View source for Cupola (geometry)

{{Short description|Solid made by joining an n- and 2n-gon with triangles and squares}}
{{other uses|Cupola (disambiguation)}}
{{More citations needed|date=March 2024}}
{{Infobox polyhedron
| name          = Set of cupolae
| image         = Pentagonal cupola.png
| caption       = Pentagonal example
| type          =
| euler         = 
| faces         = {{mvar|n}} [[triangles]],<br>{{mvar|n}} [[Square (geometry)|square]]s,<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          = Semibisected trapezohedron
| properties    = [[Convex polytope|convex]], [[prismatoid]]
| vertex_figure = 
| net           = 
| net_caption   = 
}}

In [[geometry]], a '''cupola''' is a [[Polyhedron|solid]] formed by joining two [[polygon]]s, one (the base) with twice as many [[Edge (geometry)|edges]] as the other, by an alternating band of [[isosceles triangle]]s and [[rectangle]]s. If the triangles are [[equilateral triangle|equilateral]] and the rectangles are [[Square (geometry)|square]]s, 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]], [[rhombicuboctahedron]], and [[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]] {{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]]s.

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

==Examples==
{{Cupolae}}

[[Image:Tile 3464.svg|thumb|Plane "[[hexagon]]al cupolae" in the [[rhombitrihexagonal tiling]]]]
The triangular, square, and pentagonal cupolae are the only non-trivial convex cupolae with regular faces: The "[[hexagon]]al cupola" is a plane figure, and the [[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 tetracontagonal cupola has:
{{legend|blue|40 [[isosceles triangle]]s;}}
{{legend|yellow|40 rectangles;}}
{{legend|red|A top regular [[tetracontagon]];}} 

and a bottom regular [[octacontagon]] (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>

for {{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} \\[5pt]
  =\ & 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] } \\[5pt]
  =\ & 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]} \\[5pt]
  =\ & 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} } \\[5pt]
  =\ & r_t \sqrt{ \left[ \cos^2\tfrac{\pi}{n} - 2\cos\tfrac{\pi}{n} + 1 \right] + \sin^2\tfrac{\pi}{n} } \\[5pt]
  =\ & 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}}
{{Star-cupoloids}}
Star cupolae exist for any top base {{math|{{mset|''n''/''d''}}}} where {{math|6/5 < ''n''/''d'' < 6}} and {{mvar|d}} is odd. At these limits, the cupolae collapse into plane figures. Beyond these limits, the triangles and squares can no longer span the distance between the two base polygons (it can still be made with non-equilateral [[isosceles triangle]]s and non-square rectangles). If {{mvar|d}} is even, the bottom base {{math|{{mset|2''n''/''d''}}}} becomes degenerate; then we can form a ''cupoloid'' or ''semicupola'' by withdrawing this degenerate face and letting the triangles and squares connect to each other here (through single edges) rather than to the late bottom base (through its double edges). In particular, the [[tetrahemihexahedron]] may be seen as a {{math|{{mset|3/2}}}}-cupoloid.

The cupolae are all [[Orientability|orientable]], while the cupoloids are all non-orientable. For a cupoloid, if {{math|''n''/''d'' > 2}}, then the triangles and squares do not cover the entire (single) base, and a small membrane is placed in this base {{math|{{mset|''n''/''d''}}}}-gon that simply covers empty space. Hence the {{math|{{mset|5/2}}}}- and {{math|{{mset|7/2}}}}-cupoloids pictured above have membranes (not filled in), while the {{math|{{mset|5/4}}}}- and {{math|{{mset|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} \left( \frac{\pi d}{n} \right)}}.</math>
In particular, {{math|1=''h'' = 0}} at the limits {{math|1=''n''/''d'' = 6}} and {{math|1=''n''/''d'' = 6/5}}, and {{mvar|h}} is maximized at {{math|1=''n''/''d'' = 2}} (in the [[digonal cupola]]: 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|2''n''/''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 base {{math|{{mset|2''n''/''d''}}}}-gon has been withdrawn.

==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.svg|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.svg|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]]<br>4 [[triangular prism]]s<br>6 [[triangular prism]]s<br>4 [[triangular pyramid]]s<br>1 [[cuboctahedron]]
|align=center |28
|&nbsp;1 [[cube]]<br>&nbsp;6 [[cube|square prisms]]<br>12 [[triangular prism]]s<br>&nbsp;8 [[triangular pyramid]]s<br>&nbsp;1&nbsp;[[rhombicuboctahedron]]
|align=center |28
|&nbsp;1 [[octahedron]]<br>&nbsp;8 [[triangular prism]]s<br>12 [[triangular prism]]s<br>&nbsp;6 [[square pyramid]]s<br>&nbsp;1&nbsp;[[rhombicuboctahedron]]
|align=center |64
|&nbsp;1 [[dodecahedron]]<br>12 [[pentagonal prism]]s<br>30 [[triangular prism]]s<br>20 [[triangular pyramid]]s<br>&nbsp;1&nbsp;[[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]]<br>20 [[triangular prism]]s<br>30 [[triangular prism]]s<br>12 [[pentagonal pyramid]]s<br> 1 [[rhombicosidodecahedron]]-->
|- align=center 
!Related<BR>uniform<BR>polychora
|colspan="2" |[[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}}
*[[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]]