | 30 | * | * | 1 | 1 | 0 |
| f$_2$ | 3 | 0 | 3 | * | * | 20 | * | 1 | 0 | 1 |
| f$_2$ | 3 | 0 | 3 | * | * | * | 20 | 0 | 1 | 1 |
| f$_3$ | 12 | 12 | 12 | 4 | 6 | 4 | 0 | 5 | * | * |
| f$_3$ | 6 | 3 | 6 | 0 | 3 | 0 | 2 | * | 10 | * |
| f$_3$ | 6 | 0 | 12 | 0 | 0 | 4 | 4 | * | * | 5 |

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |
| Wireframe | Ten triangular prisms colored green | Five octahedra colored blue |

### Coordinates

The Cartesian coordinates of the vertices of the origin-centered cantellated 5-cell having edge length 2 are:
Coordinates :
| $\left(2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)$ $\left(2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)$ $\left(2{\sqrt {\frac {2}{5}}},\ 0,\ \pm {\sqrt {3}},\ \pm 1\right)$ $\left(2{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ \pm 2\right)$ $\left(2{\sqrt {\frac {2}{5}}},\ -2{\sqrt {\frac {2}{3}}},\ {\frac {2}{\sqrt {3}}},\ 0\right)$ $\left(2{\sqrt {\frac {2}{5}}},\ -2{\sqrt {\frac {2}{3}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 1\right)$ $\left({\frac {-1}{\sqrt {10}}},\ {\sqrt {\frac {3}{2}}},\ \pm {\sqrt {3}},\ \pm 1\right)$ | $\left({\frac {-1}{\sqrt {10}}},\ {\sqrt {\frac {3}{2}}},\ 0,\ \pm 2\right)$ $\left({\frac {-1}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ \pm 2\right)$ $\left({\frac {-1}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {-4}{\sqrt {3}}},\ 0\right)$ $\left({\frac {-1}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)$ $\left({\frac {-1}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)$ $\left(-3{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ 0\right)\pm \left(0,\ {\sqrt {\frac {2}{3}}},\ {\frac {2}{\sqrt {3}}},\ 0\right)$ $\left(-3{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ 0\right)\pm \left(0,\ {\sqrt {\frac {2}{3}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 1\right)$ |
The vertices of the cantellated 5-cell can be most simply positioned in 5-space as permutations of:
    (0,0,1,1,2)
This construction is from the positive orthant facet of the cantellated 5-orthoplex.

### Related polytopes

The convex hull of two cantellated 5-cells in opposite positions is a nonuniform polychoron composed of 100 cells: three kinds of 70 octahedra (10 rectified tetrahedra, 20 triangular antiprisms, 40 triangular antipodiums), 30 tetrahedra (as tetragonal disphenoids), and 60 vertices. Its vertex figure is a shape topologically equivalent to a cube with a triangular prism attached to one of its square faces.
Vertex figure

## Cantitruncated 5-cell

Cantitruncated 5-cell :
| Schlegel diagram with Truncated tetrahedral cells shown |||
| Type | Uniform 4-polytope | Uniform 4-polytope |
| Schläfli symbol | t$_{0,1,2}${3,3,3} tr{3,3,3} | t$_{0,1,2}${3,3,3} tr{3,3,3} |
| Coxeter diagram | | |
| Cells | 20 | 5 (4.6.6) 10 (3.4.4) 5 (3.6.6) |
| Faces | 80 | 20{3} 30{4} 30{6} |
| Edges | 120 | 120 |
| Vertices | 60 | 60 |
| Vertex figure | sphenoid | sphenoid |
| Symmetry group | A$_4$, [3,3,3], order 120 | A$_4$, [3,3,3], order 120 |
| Properties | convex, isogonal | convex, isogonal |
| Uniform index | 6 7 8 | 6 7 8 |
The cantitruncated 5-cell or great rhombated pentachoron is a uniform 4-polytope. It is composed of 60 vertices, 120 edges, 80 faces, and 20 cells. The cells are: 5 truncated octahedra, 10 triangular prisms, and 5 truncated tetrahedra. Each vertex is surrounded by 2 truncated octahedra, one triangular prism, and one truncated tetrahedron.

