View source for Cupola (geometry)

{{Short description|Solid made by joining an n- and 2n-gon with triangles and squares}}
{{Infobox polyhedron
| name          = Set of cupolae
| image         = Pentagonal cupola.png
| caption       = Pentagonal example
| type          =
| euler         = 
| faces         = {{mvar|n}} triangles,<br>{{mvar|n}} squares,<br>1 [[Polygon|{{mvar|n}}-gon]],<br>1 {{math|2''n''}}-gon
| edges         = {{math|5''n''}}
| vertices      = {{math|3''n''}}
| vertex_config = 
| schläfli      = {{math|{''n''} {{!}}{{!}} t{''n''} }}
| wythoff       = 
| conway        = 
| coxeter       = 
| symmetry      = {{math|[[Symmetry group#Three dimensions|C{{sub|''n''v}}]], [1,''n''], (*''nn''),}} order&nbsp;{{math|2''n''}}
| rotation_group = {{math|C{{sub|''n''}}, [1,''n'']<sup>+</sup>, (''nn''),}} order&nbsp;{{mvar|n}}
| surface_area  = 
| volume        =  
| angle         =  
| dual          = ?
| properties    = [[Convex polytope|convex]], [[Prismatoid|prismatoid]]
| vertex_figure = 
| net           = 
| net_caption   = 
}}

In [[Geometry|geometry]], a '''cupola''' is a [[Polyhedron|solid]] formed by joining two [[Polygon|polygon]]s, one (the base) with twice as many [[Edge (geometry)|edges]] as the other, by an alternating band of [[Isosceles triangle|isosceles triangle]]s and [[Rectangle|rectangle]]s. If the triangles are [[Equilateral triangle|equilateral]] and the rectangles are squares, while the base and its opposite face are regular polygons, the [[Triangular cupola|triangular]], [[Square cupola|square]], and [[Pentagonal cupola|pentagonal]] cupolae all count among the [[Johnson solid]]s, and can be formed by taking sections of the [[Cuboctahedron|cuboctahedron]], [[Rhombicuboctahedron|rhombicuboctahedron]], and [[Rhombicosidodecahedron|rhombicosidodecahedron]], respectively.

A cupola can be seen as a [[Prism (geometry)|prism]] where one of the polygons has been collapsed in half by merging alternate vertices.

A cupola can be given an extended [[Schläfli symbol]] {{math| {''n''} {{!}}{{!}} t{''n''},}} representing a [[Regular polygon|regular polygon]] {{math|{''n''} }} joined by a parallel of its [[Truncation (geometry)#Truncation of polygons|truncation]], {{math|t{''n''} }} or {{math|{2''n''}.}}

Cupolae are a subclass of the [[Prismatoid|prismatoid]]s.

Its dual contains a shape that is sort of a weld between half of an {{mvar|n}}-sided [[Trapezohedron|trapezohedron]] and a {{math|2''n''}}-sided pyramid.

==Examples==

[[Image:Tile 3464.svg|thumb|Plane "[[Hexagon|hexagon]]al cupolae" in the [[Rhombitrihexagonal tiling|rhombitrihexagonal tiling]]]]
The above-mentioned three polyhedra are the only non-trivial convex cupolae with regular faces: The "[[Hexagon|hexagon]]al cupola" is a plane figure, and the [[Triangular prism|triangular prism]] might be considered a "cupola" of degree&nbsp;2 (the cupola of a line segment and a square). However, cupolae of higher-degree polygons may be constructed with [[Regular polygon#Taxonomic classification|irregular]] triangular and rectangular faces.

