# Bidecachoron

Bidecachoron | |
---|---|

Rank | 4 |

Type | Noble |

Space | Spherical |

Notation | |

Bowers style acronym | Bideca |

Coxeter diagram | o3m3m3o () |

Elements | |

Cells | 30 tetragonal disphenoids |

Faces | 60 isosceles triangles |

Edges | 20+20 |

Vertices | 10 |

Vertex figure | Triakis tetrahedron |

Measures (based on two pentachora of edge length 1) | |

Edge lengths | Lacing edges (20): |

Edges of pentachora (20): 1 | |

Circumradius | |

Inradius | |

Dichoral angle | |

Central density | 1 |

Related polytopes | |

Army | Bideca |

Regiment | Bideca |

Dual | Decachoron |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | A_{4}×2, order 240 |

Convex | Yes |

Nature | Tame |

The **bidecachoron** or **bideca**, also known as the **tetradisphenoidal triacontachoron**, is a convex noble polychoron with 30 tetragonal disphenoids as cells. 12 cells join at each vertex, with the vertex figure being a triakis tetrahedron. It can be constructed as the convex hull of a pentachoron and its central inversion (or, equivalently, its dual). It is also the 10-3 step prism.

The ratio between the longest and shortest edges is 1: ≈ 1:1.29099.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a bidecachoron, based on two pentachora of edge length 1, centered at the origin, are given by:

- ,
- ,
- ,
- .

## Variations[edit | edit source]

The bidecachoron has a number of variants that remain either isotopic or isogonal:

- Disphenoidal triacontachoron (cells are digonal disphenoids, isotopic)
- 10-3 step prism (10 tetragonal and 20 phyllic disphenoids, step prism symmetry)

## Isogonal derivatives[edit | edit source]

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

- Tetragonal disphenoid (30): Decachoron
- Isosceles triangle (60): Rectified decachoron
- Edge (20): Small prismatodecachoron
- Edge (20): Biambodecachoron

## External links[edit | edit source]

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

- Klitzing, Richard. "bideca".