# Pentagonal-gyroprismatic triacosihexecontachoron

Pentagonal-gyroprismatic triacosihexecontachoron | |
---|---|

Rank | 4 |

Type | Noble |

Elements | |

Cells | 360 pentagonal gyroprisms |

Faces | 1800 isosceles triangles, 360 pentagons |

Edges | 600+1800 |

Vertices | 600 |

Vertex figure | Triangular gyrotegum |

Measures (edge length 1) | |

Central density | 1 |

Related polytopes | |

Dual | Triangular-antiprismatic hexacosichoron |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | H_{3}●I_{2}(30), order 3600 |

Convex | Yes |

Nature | Tame |

The **pentagonal-gyroprismatic triacosihexecontachoron**, also known as the **doeswirl 360**, is a noble polychoron with 360 pentagonal gyroprisms for cells and 600 vertices. 6 cells join at each vertex.

It is the first in an infinite family of isogonal dodecahedral swirlchora (the triacontafold dodecaswirlchoron) and also the third in an infinite family of isochoric dodecahedral swirlchora (the dodecaswirlic triacosihexecontachoron).

Each cell of this polychoron is a chiral variant of the pentagonal antiprism. If the edges of the base pentagons are of length 1, half the side edges are also of length 1, while the other half are of length .

The ratio between the longest and shortest edges is 1: ≈ 1:1.80182.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a pentagonal-antiprismatic triacosihexecontachoron of circumradius 1, centered at the origin, are given by, along with 72°, 144°, 216° and 288° rotations in the *xy* axis of:

- ±(
*c1**sin(*k*π/15),*c1**cos(*k*π/15),*c2**cos(*k*π/15),*c2**sin(*k*π/15)), - ±(
*c2**sin(*k*π/15),*c2**cos(*k*π/15), -*c1**cos(*k*π/15), -*c1**sin(*k*π/15)), - ±(
*c3**sin((*k*+n/2)π/15),*c3**cos((*k*+n/2)π/15),*c4**cos((*k*+n/2)π/15),*c4**sin((*k*+n/2)π/15)), - ±(
*c4**sin((*k*+n/2)π/15),*c4**cos((*k*+n/2)π/15), -*c3**cos((*k*+n/2)π/15), -*c3**sin((*k*+n/2)π/15)),

where *c1* = √450-30√75+30√5/30, *c2* = √450+30√75+30√5/30, *c3* = √450-30√75-30√5/30, *c4* = √450+30√75-30√5/30 and *k* is an integer from 0 to 14.

## Isogonal derivatives[edit | edit source]

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

- Pentagonal antiprism (360): Triangular-antiprismatic hexacosichoron
- Pentagon (360): Triangular-antiprismatic hexacosichoron
- Edge (600): Pentagonal-antiprismatic triacosihexecontachoron

## External links[edit | edit source]

- Klitzing, Richard. "pap-360".