==Coordinates of the vertices==
[[File:Cupola_40.png|thumb|left|A 40-sided cupola has:
{{legend|blue|40 [[Isosceles triangle|isosceles triangle]]s;}}
{{legend|yellow|40 rectangles;}}
{{legend|red|A top regular [[Tetracontagon|40-gon]];}} 
and a bottom regular [[Octacontagon|80-gon]] (hidden).]]
The definition of the cupola does not require the base (or the side opposite the base, which can be called the top) to be a regular polygon, but it is convenient to consider the case where the cupola has its maximal symmetry, {{math|C<sub>''n''v</sub>}}. In that case, the top is a regular {{mvar|n}}-gon, while the base is either a regular {{math|2''n''}}-gon or a {{math|2''n''}}-gon which has two different side lengths alternating and the same angles as a regular {{math|2''n''}}-gon. It is convenient to fix the coordinate system so that the base lies in the {{mvar|xy}}-plane, with the top in a plane parallel to the {{mvar|xy}}-plane. The {{mvar|z}}-axis is the {{mvar|n}}-fold axis, and the mirror planes pass through the {{mvar|z}}-axis and bisect the sides of the base. They also either bisect the sides or the angles of the top polygon, or both. (If {{mvar|n}} is even, half of the mirror planes bisect the sides of the top polygon and half bisect the angles, while if {{mvar|n}} is odd, each mirror plane bisects one side and one angle of the top polygon.) The vertices of the base can be designated {{tmath|V_1}} through {{tmath|V_{2n},}} while the vertices of the top polygon can be designated {{tmath|V_{2n+1} }} through {{tmath|V_{3n}.}} With these conventions, the coordinates of the vertices can be written as:
<math display=block>\begin{array}{rllcc}
  V_{2j-1} :& \biggl( r_b \cos\left(\frac{2\pi(j-1)}{n} + \alpha\right), & r_b \sin\left(\frac{2\pi(j-1)}{n} + \alpha\right), & 0 \biggr) \\[2pt]
 V_{2j} :& \biggl( r_b \cos\left(\frac{2\pi j}{n} - \alpha\right), & r_b \sin\left(\frac{2\pi j}{n} - \alpha\right), & 0 \biggr) \\[2pt]
  V_{2n+j} :& \biggl( r_t \cos\frac{\pi j}{n}, & r_t \sin\frac{\pi j}{n}, & h \biggr)
\end{array}</math>

where {{math|1=''j'' = 1, 2, ..., ''n''}}.

Since the polygons {{tmath|V_1 V_ 2 V_{2n+2} V_{2n+1},}} etc. are rectangles, this puts a constraint on the values of {{tmath|r_b, r_t, \alpha.}} The distance <math>\bigl|V_1 V_2 \bigr|</math> is equal to
<math display=block>\begin{align}
  & r_b \sqrt{ \left[\cos\left(\tfrac{2\pi}{n} - \alpha\right) - \cos \alpha\right]^2 + \left[\sin\left(\tfrac{2\pi}{n} - \alpha\right) - \sin\alpha\right]^2} \\
  =\ & r_b \sqrt{ \left[\cos^2 \left(\tfrac{2\pi}{n} - \alpha\right) - 2\cos\left(\tfrac{2pi}{n} - \alpha\right)\cos\alpha + \cos^2 \alpha \right] + \left[\sin^2 \left(\tfrac{2\pi}{n} - \alpha\right) - 2\sin\left(\tfrac{2\pi}{n} - \alpha\right) \sin\alpha + \sin^2 \alpha \right] } \\
  =\ & r_b \sqrt{ 2\left[1 - \cos\left(\tfrac{2\pi}{n} - \alpha\right) \cos\alpha - \sin\left(\tfrac{2\pi}{n} - \alpha\right)\sin\alpha \right]} \\
  =\ & r_b \sqrt{ 2\left[1 - \cos\left(\tfrac{2\pi}{n} - 2\alpha\right)\right]}
\end{align}</math>

while the distance <math>\bigl| V_{2n+1}V_{2n+2} \bigr|</math> is equal to
<math display=block>\begin{align}
  & r_t \sqrt{ \left[ \cos\tfrac{\pi}{n} - 1 \right]^2 + \sin^2 \tfrac{\pi}{n} } \\
  =\ & r_t \sqrt{ \left[ \cos^2\tfrac{\pi}{n} - 2\cos\tfrac{\pi}{n} + 1 \right] + \sin^2\tfrac{\pi}{n} } \\
  =\ & r_t \sqrt{2 \left[1 - \cos\tfrac{\pi}{n} \right]}
\end{align}</math>

These are to be equal, and if this common edge is denoted by {{mvar|s}},
<math display=block>\begin{align}
  r_b &= \frac{s}{ \sqrt{2\left[1 - \cos\left(\tfrac{2\pi}{n} - 2\alpha \right) \right] }} \\[4pt]
  r_t &= \frac{s}{ \sqrt{2\left[1 - \cos\tfrac{\pi}{n} \right] }}
\end{align}</math>

These values are to be inserted into the expressions for the coordinates of the vertices given earlier.

