# Pentakis dodecahedron

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

Rank | 3 |

Type | Uniform dual |

Notation | |

Bowers style acronym | Pakid |

Coxeter diagram | o5m3m () |

Conway notation | kD |

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 |

Number of external pieces | 60 |

Level of complexity | 3 |

Related polytopes | |

Army | Pakid |

Regiment | Pakid |

Dual | Truncated icosahedron |

Conjugate | Great stellapentakis dodecahedron |

Abstract & topological properties | |

Flag count | 360 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{3}, order 120 |

Flag orbits | 3 |

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 gives a fully symmetric variant of this polyhedron. The lower limit is times that of the cube, where the icosahedron's vertices will coincide with the dodecahedron's face centers.

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"