# Gyroelongated pyramid

In geometry, the **gyroelongated pyramids** (also called augmented antiprisms) are an infinite set of polyhedra, constructed by adjoining an n-gonal pyramid to an n-gonal antiprism.

Set of gyroelongated pyramids | |
---|---|

Example pentagonal form | |

Faces | 3n triangles 1 n-gon |

Edges | 5n |

Vertices | 2n+1 |

Symmetry group | C_{nv}, [n], (*nn) |

Rotational group | C_{n}, [n]^{+}, (nn) |

Dual polyhedron | ? |

Properties | convex |

There are two *gyroelongated pyramids* that are Johnson solids made from regular triangles and square, and pentagons. A triangular and hexagonal form can be constructed with coplanar faces. Others can be constructed allowing for isosceles triangles.

## Forms

Image | Name | Faces |
---|---|---|

Gyroelongated triangular pyramid (Coplanar faces) | 9+1 triangles | |

Gyroelongated square pyramid (J10) | 12 triangles, 1 squares | |

Gyroelongated pentagonal pyramid (J11) | 15 triangles, 1 pentagon | |

Gyroelongated hexagonal pyramid (Coplanar faces) | 18 triangles, 1 hexagon |

## References

- Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics,
**18**, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others. - Victor A. Zalgaller (1969).
*Convex Polyhedra with Regular Faces*. Consultants Bureau. No ISBN. The first proof that there are only 92 Johnson solids.

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