# Great dodecahedron cone

The **great dodecahedron cone**, or **gadcone**, is a nonconvex orbiform polyhedron and an edge-faceting of the icosahedron. Its faces are 5 pentagons and 5 triangles. It is named as such because the 5 pentagons are arranged around one vertex similar to the configuration found in the great dodecahedron.

Great dodecahedron cone | |
---|---|

Rank | 3 |

Notation | |

Bowers style acronym | Gadcone |

Elements | |

Faces | 5 triangles, 5 pentagons |

Edges | 5+5+10 |

Vertices | 1+5+5 |

Vertex figures | 1 pentagram, edge length (1+√5)/2 |

5 butterflies, edge lengths 1 and (1+√5)/2 | |

5 isosceles triangles, edge lengths 1, (1+√5)/2, (1+√5)/2 | |

Measures (edge length 1) | |

Circumradius | |

Dihedral angles | 3–5 #1: |

5–5: | |

3–5 #2: | |

Related polytopes | |

Army | Gyepip |

Regiment | Gadcone |

Conjugate | Stelladodecahedron cone |

Convex hull | Gyroelongated pentagonal pyramid |

Abstract & topological properties | |

Flag count | 80 |

Euler characteristic | 1 |

Orientable | No |

Genus | 1 |

Properties | |

Symmetry | H_{2}×I, order 10 |

Convex | No |

Nature | Tame |

It is the 5-5-3 acrohedron generated by Green's rules.

## Vertex coordinates Edit

Its vertex coordinates are the same as those of the icosahedron with any 1 vertex removed, which is equivalent to those of the gyroelongated pentagonal pyramid.

## External links Edit

- Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#5 under ike).

- McNeil, Jim. "n-n-3 acrohedra".
- Klitzing, Richard. "ike-facetings".