== Star-cupolae ==
Star cupolae exist for all bases {{math|{''n''/''d''} }} where {{math|6/5 < ''n''/''d'' < 6}} and {{mvar|d}} is odd. At the limits the cupolae collapse into plane figures: beyond the limits the triangles and squares can no longer span the distance between the two polygons (it can still be made if the triangles or squares are irregular.). When {{mvar|d}} is even, the bottom base {{math|{2''n''/''d''} }} becomes degenerate: we can form a ''cupoloid'' or ''semicupola'' by withdrawing this degenerate face and instead letting the triangles and squares connect to each other here. In particular, the [[Tetrahemihexahedron|tetrahemihexahedron]] may be seen as a {3/2}-cupoloid. The cupolae are all [[Orientability|orientable]], while the cupoloids are all nonorientable. When {{math|''n''/''d'' > 2}} in a cupoloid, the triangles and squares do not cover the entire base, and a small membrane is left in the base that simply covers empty space. Hence the {5/2} and {7/2} cupoloids pictured above have membranes (not filled in), while the {5/4} and {7/4} cupoloids pictured above do not.

The height {{mvar|h}} of an {{math|{{mset|''n''/''d''}}}}-cupola or cupoloid is given by the formula
<math display=block>h = \sqrt{1-\frac{1}{4 \sin^{2} \frac{\pi d}{n}}}</math>
In particular, {{math|1=''h'' = 0}} at the limits of {{math|1=''n''/''d'' = 6}} and {{math|1=''n''/''d'' = 6/5}}, and {{mvar|h}} is maximized at {{math|1=''n''/''d'' = 2}} (the triangular prism, where the triangles are upright).<ref>{{cite web|url=http://www.orchidpalms.com/polyhedra/cupolas/cupola1.html|title=cupolas|website=www.orchidpalms.com|access-date=21 April 2018}}</ref><ref>{{cite web|url=http://www.orchidpalms.com/polyhedra/cupolas/cupola2.html|title=semicupolas|website=www.orchidpalms.com|access-date=21 April 2018}}</ref>

In the images above, the star cupolae have been given a consistent colour scheme to aid identifying their faces: the base {{math|{{mset|''n''/''d''}}}}-gon is red, the base {{math|{{mset|''2n''/''d''}}}}-gon is yellow, the squares are blue, and the triangles are green. The cupoloids have the base {{math|{{mset|''n''/''d''}}}}-gon red, the squares yellow, and the triangles blue, as the other base has been withdrawn.

== Anticupola==
{{Infobox polyhedron
| name          = Set of anticupolae
| image         = Pentagonal anticupola.png
| caption       = Pentagonal example
| type          =
| euler         = 
| faces         = {{math|3''n''}} triangles,<br>1 [[Polygon|{{mvar|n}}-gon]],<br>1 {{math|2''n''}}-gon
| edges         = {{math|6''n''}}
| vertices      = {{math|3''n''}}
| vertex_config = 
| schläfli      = {{math|{''n''} {{!}}{{!}} t{''n''} }}
| wythoff       = 
| conway        = 
| coxeter       = 
| symmetry      = {{math|[[Symmetry group#Three dimensions|C{{sub|''n''v}}]], [1,''n''], (*''nn''),}} order&nbsp;{{math|2''n''}}
| rotation_group = {{math|C{{sub|''n''}}, [1,''n'']<sup>+</sup>, (''nn''),}} order&nbsp;{{mvar|n}}
| surface_area  = 
| volume        =  
| angle         =  
| dual          = ?
| properties    = [[Convex polytope|convex]], [[Prismatoid|prismatoid]]
| vertex_figure = 
| net           = 
| net_caption   = 
}}

An {{mvar|n}}-gonal '''anticupola''' is constructed from a regular {{math|2''n''}}-gonal base, {{math|3''n''}} triangles as two types, and a regular {{mvar|n}}-gonal top. For {{math|1=''n'' = 2}}, the top digon face is reduced to a single edge. The vertices of the top polygon are aligned with vertices in the lower polygon. The symmetry is {{math|C<sub>''n''v</sub>}}, order&nbsp;{{math|2''n''}}.

An anticupola can't be constructed with all regular faces, {{fact|date=August 2020}} although some can be made regular. If the top {{mvar|n}}-gon and triangles are regular, the base {{math|2''n''}}-gon can not be planar and regular. In such a case, {{math|1=''n'' = 6}} generates a regular hexagon and surrounding equilateral triangles of a snub hexagonal tiling, which can be closed into a zero volume polygon with the base a symmetric 12-gon shaped like a larger hexagon, having adjacent pairs of [[Collinearity|colinear]] edges.

Two anticupola can be augmented together on their base as a bianticupola.

