View source for Gyroelongated bipyramid

{{Short description|Polyhedron formed by capping an antiprism with pyramids}}
{{Infobox polyhedron
| image         = Pentagonal gyroelongated bipyramid.png
| caption       = The pentagonal gyroelongated bipyramid is the regular [[icosahedron]].
| euler         = 
| faces         = {{math|4''n''}} [[triangle]]s
| edges         = {{math|6''n''}}
| vertices      = {{math|2''n'' + 2}}
| schläfli      = 
| wythoff       = 
| coxeter       = 
| symmetry      = {{math|[[Symmetry group#Three dimensions|''D''{{sub|''n''d}}]], [2{{sup|+}},2''n''], (2*''n''),}} order {{math|4''n''}}
| rotation_group     = {{math|''D''{{sub|''n''}}, [2,''n'']{{sup|+}}, (22''n''),}} order {{math|2''n''}}
| surface_area  = 
| volume        =
| dual          = [[truncated trapezohedron|truncated trapezohedra]]
| properties    = [[Convex polytope|convex]]
}}

In [[geometry]], the '''gyroelongated bipyramids''' are an infinite set of [[polyhedra]], constructed by elongating an {{nowrap|{{mvar|n}}-gonal}} [[bipyramid]] by inserting an {{nowrap|{{mvar|n}}-gonal}} [[antiprism]] between its [[Pyramid (geometry)|congruent halves]].

==Forms==
Three members of the set can be [[deltahedra]], that is, constructed entirely of [[equilateral triangle]]s: 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.

{|class=wikitable width=640
!''n''
!3
!4
!5
!6
!''n''
|-
!Type
!Coplanar
!Equilateral
!Regular
!Coplanar
!
|-
!Shape
![[Gyroelongated triangular bipyramid]]
![[Gyroelongated square bipyramid]]
!Gyroelongated pentagonal bipyramid<BR>([[icosahedron]])
![[Gyroelongated hexagonal bipyramid]]
!Gyroelongated bipyramid
|- align=center
!Image
|[[File:Gyroelongated triangular bipyramid.png|100px]]
|[[Image:Gyroelongated square dipyramid.png|100px]]
|[[Image:Pentagonal gyroelongated bipyramid.png|100px]]
|[[File:Augmented hexagonal antiprism flat.png|100px]]
|
|- align=center
!Faces
|[[dodecahedron#Other dodecahedra|12]]
|[[hexadecahedron#Other hexadecahedra|16]]
|[[icosahedron#Other icosahedra|20]]
|[[icositetrahedron#Other icositetrahedron|24]]
|4''n''
|- align=center
!Dual
![[Triangular truncated trapezohedron]]
![[Square truncated trapezohedron]]
!Pentagonal truncated trapezohedron<br>([[Dodecahedron]])
![[Hexagonal truncated trapezohedron]]
![[Truncated trapezohedron|Truncated trapezohedra]]
|}

== See also ==
* [[Elongated bipyramid]]
* [[Gyroelongated pyramid]]
* [[Elongated pyramid]]
* [[Diminished trapezohedron]]

==External links==
*[http://www.georgehart.com/virtual-polyhedra/conway_notation.html Conway Notation for Polyhedra] Try: "k'''n'''A'''n'''", where '''n'''=4,5,6... example "k5A5" is an icosahedron.

{{Polyhedron navigator}}

[[Category:Bipyramids]]
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