### Configuration

Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
| f$_k$ | f$_0$ | f$_1$ | f$_1$ | f$_1$ | f$_2$ | f$_2$ | f$_2$ | f$_2$ | f$_3$ | f$_3$ | f$_3$ |
| f$_0$ | 60 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 1 | 1 |
| f$_1$ | 2 | 30 | * | * | 1 | 2 | 0 | 0 | 2 | 1 | 0 |
| f$_1$ | 2 | * | 30 | * | 1 | 0 | 2 | 0 | 2 | 0 | 1 |
| f$_1$ | 2 | * | * | 60 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| f$_2$ | 6 | 3 | 3 | 0 | 10 | * | * | * | 2 | 0 | 0 |
| f$_2$ | 4 | 2 | 0 | 2 | * | 30 | * | * | 1 | 1 | 0 |
| f$_2$ | 6 | 0 | 3 | 3 | * | * | 20 | * | 1 | 0 | 1 |
| f$_2$ | 3 | 0 | 0 | 3 | * | * | * | 20 | 0 | 1 | 1 |
| f$_3$ | 24 | 12 | 12 | 12 | 4 | 6 | 4 | 0 | 5 | * | * |
| f$_3$ | 6 | 3 | 0 | 6 | 0 | 3 | 0 | 2 | * | 10 | * |
| f$_3$ | 12 | 0 | 6 | 12 | 0 | 0 | 4 | 4 | * | * | 5 |

### Alternative names

* Cantitruncated pentachoron
* Cantitruncated 4-simplex
* Great prismatodispentachoron
* Truncated dispentachoron
* Great rhombated pentachoron (Acronym: grip) (Jonathan Bowers)

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |
* Stereographic projection with its 10 triangular prisms.

