Gyroelongated bipyramid

Short description: Polyhedron formed by capping an antiprism with pyramids

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

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.

## 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

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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.

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Convex polyhedra

Platonic solids (regular)|

* tetrahedron
* cube
* octahedron
* dodecahedron
* icosahedron

Archimedean solids
(semiregular or uniform)

* truncated tetrahedron
* cuboctahedron
* truncated cube
* truncated octahedron
* rhombicuboctahedron
* truncated cuboctahedron
* snub cube
* icosidodecahedron
* truncated dodecahedron
* truncated icosahedron
* rhombicosidodecahedron
* truncated icosidodecahedron
* snub dodecahedron

Catalan solids
(duals of Archimedean)

* triakis tetrahedron
* rhombic dodecahedron
* triakis octahedron
* tetrakis hexahedron
* deltoidal icositetrahedron
* disdyakis dodecahedron
* pentagonal icositetrahedron
* rhombic triacontahedron
* triakis icosahedron
* pentakis dodecahedron
* deltoidal hexecontahedron
* disdyakis triacontahedron
* pentagonal hexecontahedron

Dihedral regular|

* dihedron
* hosohedron

Dihedral uniform|

* prisms
* antiprisms

duals:|

* bipyramids
* trapezohedra

Dihedral others|

* pyramids
* truncated trapezohedra
* gyroelongated bipyramid
* cupola
* bicupola
* pyramidal frusta
* bifrustum
* rotunda
* birotunda
* prismatoid
* scutoid

Degenerate polyhedra are in italics.

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