{| class=wikitable width=400
|+ Family of convex anticupolae
|-
! {{mvar|n}} ||2 !!3 !!4 !!5 !!6...
|-
!Name || {{math|s{2} <nowiki>||</nowiki> t{2} }} || {{math|s{3} <nowiki>||</nowiki> t{3} }} || {{math|s{4} <nowiki>||</nowiki> t{4} }} || {{math|s{5} <nowiki>||</nowiki> t{5} }} || {{math|s{6} <nowiki>||</nowiki> t{6} }} 
|- align=center
!Image
|valign=bottom|[[File:Digonal anticupola.png|80px]]<BR>Digonal
|valign=bottom|[[File:triangular anticupola.png|80px]]<BR>Triangular
|valign=bottom|[[File:square anticupola.png|80px]]<BR>Square
|valign=bottom|[[File:Pentagonal anticupola.png|80px]]<BR>Pentagonal
|valign=bottom|[[File:Hexagonal anticupola.png|80px]]<BR>Hexagonal
|- align=center
!Transparent
|[[File:Digonal anticupola-trans.png|70px]]
|[[File:triangular anticupola-trans.png|80px]]
|[[File:square anticupola-trans.png|80px]]
|[[File:Pentagonal anticupola-trans.png|80px]]
|[[File:Hexagonal anticupola-trans.png|80px]]
|- align=center
!Net
|[[File:Digonal anticupola_net.png|80px]]
|[[File:triangular anticupola_net.png|80px]]
|[[File:square anticupola_net.png|80px]]
|[[File:Pentagonal anticupola_net.png|80px]]
|[[File:Hexagonal anticupola_net.png|80px]]
|}

