# Triangular-gyroprismatic hexacosichoron

Triangular-gyroprismatic hexacosichoron | |
---|---|

File:Triangular-gyroprismatic hexacosichoron.png | |

Rank | 4 |

Type | Noble |

Elements | |

Cells | 600 triangular gyroprisms |

Faces | 1800 isosceles triangles, 600 triangles |

Edges | 360+1800 |

Vertices | 360 |

Vertex figure | Pentagonal gyrotegum |

Measures (edge length 1) | |

Central density | 1 |

Related polytopes | |

Dual | Pentagonal-antiprismatic triacosihexecontachoron |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

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

Convex | Yes |

Nature | Tame |

The **triangular-gyroprismatic hexacosichoron**, also known as the **ikeswirl 600**, is a noble polychoron with 600 triangular gyroprisms for cells and 360 vertices. 10 cells join at each vertex.

It is the third in an infinite family of isogonal icosahedral swirlchora (the triacontafold icosaswirlchoron) and also the first in an infinite family of isochoric icosahedral swirlchora (the icosaswirlic hexacosichoron).

Each cell of this polychoron is a chiral variant of the triangular antiprism. If the edges of the base triangles 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: ≈ 1:2.65475.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a triangular-antiprismatic hexacosichoron of circumradius 1, centered at the origin, are given by:

- ±(0, 0, sin(
*k*π/15), cos(*k*π/15)), - ±(cos(
*k*π/15), sin(*k*π/15), 0, 0),

along with 72°, 144°, 216° and 288° rotations in the *xy* axis of:

- ±(2sin(
*k*π/15)/√10+2√5, 2cos(*k*π/15)/√10+2√5, 2cos(*k*π/15)/√10-2√5, 2sin(*k*π/15)/√10-2√5), - ±(2sin(
*k*π/15)/√10-2√5, 2cos(*k*π/15)/√10-2√5, -2cos(*k*π/15)/√10+2√5, -2sin(*k*π/15)/√10+2√5),

where *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:

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

## External links[edit | edit source]

- Klitzing, Richard. "trap-600".