View source for Elongated triangular gyrobicupola

{{Short description|36th Johnson solid}}
{{Infobox polyhedron
 | image = Elongated triangular gyrobicupola.png
 | type = [[Johnson solid|Johnson]]<br>{{math|[[elongated triangular orthobicupola|''J''{{sub|35}}]] – '''''J''{{sub|36}}''' – [[elongated square gyrobicupola|''J''{{sub|37}}]]}}
 | faces = 8 [[triangle]]s<br>12 [[Square (geometry)|square]]s
 | edges = 36
 | vertices = 18
 | symmetry = <math> D_{3h} </math>
 | vertex_config = <math> \begin{align}
 &6 \times (3 \times 4 \times 3 \times 4) + \\
 &12 \times (3 \times 4^3)
\end{align} </math>
 | properties = [[convex set|convex]]
 | net = Johnson solid 36 net.png
}}

In [[geometry]], the '''elongated triangular gyrobicupola''' is a polyhedron constructed by attaching two regular [[triangular cupola]]s to the base of a regular [[hexagonal prism]], with one of them rotated in <math> 60^\circ </math>. It is an example of [[Johnson solid]].

== Construction ==
The elongated triangular gyrobicupola is similarly can be constructed as the [[elongated triangular orthobicupola]], started from a [[hexagonal prism]] by attaching two regular [[triangular cupola]]e onto its base, covering its hexagonal faces.{{r|rajwade}} This construction process is known as [[Elongation (geometry)|elongation]], giving the resulting polyhedron has 8 [[equilateral triangle]]s and 12 squares.{{r|berman}} The difference between those two polyhedrons is one of two triangular cupolas in the elongated triangular gyrobicupola is rotated in <math> 60^\circ </math>. A [[Convex set|convex]] polyhedron in which all faces are [[Regular polygon|regular]] is [[Johnson solid]], and the elongated triangular gyrobicupola is one among them, enumerated as 36th Johnson solid <math> J_{36} </math>.{{r|francis}}

== Properties ==
An elongated triangular gyrobicupola with a given edge length <math> a </math> has a surface area by adding the area of all regular faces:{{r|berman}}
<math display="block"> \left(12 + 2\sqrt{3}\right)a^2 \approx 15.464a^2. </math>
Its volume can be calculated by cutting it off into two triangular cupolae and a hexagonal prism with regular faces, and then adding their volumes up:{{r|berman}}
<math display="block"> \left(\frac{5\sqrt{2}}{3} + \frac{3\sqrt{3}}{2}\right)a^3 \approx 4.955a^3. </math>

Its [[Point groups in three dimensions|three-dimensional symmetry groups]] is the [[prismatic symmetry]], the [[dihedral group]] <math> D_{3d} </math> of order 12.{{clarification needed|date=March 2024|reason=Need to explicitly explain the meaning of this symmetry}} Its [[dihedral angle]] can be calculated by adding the angle of the triangular cupola and hexagonal prism. The dihedral angle of a hexagonal prism between two adjacent squares is the [[internal angle]] of a regular hexagon <math> 120^\circ = 2\pi/3</math>, and that between its base and square face is <math> \pi/2 = 90^\circ </math>. The dihedral angle of a regular triangular cupola between each triangle and the hexagon is approximately <math> 70.5^\circ </math>, that between each square and the hexagon is <math> 54.7^\circ </math>, and that between square and triangle is <math> 125.3^\circ </math>. The dihedral angle of an elongated triangular orthobicupola between the triangle-to-square and square-to-square, on the edge where the triangular cupola and the prism is attached, is respectively:{{r|johnson}}
<math display="block"> \begin{align}
 \frac{\pi}{2} + 70.5^\circ &\approx 160.5^\circ, \\
 \frac{\pi}{2} + 54.7^\circ &\approx 144.7^\circ.
\end{align} </math>

==Related polyhedra and honeycombs==

The elongated triangular gyrobicupola forms space-filling [[Honeycomb (geometry)|honeycomb]]s with [[Tetrahedron|tetrahedra]] and [[square pyramid]]s.<ref>{{Cite web|url=http://woodenpolyhedra.web.fc2.com/J36.html|title = J36 honeycomb}}</ref><br>

== 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="francis">{{cite journal
 | last = Francis | first = Darryl
 | title = Johnson solids & their acronyms
 | journal = Word Ways
 | date = August 2013
 | volume = 46 | issue = 3 | page = 177
 | url = https://go.gale.com/ps/i.do?id=GALE%7CA340298118
}}</ref>

<ref name="johnson">{{cite journal
 | last = Johnson | first = Norman W. | authorlink = 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="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>

}}

==External links==
* {{MathWorld2|title2=Johnson solid|urlname2=JohnsonSolid| urlname=ElongatedTriangularGyrobicupola | title=Elongated triangular gyrobicupola}}

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