# 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.
Faces20 triangles
Edges6n
Vertices2n+2
Symmetry groupDnd, [2+,2n], (2*n), order 4n
Rotation groupDn, [2,n]+, (22n), order 2n
Dual polyhedrontruncated trapezohedra
Propertiesconvex

## 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