# Gyroelongated pentagonal pyramid

The **gyroelongated pentagonal pyramid** (OBSA: **gyepip**) is one of the 92 Johnson solids (J_{11}). It consists of 5+5+5 triangles and 1 pentagon. It can be constructed by attaching a pentagonal antiprism to the base of the pentagonal pyramid.

Gyroelongated pentagonal pyramid | |
---|---|

Rank | 3 |

Type | CRF |

Notation | |

Bowers style acronym | Gyepip |

Coxeter diagram | oxo5oox&#xt |

Elements | |

Faces | |

Edges | 5+5+5+10 |

Vertices | 1+5+5 |

Vertex figures | 1+5 pentagons, edge length 1 |

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

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 3–3: |

3–5: | |

Central density | 1 |

Number of external pieces | 16 |

Level of complexity | 10 |

Related polytopes | |

Army | Gyepip |

Regiment | Gyepip |

Dual | Order-5 monotruncated pentagonal trapezohedron |

Conjugate | Replenished great icosahedron |

Abstract & topological properties | |

Flag count | 100 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | H_{2}×I, order 10 |

Flag orbits | 10 |

Convex | Yes |

Nature | Tame |

Alternatively, it can be constructed by diminishing one vertex from the regular icosahedron, which is why this polyhedron can also be called the **diminished icosahedron**. This means the icosahedron can also be thought of as a gyroelongated pentagonal bipyramid.

## Vertex coordinates edit

A gyroelongated pentagonal pyramid of edge length 1 has the following vertices:

- ,
- ,
- ,
- .

Alternative coordinates can be obtained by removing one vertex from the regular icosahedron:

- ,
- ,
- ,
- .

## External links edit

- Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#8 under ike).

- Klitzing, Richard. "gyepip".
- Quickfur. "The Gyroelongated Pentagonal Pyramid".

- Wikipedia contributors. "Gyroelongated pentagonal pyramid".
- McCooey, David. "Gyroelongated Pentagonal Pyramid"