# Small rhombated pentachoron

Small rhombated pentachoron | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Srip |

Coxeter diagram | x3o3x3o () |

Elements | |

Cells | 5 octahedra, 10 triangular prisms, 5 cuboctahedra |

Faces | 10+20+20 triangles, 30 squares |

Edges | 30+60 |

Vertices | 30 |

Vertex figure | Square wedge, edge lengths 1 (base square and top edge) and √2 (side edges) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Oct–3–trip: |

Co–4–trip: | |

Co–3–oct: | |

Co–3–co: | |

Central density | 1 |

Number of external pieces | 20 |

Level of complexity | 9 |

Related polytopes | |

Army | Srip |

Regiment | Srip |

Dual | Notched triacontachoron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 1080 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | A_{4}, order 120 |

Convex | Yes |

Nature | Tame |

The **small rhombated pentachoron**, or **srip**, also commonly called the **cantellated 5-cell** or **cantellated pentachoron**, is a convex uniform polychoron that consists of 5 regular octahedra, 10 triangular prisms, and 5 cuboctahedra. 1 octahedron, 2 triangular prisms, and 2 cuboctahedra join at each vertex. As one of its names suggests, it can be formed by cantellating the pentachoron. It can also be formed by rectification of the rectified pentachoron.

## Gallery[edit | edit source]

## Vertex coordinates[edit | edit source]

The vertices of a small rhombated pentachoron of edge length 1 are given by:

Much simpler coordinates can be given in five dimensions, as all permutations of:

## Representations[edit | edit source]

A small rhombated pentachoron has the following Coxeter diagrams:

- x3o3x3o (full symmetry)
- oxx3xxo3oox&#xt (A
_{3}axial, octahedron-first) - x(uo)xo x(ou)xx3o(xo)xo&#xt (A
_{2}×A_{1}axial, triangular prism-first)

## Semi-uniform variant[edit | edit source]

The small rhombated pentachoron has a semi-uniform variant of the form x3o3y3o that maintains its full symmetry. This variant uses 5 octahedra of size y, 5 rhombitetratetrahedra of form x3o3y, and 10 triangular prisms of form x y3o as cells, with 2 edge lengths.

With edges of length a (surrounds 2 rhombitetratetrahedra) and b (of octahedra), its circumradius is given by and its hypervolume is given by .

## Related polychora[edit | edit source]

The small rhombated pentachoron is the colonel of the largest regiment of uniform polychora with A_{4} symmetry, which has a total of 7 members. Its facetings include the retrosphenoverted trispentachoron, small rhombic dispentachoron, pseudorhombic prismatopentachoron, grand rhombic prismatopentachoron, prismatopentintercepted dispentachoron, and prismatointercepted prismatodispentachoron.

When viewed in A_{3} axial symmetry, the small rhombated pentachoron can be cut into 2 segmentochora, namely cuboctahedron atop truncated tetrahedron and octahedron atop truncated tetrahedron, join at the truncated tetrahedral bases.

The triangular pucofastegium occurs as the triangle-first cap of the small rhombated pentachoron.

Uniform polychoron compounds composed of small rhombated pentachora include:

Name | OBSA | CD diagram | Picture |
---|---|---|---|

Pentachoron | pen | ||

Truncated pentachoron | tip | ||

Rectified pentachoron | rap | ||

Decachoron | deca | ||

Rectified pentachoron | rap | ||

Truncated pentachoron | tip | ||

Pentachoron | pen | ||

Small rhombated pentachoron | srip | ||

Great rhombated pentachoron | grip | ||

Small rhombated pentachoron | srip | ||

Great rhombated pentachoron | grip | ||

Small prismatodecachoron | spid | ||

Prismatorhombated pentachoron | prip | ||

Prismatorhombated pentachoron | prip | ||

Great prismatodecachoron | gippid |

## External links[edit | edit source]

- Bowers, Jonathan. "Category 6: Sphenoverts" (#133).

- Bowers, Jonathan. "Pennic and Decaic Isogonals".

- Klitzing, Richard. "srip".

- Quickfur. "The Cantellated 5-cell".

- Wikipedia Contributors. "Cantellated 5-cell".