# Gyroelongated bipyramid

In geometry, the **gyroelongated bipyramids** are an infinite set of polyhedra, constructed by elongating an *n*-gonal bipyramid by inserting an *n*-gonal antiprism between its congruent halves.

Set of gyroelongated bipyramids | |
---|---|

The pentagonal gyroelongated bipyramid is the regular icosahedron. | |

Faces | 20 triangles |

Edges | 6n |

Vertices | 2n+2 |

Symmetry group | D_{nd}, [2^{+},2n], (2*n), order 4n |

Rotation group | D_{n}, [2,n]^{+}, (22n), order 2n |

Dual polyhedron | truncated trapezohedra |

Properties | convex |

## Forms

Two members of the set can be deltahedra, that is, constructed entirely of equilateral triangles: the gyroelongated square bipyramid, a Johnson solid, and the icosahedron, a Platonic solid. The *gyroelongated triangular bipyramid* can be made with equilateral triangles, but is not a deltahedron because it has coplanar faces, i.e. is not strictly convex. With pairs of triangles merged into rhombi, it can be seen as a trigonal trapezohedron. The other members can be constructed with isosceles triangles.

n |
3 | 4 | 5 | 6 | n |
---|---|---|---|---|---|

Type | Coplanar | Equilateral | Regular | Coplanar | |

Shape | Gyroelongated triangular bipyramid | Gyroelongated square bipyramid | Gyroelongated pentagonal bipyramid (icosahedron) |
Gyroelongated hexagonal bipyramid | Gyroelongated bipyramid |

Image | |||||

Faces | 12 | 16 | 20 | 24 | 4n |

Dual | Triangular truncated trapezohedron | Square truncated trapezohedron | Pentagonal truncated trapezohedron (Dodecahedron) |
Hexagonal truncated trapezohedron | Truncated trapezohedra |

## External links

- Conway Notation for Polyhedra Try: "k
**n**A**n**", where**n**=4,5,6... example "k5A5" is an icosahedron.

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