View source for Elongated bipyramid

{{short description|Polyhedron formed by capping a prism with pyramids}}
{{Infobox polyhedron
| image         = Elongated hexagonal dipyramid.png
| caption       = Example: elongated hexagonal bipyramid
| type          = 
| euler         = 
| faces        = {{mvar|2n}} [[Triangle|triangle]]s<BR>{{mvar|n}} squares
| edges         = {{math|5''n''}}
| vertices      = {{math|2''n'' + 2}}
| vertex_config = 
| schläfli      = 
| wythoff       = 
| coxeter       = 
| symmetry      = {{math|''D''<sub>''nh''</sub>, [''n'',2], (*''n''22)}}
| rotation_group = {{math|''D''<sub>''n''</sub>, [''n'',2]<sup>+</sup>, (''n''22)}}
| surface_area  = 
| volume        = 
| dual          = [[Bifrustum|bifrustum]]s
| properties    = [[Convex polytope|convex]]
}}

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

There are three ''elongated bipyramids'' that are [[Johnson solid]]s:
* [[Elongated triangular bipyramid]] ({{math|''J''{{sub|14}}}}),
* [[Elongated square bipyramid]] ({{math|''J''{{sub|15}}}}), and
* [[Elongated pentagonal bipyramid]] ({{math|''J''{{sub|16}}}}).
Higher forms can be constructed with [[Isosceles triangle|isosceles triangle]]s.

== Forms ==
{| class="wikitable"
!Name
! [[Elongated triangular bipyramid|elongated triangular bipyramid]] <br>{{math|''J''{{sub|14}}}}
![[Elongated square bipyramid|elongated square bipyramid]]<br>{{math|''J''{{sub|15}}}}
![[Elongated pentagonal bipyramid|elongated pentagonal bipyramid]]<br>{{math|''J''{{sub|16}}}}
![[Elongated hexagonal bipyramid|elongated<br>hexagonal<BR>bipyramid]]
|-
!Type
!colspan=3|Equilateral
!Irregular
|-
!Image
| [[Image:Elongated triangular dipyramid.png|100px]]
| [[Image:Elongated square dipyramid.png|100px]] 
| [[Image:Elongated pentagonal dipyramid.png|100px]]
| [[Image:Elongated hexagonal dipyramid.png|100px]]
|- align=center
!Faces
| 6 triangles,<BR>3 squares
| 8 triangles,<BR>4 squares 
| 10 triangles,<BR>5 squares
| 12 triangles,<BR>6 squares
|-
!Dual
|[[Triangular bifrustum|triangular bifrustum]]
|[[Square bifrustum|square bifrustum]]
|[[Pentagonal bifrustum|pentagonal bifrustum]]
|[[Hexagonal bifrustum|hexagonal bifrustum]]
|}

== Applications ==
Elongated bipyramids are sometimes used as dice, especially to make dice with atypical side count, such as 5 or 7. Such a die has numbers written on the square faces, which are usually heightened into rectangles for convenience in rolling. Whichever number comes face-up when the die is rolled is the side that is to be read.

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

==References==
*[[Biography:Norman Johnson (mathematician)|Norman W. Johnson]], "Convex Solids with Regular Faces", Canadian Journal of Mathematics, '''18''', 1966, pages 169&ndash;200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
*{{cite book|author=Victor A. Zalgaller|title=Convex Polyhedra with Regular Faces|publisher=Consultants Bureau|year=1969|id=No ISBN}}  The first proof that there are only 92 Johnson solids.

{{Polyhedron navigator}}

[[Category:Pyramids and bipyramids]]



{{Sourceattribution|Elongated bipyramid}}