# Great pentagonal hexecontahedron

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Great pentagonal hexecontahedron | |
---|---|

Rank | 3 |

Type | Uniform dual |

Notation | |

Coxeter diagram | p5/2p3p |

Elements | |

Faces | 60 mirror-symmetric pentagons |

Edges | 30+60+60 |

Vertices | 12+20+60 |

Vertex figure | 20+60 triangles, 12 pentagrams |

Measures (edge length 1) | |

Inradius | ≈ 0.50974 |

Dihedral angle | ≈ 104.43227° |

Central density | 7 |

Related polytopes | |

Dual | Great snub icosidodecahedron |

Conjugates | Pentagonal hexecontahedron, great inverted pentagonal hexecontahedron, great pentagrammic hexecontahedron |

Convex core | Non-Catalan pentagonal hexecontahedron |

Abstract & topological properties | |

Flag count | 600 |

Euler characteristic | 2 |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{3}+, order 60 |

Convex | No |

Nature | Tame |

The **great pentagonal hexecontahedron** is a uniform dual polyhedron. It consists of 60 mirror-symmetric pentagons, each with two short and three long edges.

If its dual, the great snub icosidodecahedron, has unit edge length, then the pentagon faces' short edges have approximate length 0.49069 (equal to a root of the polynomial ), and the long edges have approximate length 0.64563 (equal to a root of the polynomial ). The hexagons have four interior angles of , and one of , where is the negative root of the polynomial , and is the golden ratio.

A dihedral angle can be given as .

The inradius R ≈ 0.50974 of the great pentagonal hexecontahedron with unit edge length is equal to the square root of a real root of .

## External links[edit | edit source]

- Wikipedia contributors. "Great pentagonal hexecontahedron".
- McCooey, David. "Great Pentagonal Hexecontahedron"