# Rectified pentachoron

Rectified pentachoron | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Rap |

Coxeter diagram | o3x3o3o () |

Elements | |

Cells | 5 tetrahedra, 5 octahedra |

Faces | 10+20 triangles |

Edges | 30 |

Vertices | 10 |

Vertex figure | Triangular prism, edge length 1 |

Edge figure | tet 3 oct 3 oct 3 |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Oct–3–tet: |

Oct-3-oct: | |

Height | |

Central density | 1 |

Number of external pieces | 10 |

Level of complexity | 3 |

Related polytopes | |

Army | Rap |

Regiment | Rap |

Dual | Joined pentachoron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 360 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | A_{4}, order 120 |

Convex | Yes |

Nature | Tame |

The **rectified pentachoron**, or **rap**, also commonly called the **rectified 5-cell**, is a convex uniform polychoron that consists of 5 regular tetrahedra and 5 regular octahedra. Two tetrahedra and three octahedra join at each triangular prismatic vertex. As the name suggests, it can be constructed by rectification of the pentachoron.

It is the vertex figure of the demipenteract.

It is also a convex segmentochoron (designated K-4.5 in Richard Klitzing's list), formed as a tetrahedron atop an octahedron. If the edge lengths of this subsymmetrical variation are changed, the result is various non-uniform tetrahedron atop octahedron polychora with 4 triangle antipodiums and 4 triangle pyramids as sides.

It is also isogonal under the 5-2 step prism subsymmetry, and can be considered to be a 5-2 double step prism.

## Gallery[edit | edit source]

## Vertex coordinates[edit | edit source]

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

Simpler coordinates can be given by all even sign changes of first 3 coordinates of:

and all permutations of the first three coordinates of:

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

## Representations[edit | edit source]

A rectified pentachoron has the following Coxeter diagrams:

- o3x3o3o (full symmetry)
- xo3ox3oo&#x (A3 axial, as tetrahedron atop octahedron
- oxo oxo3oox&#xt (A2×A1 axial, vertex-first)
- oxoo3xoxo&#xr (A2 axial)
- oxoox oxoxo&#xr (A1×A1 axial)

## Segmentochoron display[edit | edit source]

Tetrahedron atop octahedron

## Related polychora[edit | edit source]

The rectified pentachoron is the colonel of a three-member regiment that also includes the facetorectified pentachoron and the prismatointercepted pentachoron.

The rectified pentachoron can be diminished by cutting off triangular prismatic pyramids, with each removing 2 tetrahedra and diminishing the octahedra down to square pyramids. If one pyramid is removed, the result is the triangular antifastegium. If two non-adjacent pyramids are removed, such that one of the octahedra gets reduced down to an equatorial square only, the result is the bidiminished rectified pentachoron.

An octahedral pyramid can be attached to one of the rectified pentachoron's octahedral cells to create the augmented rectified pentachoron, notable as a non-uniform Blind polytope.

Uniform polychoron compounds composed of rectified pentachora include:

- Rectified stellated decachoron (2)
- Compound of 12 rectified pentachora (12)
- Rectified medial hexacosichoron (120)

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

Pentachoron | pen | x3o3o3o () | |

Truncated pentachoron | tip | x3x3o3o () | |

Rectified pentachoron | rap | o3x3o3o () | |

Decachoron | deca | o3x3x3o () | |

Rectified pentachoron | rap | o3o3x3o () | |

Truncated pentachoron | tip | o3o3x3x () | |

Pentachoron | pen | o3o3o3x () | |

Small rhombated pentachoron | srip | x3o3x3o () | |

Great rhombated pentachoron | grip | x3x3x3o () | |

Small rhombated pentachoron | srip | o3x3o3x () | |

Great rhombated pentachoron | grip | o3x3x3x () | |

Small prismatodecachoron | spid | x3o3o3x () | |

Prismatorhombated pentachoron | prip | x3x3o3x () | |

Prismatorhombated pentachoron | prip | x3o3x3x () | |

Great prismatodecachoron | gippid | x3x3x3x () |

### Isogonal derivatives[edit | edit source]

Substitution by vertices of these following elements will produce these convex isogonal polychora:

- Octahedron (5): Pentachoron
- Tetrahedron (5): Pentachoron
- Triangle (10): Rectified pentachoron
- Triangle (20): Semi-uniform small prismatodecachoron without doubled symmetry
- Edge (30): Uniform small rhombated pentachoron

## External links[edit | edit source]

- Bowers, Jonathan. "Category 3: Triangular Rectates" (#39).

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

- Klitzing, Richard. "rap".

- Quickfur. "The Rectified 5-cell".

- Wikipedia Contributors. "Rectified 5-cell".
- Hi.gher.Space Wiki Contributors. "Pyrorectichoron".