Set of gyroelongated bipyramids
The pentagonal gyroelongated bipyramid is the regular icosahedron Icosahedron In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.... .
Faces	4n triangle Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted .... s
Edges	6n
Vertices	2n+2
Symmetry group	Dnd, [2+,2n], (2*n)
Dual polyhedron Dual polyhedron In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...	truncated trapezohedra Truncated trapezohedron An n-agonal truncated trapezohedron is a polyhedron formed by a n-agonal trapezohedron with n-agonal pyramids truncated from its two polar axis vertices....
Properties	convex

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the gyroelongated bipyramids are an infinite set of polyhedra

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

, constructed by elongating an n-gonal bipyramid

Bipyramid

An n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.The...

 by inserting an n-gonal antiprism

Antiprism

In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...

 between its congruent halves.

Forms

Two members of the set can be deltahedra

Deltahedron

A deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta , which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces...

, that is, constructed entirely of equilateral triangles: the gyroelongated square bipyramid, a Johnson solid

Johnson solid

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...

, and the icosahedron

Icosahedron

In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

, a Platonic solid

Platonic solid

In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

. 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. The other members can be constructed with isosceles triangles.

n	3	4	5	6	n
Shape	Gyroelongated triangular bipyramid	Gyroelongated square bipyramid	Gyroelongated pentagonal bipyramid (icosahedron Icosahedron In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.... )	Gyroelongated hexagonal bipyramid	Gyroelongated bipyramid
Image
Dual	Triangular truncated trapezohedron	Square truncated trapezohedron Square truncated trapezohedron The square truncated trapezohedron is the second in an infinite series of truncated trapezohedron polyhedra. It has 8 pentagon and 2 square faces.This polyhedron can be constructed by taking a tetragonal trapezohedron and truncating the polar axis vertices....	Pentagonal truncated trapezohedron (Dodecahedron)	Hexagonal truncated trapezohedron Hexagonal truncated trapezohedron The hexagonal truncated trapezohedron is the fourth in an infinite series of truncated trapezohedron polyhedra. It has 12 pentagon and 2 hexagon faces....	Truncated trapezohedra Truncated trapezohedron An n-agonal truncated trapezohedron is a polyhedron formed by a n-agonal trapezohedron with n-agonal pyramids truncated from its two polar axis vertices....

External links

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