# Gyroelongated 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 |

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.

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 | edit source]

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 | edit source]

- 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"