Editing Triangular cupola

{{Short description|Cupola with hexagonal base}}
{{Infobox polyhedron
 | image = triangular_cupola.png
 | type = [[Johnson solid|Johnson]]<br>{{math|[[pentagonal pyramid|''J''{{sub|2}}]] – '''''J''{{sub|3}}''' – [[square cupola|''J''{{sub|4}}]]}}
 | faces = 4 [[triangle]]s<br>3 [[Square (geometry)|square]]s<br>1 [[hexagon]]
 | edges = 15
 | vertices = 9
 | symmetry = <math> C_{3v} </math>
 | vertex_config = <math> \begin{align}
 &6 \times (3 \times 4 \times 6) \, + \\
 &3 \times (3 \times 4 \times 3 \times 4)
\end{align} </math>
 | properties = [[convex set|convex]]
 | net = Triangular cupola (symmetric net).svg
}}

In [[geometry]], the '''triangular cupola''' is the [[Cupola (geometry)|cupola]] with [[hexagon]] as its base and [[triangle]] as its top. If the edges are equal in length, the triangular cupola is the Johnson solid. It can be seen as half a [[cuboctahedron]]. The triangular cupola can be applied to construct many polyhedrons.

== Properties ==
The triangular cupola has 4 [[Triangle|triangles]], 3 [[Square (geometry)|squares]], and 1 [[hexagon]] as their faces; the hexagon is the base and one of the four triangles is the top. If all of the edges are equal in length, the [[Equilateral triangle|triangles]] and the hexagon becomes [[Regular polygon|regular]].{{r|berman|uehara}} The [[dihedral angle]] between each triangle and the hexagon is approximately 70.5°, that between each square and the hexagon is 54.7°, and that between square and triangle is 125.3°.{{r|johnson}} A [[Convex polytope|convex]] polyhedron in which all of the faces are regular is a [[Johnson solid]], and the triangular cupola is among them, enumerated as the third Johnson solid <math> J_{3} </math>.{{r|uehara}}

Given that <math> a </math> is the edge length of a triangular cupola. Its surface area <math> A </math> can be calculated by adding the area of four equilateral triangles, three squares, and one hexagon:{{r|berman}}
<math display="block"> A = \left(3+\frac{5\sqrt{3}}{2} \right) a^2 \approx 7.33a^2. </math>
Its height <math> h </math> and volume <math> V </math> is:{{r|pye|berman}}
<math display="block"> \begin{align}
 h &= \frac{\sqrt{6}}{3} a\approx 0.82a, \\
 V &= \left(\frac{5}{3\sqrt{2}}\right)a^3 \approx 1.18a^3.
\end{align}
</math>

[[File:J3 triangular cupola.stl|thumb|3D model of a triangular cupola]]
It has an [[axis of symmetry]] passing through the center of its both top and base, which is symmetrical by rotating around it at one- and two-thirds of a full-turn angle. It is also mirror-symmetric relative to any perpendicular plane passing through a bisector of the hexagonal base. Therefore, it has [[pyramidal symmetry]], the [[cyclic group]] <math> C_{3\mathrm{v}} </math> of order 6.{{r|johnson}}

== Related polyhedra ==
The triangular cupola can be found in the construction of many polyhedrons. An example is the [[cuboctahedron]] in which the triangular cupola may be considered as its hemisphere.{{r|cromwell}} A construction that involves the attachment of its base to another polyhedron is known as [[Augmentation (geometry)|augmentation]]; attaching it to [[Prism (geometry)|prisms]] or [[Antiprism|antiprisms]] is known as [[Elongation (geometry)|elongation]] or [[gyroelongation]].{{r|demey|slobodan}} Some of the other Johnson solids constructed in such a way are [[elongated triangular cupola]] <math> J_{18} </math>, [[gyroelongated triangular cupola]] <math> J_{22} </math>, [[triangular orthobicupola]] <math> J_{27} </math>, [[elongated triangular orthobicupola]] <math> J_{35} </math>, [[elongated triangular gyrobicupola]] <math> J_{36} </math>, [[gyroelongated triangular bicupola]] <math> J_{44} </math>, [[augmented truncated tetrahedron]] <math> J_{65} </math>.{{r|rajwade}}

