# Decachoron

Decachoron | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Deca |

Coxeter diagram | o3x3x3o () |

Elements | |

Cells | 10 truncated tetrahedra |

Faces | 20 triangles, 20 hexagons |

Edges | 60 |

Vertices | 30 |

Vertex figure | Tetragonal disphenoid, edge lengths 1 (base) and √3 (sides) |

Edge figure | tut 6 tut 6 tut 3 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Hypervolume | |

Dichoral angles | Tut–6–tut: |

Tut–3–tut: | |

Central density | 1 |

Number of external pieces | 10 |

Level of complexity | 3 |

Related polytopes | |

Army | Deca |

Regiment | Deca |

Dual | Bidecachoron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 720 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | A_{4}×2, order 240 |

Convex | Yes |

Nature | Tame |

The **decachoron**, or **deca**, also commonly called the **10-cell**, **bitruncated 5-cell** or **bitruncated pentachoron**, is a convex noble uniform polychoron that consists of 10 truncated tetrahedra as cells. Four cells join at each vertex. It is the medial stage of the truncation series between a regular pentachoron and itself. Equivalently, it is also the stellation core of the compound of two dual pentachora, the stellated decachoron.

It is also the 10-3 gyrochoron.

## Gallery[edit | edit source]

## Vertex coordinates[edit | edit source]

The vertices of a decachoron of edge length 1 are given by the following points:

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

## Representations[edit | edit source]

A decachoron has the following Coxeter diagrams:

- o3x3x3o (full symmetry)
- oox3xux3xoo&#xt (A3 axial, cell-first)
- oxuxo ooxux3xuxoo&#xt (A2×A1 axial, triangle-first)

## Variations[edit | edit source]

The following variants of the decachoron exist:

- Pentapentachoron - semi-uniform with 2 types of cells
- 10-3 gyrochoron - has step prism symmetry, isochoric

## Related polychora[edit | edit source]

The tripesic hexacosichoron is a uniform polychoron compound composed of 60 decachora.

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 |

### Isogonal derivatives[edit | edit source]

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

- Truncated tetrahedron (10): Bidecachoron
- Triangle (20): Biambodecachoron
- Hexagon (20): Small prismatodecachoron
- Edge (60): Rectified decachoron

## External links[edit | edit source]

- Bowers, Jonathan. "Category 7: Bitruncates" (#298).

- Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".

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

- Klitzing, Richard. "deca".

- Quickfur. "The Bitruncated 5-cell".

- Wikipedia Contributors. "Bitruncated 5-cell".