{{-}}

==Hypercupolae==
The '''hypercupolae''' or '''polyhedral cupolae''' are a family of convex nonuniform polychora (here four-dimensional figures), analogous to the cupolas. Each one's bases are a [[Platonic solid]] and its [[Expansion (geometry)|expansion]].<ref name="Segmentochora">[http://www.bendwavy.org/klitzing/pdf/artConvSeg_8.pdf Convex Segmentochora] Dr. Richard Klitzing, Symmetry: Culture and Science, Vol. 11, Nos. 1-4, 139-181, 2000</ref>
{| class="wikitable"
|- style="background-color: #e7dcc3;"
!Name
!colspan="2"|[[Tetrahedral cupola]]
!colspan="2"|[[Cubic cupola]]
!colspan="2"|[[Octahedral cupola]]
!colspan="2"|[[Dodecahedral cupola]]
!colspan="2"|Hexagonal tiling cupola
<!--!colspan="2"|Icosahedral cupola -->
|- align=center 
![[Schläfli symbol]]
!colspan=2|{{math|{3,3} <nowiki>||</nowiki> rr{3,3} }}
!colspan=2|{{math|{4,3} <nowiki>||</nowiki> rr{4,3} }}
!colspan=2|{{math|{3,4} <nowiki>||</nowiki> rr{3,4} }}
!colspan=2|{{math|{5,3} <nowiki>||</nowiki> rr{5,3} }}
!colspan=2|{{math|{6,3} <nowiki>||</nowiki> rr{6,3} }}
|- align=center 
!Segmentochora<BR>index<ref name="Segmentochora" />
!colspan="2"|K4.23
!colspan="2"|K4.71
!colspan="2"|K4.107
!colspan="2"|K4.152
!colspan="2"|
|- align=center 
!circumradius
|colspan="2"|<math>1</math>
|colspan="2"|<math display=inline>\sqrt{\frac{3+\sqrt 2}{2}} \approx 1.485634</math>
|colspan="2"|<math display=inline>\sqrt{2+\sqrt 2} \approx 1.847759</math>
|colspan="2"|<math display=inline>3+\sqrt 5 \approx 5.236068</math>
|colspan="2"|
|- align=center 
!Image
|colspan="2" |[[File:4D Tetrahedral Cupola-perspective-cuboctahedron-first.png|150px]] 
|colspan="2" |[[File:4D Cubic Cupola-perspective-cube-first.png|150px]] 
|colspan="2" |[[File:4D octahedral cupola-perspective-octahedron-first.png|150px]] 
|colspan="2" |[[File:Dodecahedral_cupola.png|150px]]
|colspan="2"|
|- align=center 
!Cap cells
|colspan="2"|[[File:Uniform_polyhedron-33-t0.png|60px]][[File:Uniform_polyhedron-33-t02.png|60px]]
|colspan="2"|[[File:Uniform_polyhedron-43-t0.png|60px]][[File:Uniform_polyhedron-43-t02.png|60px]]
|colspan="2"|[[File:Uniform_polyhedron-43-t2.png|60px]][[File:Uniform_polyhedron-43-t02.png|60px]]
|colspan="2"|[[File:Uniform_polyhedron-53-t0.png|60px]][[File:Uniform_polyhedron-53-t02.png|60px]]
|colspan="2"|[[File:Uniform_tiling 63-t0.png|60px]][[File:Uniform_tiling 63-t02.png|60px]]
<!--|colspan="2"|[[File:Uniform_polyhedron-53-t2.png|60px]][[File:Uniform_polyhedron-53-t02.png|60px]]-->
|- align=center 
!Vertices
|colspan="2" |16
|colspan="2" |32
|colspan="2" |30
|colspan="2" |80
|colspan="2" |&infin;
<!--|colspan="2" |72-->
|- align=center 
!Edges
|colspan="2" |42
|colspan="2" |84
|colspan="2" |84
|colspan="2" |210
|colspan="2" |&infin;
<!--|colspan="2" |210-->
|-
!Faces
|align=center |42
|24 triangles<br>18 squares
|align=center |80
|32 triangles<br>48 squares
|align=center |82
|40 triangles<br>42 squares
|align=center |194
|80 triangles<br>90 squares<br>24 pentagons
|align=center |&infin; ||
<!--|align=right |202
|100 [[Triangle|{3}]] + 90 [[Square|{4}]] + 12 [[Pentagon|{5}]]-->
|-
!Cells
|align=center |16
|1 [[Tetrahedron|tetrahedron]]<br>4 [[Triangular prism|triangular prism]]s<br>6 [[Triangular prism|triangular prism]]s<br>4 triangular pyramids<br>1 [[Cuboctahedron|cuboctahedron]]
|align=center |28
|&nbsp;1 [[Cube|cube]]<br>&nbsp;6 [[Cube|square prisms]]<br>12 [[Triangular prism|triangular prism]]s<br>&nbsp;8 triangular pyramids<br>&nbsp;1&nbsp;[[Rhombicuboctahedron|rhombicuboctahedron]]
|align=center |28
|&nbsp;1 [[Octahedron|octahedron]]<br>&nbsp;8 [[Triangular prism|triangular prism]]s<br>12 [[Triangular prism|triangular prism]]s<br>&nbsp;6 [[Square pyramid|square pyramid]]s<br>&nbsp;1&nbsp;[[Rhombicuboctahedron|rhombicuboctahedron]]
|align=center |64
|&nbsp;1 [[Dodecahedron|dodecahedron]]<br>12 [[Pentagonal prism|pentagonal prism]]s<br>30 [[Triangular prism|triangular prism]]s<br>20 triangular pyramids<br>&nbsp;1&nbsp;[[Rhombicosidodecahedron|rhombicosidodecahedron]]
|align=center |&infin; ||1 hexagonal tiling<BR>&infin; hexagonal prisms<BR>&infin; triangular prisms<BR>&infin; triangular pyramids<BR>1&nbsp;rhombitrihexagonal tiling
<!--|align=right |64
| 1 [[Icosahedron|icosahedron]]<br>20 [[Triangular prism|triangular prism]]s<br>30 [[Triangular prism|triangular prism]]s<br>12 [[Pentagonal pyramid|pentagonal pyramid]]s<br> 1 [[Rhombicosidodecahedron|rhombicosidodecahedron]]-->
|- align=center 
!Related<BR>uniform<BR>polychora
|colspan="2" |[[Runcinated 5-cell|runcinated 5-cell]]<BR>{{CDD|node_1|3|node|3|node|3|node_1}}
|colspan="2" |runcinated tesseract<BR>{{CDD|node_1|4|node|3|node|3|node_1}}
|colspan="2" |runcinated 24-cell<BR>{{CDD|node_1|3|node|4|node|3|node_1}}
|colspan="2" |runcinated 120-cell<BR>{{CDD|node_1|5|node|3|node|3|node_1}}
|colspan="2" |runcinated hexagonal tiling honeycomb<BR>{{CDD|node_1|6|node|3|node|3|node_1}}
|}

== See also ==
* Orthobicupola
* Gyrobicupola
* [[Rotunda (geometry)|Rotunda]]

==References==
{{reflist}}
*[[Biography:Norman Johnson (mathematician)|Johnson, N.W.]] ''Convex Polyhedra with Regular Faces.'' Can. J. Math. 18, 169–200, 1966.

==External links==
*{{MathWorld |urlname=Cupola |title=Cupola}}
* [http://bendwavy.org/klitzing/explain/segmentochora.htm Segmentotopes]

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[[Category:Prismatoid polyhedra]]
[[Category:Johnson solids]]

{{Sourceattribution|Cupola (geometry)}}