# Medial pentagonal hexecontahedron

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Medial pentagonal hexecontahedron | |
---|---|

Rank | 3 |

Type | Uniform dual |

Notation | |

Coxeter diagram | p5/2p5p () |

Elements | |

Faces | 60 irregular pentagons |

Edges | 30+60+60 |

Vertices | 12+12+60 |

Vertex figure | 60 triangles, 12 pentagons, 12 pentagrams |

Measures (edge length 1) | |

Inradius | ≈ 1.07828 |

Dihedral angle | ≈ 133.80098° |

Central density | 3 |

Related polytopes | |

Dual | Snub dodecadodecahedron |

Conjugate | Medial inverted pentagonal hexecontahedron |

Convex core | Non-Catalan pentakis dodecahedron |

Abstract & topological properties | |

Flag count | 600 |

Euler characteristic | –6 |

Orientable | Yes |

Genus | 4 |

Properties | |

Symmetry | H_{3}+, order 60 |

Convex | No |

Nature | Tame |

The **medial pentagonal hexecontahedron** is a uniform dual polyhedron. It consists of 60 irregular pentagons, each with two short, one medium, and two long edges. Its dual is the snub dodecadodecahedron.

If the pentagon faces have medium edge length 2, then the short edge length is , and the long edge length is , where is the smallest (most negative) real root of the polynomial . The pentagons have three interior angles of , one of , and one of , where is the golden ratio.

The inradius R ≈ 1.07828 of the medial pentagonal hexecontahedron with unit edge length is equal to the square root of a real root of .

A dihedral angle can be given as acos(α), where α ≈ -0.69216 is the negative of a real root of .

## External links[edit | edit source]

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