### Cartesian coordinates

The Cartesian coordinates of an origin-centered cantitruncated 5-cell having edge length 2 are:
Coordinates :
| $\left(3{\sqrt {\frac {2}{5}}},\ \pm {\sqrt {6}},\ \pm {\sqrt {3}},\ \pm 1\right)$ $\left(3{\sqrt {\frac {2}{5}}},\ \pm {\sqrt {6}},\ 0,\ \pm 2\right)$ $\left(3{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ 0\right)\pm \left(0,\ {\sqrt {\frac {2}{3}}},\ {\frac {5}{\sqrt {3}}},\ \pm 1\right)$ $\left(3{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ 0\right)\pm \left(0,\ {\sqrt {\frac {2}{3}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 3\right)$ $\left(3{\sqrt {\frac {2}{5}}},\ 0,\ 0,\ 0\right)\pm \left(0,\ {\sqrt {\frac {2}{3}}},\ {\frac {-4}{\sqrt {3}}},\ \pm 2\right)$ $\left({\frac {1}{\sqrt {10}}},\ {\frac {5}{\sqrt {6}}},\ {\frac {5}{\sqrt {3}}},\ \pm 1\right)$ $\left({\frac {1}{\sqrt {10}}},\ {\frac {5}{\sqrt {6}}},\ {\frac {-1}{\sqrt {3}}},\ \pm 3\right)$ $\left({\frac {1}{\sqrt {10}}},\ {\frac {5}{\sqrt {6}}},\ {\frac {-4}{\sqrt {3}}},\ \pm 2\right)$ $\left({\frac {1}{\sqrt {10}}},\ -{\sqrt {\frac {3}{2}}},\ {\sqrt {3}},\ \pm 3\right)$ $\left({\frac {1}{\sqrt {10}}},\ -{\sqrt {\frac {3}{2}}},\ -2{\sqrt {3}},\ 0\right)$ $\left({\frac {1}{\sqrt {10}}},\ {\frac {-7}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ \pm 2\right)$ $\left({\frac {1}{\sqrt {10}}},\ {\frac {-7}{\sqrt {6}}},\ {\frac {-4}{\sqrt {3}}},\ 0\right)$ $\left(-2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {4}{\sqrt {3}}},\ \pm 2\right)$ | $\left(-2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {1}{\sqrt {3}}},\ \pm 3\right)$ $\left(-2{\sqrt {\frac {2}{5}}},\ 2{\sqrt {\frac {2}{3}}},\ {\frac {-5}{\sqrt {3}}},\ \pm 1\right)$ $\left(-2{\sqrt {\frac {2}{5}}},\ 0,\ {\sqrt {3}},\ \pm 3\right)$ $\left(-2{\sqrt {\frac {2}{5}}},\ 0,\ -2{\sqrt {3}},\ 0\right)$ $\left(-2{\sqrt {\frac {2}{5}}},\ -4{\sqrt {\frac {2}{3}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)$ $\left(-2{\sqrt {\frac {2}{5}}},\ -4{\sqrt {\frac {2}{3}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)$ $\left({\frac {-9}{\sqrt {10}}},\ {\sqrt {\frac {3}{2}}},\ \pm {\sqrt {3}},\ \pm 1\right)$ $\left({\frac {-9}{\sqrt {10}}},\ {\sqrt {\frac {3}{2}}},\ 0,\ \pm 2\right)$ $\left({\frac {-9}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {2}{\sqrt {3}}},\ \pm 2\right)$ $\left({\frac {-9}{\sqrt {10}}},\ {\frac {-1}{\sqrt {6}}},\ {\frac {-4}{\sqrt {3}}},\ 0\right)$ $\left({\frac {-9}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {1}{\sqrt {3}}},\ \pm 1\right)$ $\left({\frac {-9}{\sqrt {10}}},\ {\frac {-5}{\sqrt {6}}},\ {\frac {-2}{\sqrt {3}}},\ 0\right)$ |
These vertices can be more simply constructed on a hyperplane in 5-space, as the permutations of:
    (0,0,1,2,3)
This construction is from the positive orthant facet of the cantitruncated 5-orthoplex.

### Related polytopes

A double symmetry construction can be made by placing truncated tetrahedra on the truncated octahedra, resulting in a nonuniform polychoron with 10 truncated tetrahedra, 20 hexagonal prisms (as ditrigonal trapezoprisms), two kinds of 80 triangular prisms (20 with D$_{3h}$ symmetry and 60 C$_{2v}$-symmetric wedges), and 30 tetrahedra (as tetragonal disphenoids). Its vertex figure is topologically equivalent to the octahedron.
Vertex figure

## Related 4-polytopes

These polytopes are art of a set of 9 Uniform 4-polytopes constructed from the [3,3,3] Coxeter group.
| Name | 5-cell | truncated 5-cell | rectified 5-cell | cantellated 5-cell | bitruncated 5-cell | cantitruncated 5-cell | runcinated 5-cell | runcitruncated 5-cell | omnitruncated 5-cell |
| Schläfli symbol | {3,3,3} 3r{3,3,3} | t{3,3,3} 2t{3,3,3} | r{3,3,3} 2r{3,3,3} | rr{3,3,3} r2r{3,3,3} | 2t{3,3,3} | tr{3,3,3} t2r{3,3,3} | t$_{0,3}${3,3,3} | t$_{0,1,3}${3,3,3} t$_{0,2,3}${3,3,3} | t$_{0,1,2,3}${3,3,3} |
| Coxeter diagram | | | | | | | | | |
| Schlegel diagram | | | | | | | | | |
| A$_4$ Coxeter plane Graph | | | | | | | | | |
| A$_3$ Coxeter plane Graph | | | | | | | | | |
| A$_2$ Coxeter plane Graph | | | | | | | | | |