# Pentakis dodecahedron

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Pentakis dodecahedron | |
---|---|

Rank | 3 |

Type | Uniform dual |

Space | Spherical |

Notation | |

Bowers style acronym | Pakid |

Coxeter diagram | o5m3m () |

Elements | |

Faces | 60 isosceles triangles |

Edges | 30+60 |

Vertices | 12+20 |

Vertex figure | 12 pentagons, 20 hexagons |

Measures (edge length 1) | |

Dihedral angle | |

Central density | 1 |

Related polytopes | |

Army | Pakid |

Regiment | Pakid |

Dual | Truncated icosahedron |

Conjugate | Great stellapentakis dodecahedron |

Abstract & topological properties | |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{3}, order 120 |

Convex | Yes |

Nature | Tame |

The **pentakis dodecahedron** is one of the 13 Catalan solids. It has 60 isosceles triangles as faces, with 12 order-5 and 20 order-6 vertices. It is the dual of the uniform truncated icosahedron.

It can also be obtained as the convex hull of a dodecahedron and an icosahedron, where the edges of the icosahedron are times the length of those of the dodecahedron. Using an icosahedron that is any number less than times the edge length of the dodecahedron including if the two have the same edge length) gives a fully symmetric variant of this polyhedron.

Each face of this polyhedron is an isosceles triangle with base side length times those of the side edges. These triangles have apex angle and base angles .

## External links[edit | edit source]

- Klitzing, Richard. "pakid".

- Wikipedia Contributors. "Pentakis dodecahedron".
- McCooey, David. "Pentakis Dodecahedron"