View source for Bicupola (geometry)

{{Short description|Solid made from 2 cupolae joined base-to-base}}
{{Infobox polyhedron
| name          = Set of bicupolae
| image         = [[Image:cuboctahedron.svg|240px|Triangular gyrobicupola]]
| caption       = Example: triangular gyrobicupola (cuboctahedron)
| faces         = {{math|2''n''}} triangles,<BR>{{math|2''n''}} squares<BR>2 {{nowrap|[[Polygon|{{mvar|n}}-gons]]}}
| edges         = {{math|8''n''}}
| vertices      = {{math|4''n''}}
| symmetry      = Ortho: {{math|D{{sub|''n''h}}, [2,n], *n22,}} order {{math|4''n''}}<BR>Gyro: {{math|D{{sub|''n''d}}, [2{{sup|+}},2n], 2*n,}} order {{math|4''n''}}
| rotation      = {{math|D{{sub|''n''}}, [2,n]{{sup|+}}, n22,}} order {{math|2''n''}}
| properties    =[[Convex polytope|convex]]
}}
[[Image:Gyrobifastigium.png|thumb|The [[Gyrobifastigium|gyrobifastigium]] ({{math|''J''{{sub|26}}}}) can be considered a digonal gyrobicupola.]]

In [[Geometry|geometry]], a '''bicupola''' is a solid formed by connecting two [[Cupola (geometry)|cupolae]] on their bases.

There are two classes of bicupola because each cupola (bicupola half) is bordered by alternating triangles and squares. If similar faces are attached together the result is an ''orthobicupola''; if squares are attached to triangles it is a ''gyrobicupola''.

Cupolae and bicupolae categorically exist as infinite sets of polyhedra, just like the [[Pyramid (geometry)|pyramid]]s, [[Bipyramid|bipyramid]]s, [[Prism (geometry)|prisms]], and trapezohedra.

Six bicupolae have [[Regular polygon|regular polygon]] faces: triangular, square and [[Pentagon|pentagon]]al ortho- and gyrobicupolae. The triangular gyrobicupola is an [[Archimedean solid]], the [[Cuboctahedron|cuboctahedron]]; the other five are [[Johnson solid]]s.

Bicupolae of higher order can be constructed if the flank faces are allowed to stretch into [[Rectangle|rectangle]]s and [[Isosceles triangle|isosceles triangle]]s.

Bicupolae are special in having four faces on every vertex. This means that their dual polyhedra will have all [[Quadrilateral|quadrilateral]] faces. The best known example is the [[Rhombic dodecahedron|rhombic dodecahedron]] composed of 12 rhombic faces. The dual of the ortho-form, [[Triangular orthobicupola|triangular orthobicupola]], is also a [[Dodecahedron|dodecahedron]], similar to ''rhombic dodecahedron'', but it has 6 trapezoid faces which alternate long and short edges around the circumference.

==Forms==

===Set of orthobicupolae===
{|class="wikitable"
![[List of spherical symmetry groups|Symmetry]]
!Picture
!Description
|-
!{{math|D{{sub|2h}}<BR>[2,2]<BR>*222}}
|[[Image:Digonal orthobicupola.png|100px]]
|[[Orthobifastigium]] or ''digonal orthobicupola'': 4 triangles (coplanar), 4 squares. It is [[Self-dual polyhedron|self-dual]]
|-
!{{math|D{{sub|3h}}<BR>[2,3]<BR>*223}}
|[[Image:Triangular orthobicupola.png|100px]]
|[[Triangular orthobicupola]] ({{math|''J''{{sub|27}}}}): 8 triangles, 6 squares; its dual is the [[Trapezo-rhombic dodecahedron|trapezo-rhombic dodecahedron]]
|-
!{{math|D{{sub|4h}}<BR>[2,4]<BR>*224}}
|[[Image:Square orthobicupola.png|100px]]
|[[Square orthobicupola]] ({{math|''J''{{sub|28}}}}): 8 triangles, 10 squares
|-
!{{math|D{{sub|5h}}<BR>[2,5]<BR>*225}}
|[[Image:Pentagonal orthobicupola.png|100px]]
|[[Pentagonal orthobicupola]] ({{math|''J''{{sub|30}}}}): 10 triangles, 10 squares, 2 pentagons
|-
!{{math|D{{sub|nh}}<BR>[2,''n'']<BR>*22n}}
|
|{{nowrap|{{mvar|n}}-gonal}} orthobicupola: {{math|2''n''}} triangles, {{math|2''n''}} rectangles, 2 {{nowrap|{{mvar|n}}-gons}}
|}

===Set of gyrobicupolae===
A {{mvar|n}}-gonal gyrobicupola has the same topology as a {{mvar|n}}-gonal rectified antiprism, [[Conway polyhedron notation]], {{math|aA''n''}}.
{|class="wikitable"
![[List of spherical symmetry groups|Symmetry]]
!Picture
!Description
|-
!{{math|D{{sub|2d}}<BR>[2{{sup|+}},4]<BR>2*2}}
|[[Image:Gyrobifastigium.png|100px]]
|[[Gyrobifastigium]] ({{math|''J''{{sub|26}}}}) or ''digonal gyrobicupola'': 4 triangles, 4 squares
|-
!{{math|D{{sub|3d}}<BR>[2{{sup|+}},6]<BR>2*3}}
|[[File:Cuboctahedron.png|100px]]
|''Triangular gyrobicupola'' or [[Cuboctahedron|cuboctahedron]]: 8 triangles, 6 squares; its dual is the [[Rhombic dodecahedron|rhombic dodecahedron]]
|-
!{{math|D{{sub|4d}}<BR>[2{{sup|+}},8]<BR>2*4}}
|[[Image:Square gyrobicupola.png|100px]]
|[[Square gyrobicupola]] ({{math|''J''{{sub|29}}}}): 8 triangles, 10 squares;its dual is the elongated [[Tetragonal trapezohedron|tetragonal trapezohedron]]
|-
!{{math|D{{sub|5d}}<BR>[2{{sup|+}},10]<BR>2*5}}
|[[Image:Pentagonal gyrobicupola.png|100px]]
|[[Pentagonal gyrobicupola]] ({{math|''J''{{sub|31}}}}): 10 triangles, 10 squares, 2 pentagons; its dual is the [[Rhombic icosahedron|rhombic icosahedron]]
|-
!{{math|D{{sub|nd}}<BR>[2{{sup|+}},2n]<BR>2*n}}
|
|{{nowrap|{{mvar|n}}-gonal}} gyrobicupola: {{math|2''n''}} triangles, {{math|2''n''}} rectangles, 2 {{nowrap|{{mvar|n}}-gons}}
|}

==References==
*[[Biography:Norman Johnson (mathematician)|Norman W. Johnson]], "Convex Solids with Regular Faces", Canadian Journal of Mathematics, '''18''', 1966, pages 169–200. 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}}

{{DEFAULTSORT:Bicupola}}
[[Category:Polyhedra]]

{{Sourceattribution|Bicupola (geometry)}}