The triangular cupola may also be applied in constructing [[truncated tetrahedron]], although it leaves some hollows and a regular tetrahedron as its interior. {{harvtxt|Cundy|1956}} constructed such polyhedron in a similar way as the [[rhombic dodecahedron]] constructed by attaching six [[square pyramid]]s outwards, each of which apices are in the [[Cube (geometry)|cube]]'s center. That being said, such truncated tetrahedron is constructed by attaching four triangular cupolas rectangle-by-rectangle; those cupolas in which the alternating sides of both right isosceles triangle and rectangle have the edges in terms of ratio <math display="inline> 1 : \frac{1}{2}\sqrt{2} </math>. The [[truncated octahedron]] can be constructed by attaching eight of those same triangular cupolas triangle-by-triangle.{{r|cundy}}

==References==
{{Reflist|refs=

<ref name="berman">{{cite journal
 | last = Berman | first = Martin
 | year = 1971
 | title = Regular-faced convex polyhedra
 | journal = Journal of the Franklin Institute
 | volume = 291
 | issue = 5
 | pages = 329–352
 | doi = 10.1016/0016-0032(71)90071-8
 | mr = 290245
}}</ref>

<ref name="cromwell">{{cite book
 | last = Cromwell | first = Peter R.
 | year = 1997
 | title = Polyhedra
 | url = https://archive.org/details/polyhedra0000crom/page/86/mode/1up?q=cupola&view=theater
 | page = 86
 | publisher = [[Cambridge University Press]]
 | isbn = 978-0-521-55432-9
}}</ref>

<ref name=cundy>{{cite journal
 | last = Cundy | first = H. Martyn
 | year = 1956
 | title = 2642. Unitary Construction of Certain Polyhedra
 | journal = [[The Mathematical Gazette]]
 | volume = 40 | issue = 234 | pages = 280&ndash;282
 | jstor = 3609622
 | doi = 10.2307/3609622
}}</ref>

<ref name="demey">{{cite journal
 | last1 = Demey | first1 = Lorenz
 | last2 = Smessaert | first2 = Hans
 | title = Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation
 | journal = Symmetry
 | date = 2017
 | volume = 9
 | issue = 10
 | page = 204
 | doi = 10.3390/sym9100204
 | doi-access = free
 | bibcode = 2017Symm....9..204D
}}</ref>

<ref name="johnson">{{cite journal
 | last = Johnson | first = Norman W. | author-link = Norman W. Johnson
 | year = 1966
 | title = Convex polyhedra with regular faces
 | journal = [[Canadian Journal of Mathematics]]
 | volume = 18
 | pages = 169–200
 | doi = 10.4153/cjm-1966-021-8
 | mr = 0185507
 | s2cid = 122006114
 | zbl = 0132.14603 | doi-access = free
}}</ref>

<ref name="pye">{{cite journal
 | last = Sapiña | first = R.
 | title = Area and volume of the Johnson solid <math> J_3 </math>
 | url = https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J3/calculadora-area-volumen-formulas.html
 | issn = 2659-9899
 | access-date = 2020-09-08
 | language = es
 | journal = Problemas y Ecuaciones
}}</ref>

<ref name="rajwade">{{cite book
 | last = Rajwade | first = A. R.
 | title = Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem
 | series = Texts and Readings in Mathematics
 | year = 2001
 | url = https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84
 | page = 84&ndash;89
 | publisher = Hindustan Book Agency
 | isbn = 978-93-86279-06-4
 | doi = 10.1007/978-93-86279-06-4
}}</ref>

<ref name="slobodan">{{cite journal
 | last1 = Slobodan | first1 = Mišić
 | last2 = Obradović | first2 = Marija
 | last3 = Ðukanović | first3 = Gordana
 | title = Composite Concave Cupolae as Geometric and Architectural Forms
 | year = 2015
 | journal = Journal for Geometry and Graphics
 | volume = 19
 | issue = 1
 | pages = 79–91
 | url = https://www.heldermann-verlag.de/jgg/jgg19/j19h1misi.pdf
}}</ref>

<ref name="uehara">{{cite book
 | last = Uehara | first = Ryuhei
 | year = 2020
 | title = Introduction to Computational Origami: The World of New Computational Geometry
 | url = https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62
 | page = 62
 | publisher = Springer
 | isbn = 978-981-15-4470-5
 | doi = 10.1007/978-981-15-4470-5
 | s2cid = 220150682
}}</ref>

}}

==External links==
* {{Mathworld2 | urlname =TriangularCupola| title = Triangular cupola | urlname2 = JohnsonSolid | title2 = Johnson solid}}

{{Johnson solids navigator}}

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