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.

Gyroelongated bipyramid
The pentagonal gyroelongated bipyramid is the regular icosahedron.
Faces	4n triangles
Edges	6n
Vertices	2n + 2
Symmetry group	Dnd, [2+,2n], (2*n), order 4n
Rotation group	Dn, [2,n]+, (22n), order 2n
Dual polyhedron	truncated trapezohedra
Properties	convex

Forms

Three members of the set can be deltahedra, that is, constructed entirely of equilateral triangles: the gyroelongated square bipyramid, a Johnson solid; the icosahedron, a Platonic solid; and the gyroelongated triangular bipyramid if it is made with equilateral triangles, but because it has coplanar faces 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

See also

• Elongated bipyramid
• Gyroelongated pyramid
• Elongated pyramid
• Diminished trapezohedron

External links

• Conway Notation for Polyhedra Try: "knAn", where n=4,5,6... example "k5A5" is an icosahedron.