Editing Elongated triangular bipyramid

{{Short description|14th Johnson solid; triangular prism capped with tetrahedra}}
{{Infobox polyhedron
|image=elongated_triangular_dipyramid.png
|type=[[Johnson solid|Johnson]]<br />{{math|[[pentagonal bipyramid|''J''{{sub|13}}]] – '''''J''{{sub|14}}''' – [[elongated square bipyramid|''J''{{sub|15}}]]}}
|faces=6 [[triangle]]s<br />3 [[Square (geometry)|square]]s
|edges=15
|vertices=8
|symmetry={{math|[[Dihedral symmetry|''D''{{sub|3h}}]], [3,2], (*322)}}
|rotation_group={{math|''D''{{sub|3}}, [3,2]{{sup|+}}, (322)}}
|vertex_config={{math|2(3{{sup|3}})<br />6(3{{sup|2}}.4{{sup|2}})}}
|dual=[[Triangular bifrustum]]
|properties=[[convex set|convex]]
|net=Johnson solid 14 net.png
}}
In [[geometry]], the '''elongated triangular [[bipyramid]]''' (or '''dipyramid''') or '''triakis triangular prism''' a polyhedron constructed from a [[triangular prism]] by attaching two [[tetrahedron]]s to its bases. It is an example of [[Johnson solid]].

== Construction ==
The elongated triangular bipyramid is constructed from a [[triangular prism]] by attaching two [[tetrahedron]]s onto its bases, a process known as the [[Elongation (geometry)|elongation]].{{r|rajwade}} These tetrahedrons cover the triangular faces so that the resulting polyhedron has nine faces (six of them are [[equilateral triangle]]s and three of them are [[square]]s), fifteen edges, and eight vertices.{{r|berman}} A [[Convex set|convex]] polyhedron in which all of the faces are [[regular polygon]]s is the [[Johnson solid]]. The elongated bipyramid is one of them, enumerated as the fourteenth Johnson solid <math> J_{14} </math>.{{r|uehara}}

== Properties ==
[[File:3D Johnson J14.stl|thumb|3D model of an elongated triangular bipyramid]]
The surface area of an elongated triangular bipyramid <math> A </math> is the sum of all polygonal face's area: six equilateral triangles and three squares. The volume of an elongated triangular bipyramid <math> V </math> can be ascertained by slicing it off into two tetrahedrons and a regular triangular prism and then adding their volume. The height of an elongated triangular bipyramid <math> h </math> is the sum of two tetrahedrons and a regular triangular prism' height. Therefore, given the edge length <math> a </math>, its surface area and volume is formulated as:{{r|berman|pye}}
<math display="block"> \begin{align}
 A &= \left(\frac{3\sqrt{3}}{2} + 3\right)a^2 \approx 5.598a^2, \\
 V &= \left(\frac{\sqrt{2}}{6} + \frac{\sqrt{3}}{2} \right) a^3 \approx 0.669a^3, \\
 h &= \left(\frac{2\sqrt{6}}{3} + 1 \right)\cdot a \approx \cdot 2.633a.
\end{align}
</math>

It has the same [[Point groups in three dimensions|three-dimensional symmetry group]] as the triangular prism, the [[dihedral group]] <math> D_{3 \mathrm{h}} </math> of order twelve. The [[dihedral angle]] of an elongated triangular bipyramid can be calculated by adding the angle of the tetrahedron and the triangular prism:{{r|johnson}}
* the dihedral angle of a tetrahedron between two adjacent triangular faces is <math display="inline"> \arccos \left(\frac{1}{3}\right) \approx 70.5^\circ </math>;
* the dihedral angle of the triangular prism between the square to its bases is <math display="inline"> \frac{\pi}{2} = 90^\circ </math>, and the dihedral angle between square-to-triangle, on the edge where tetrahedron and triangular prism are attached, is <math display="inline"> \arccos \left(\frac{1}{3}\right) + \frac{\pi}{2} \approx 160.5^\circ </math>;
* the dihedral angle of the triangular prism between two adjacent square faces is the internal angle of an equilateral triangle <math display="inline"> \frac{\pi}{3} = 60^\circ </math>.

== Appearances ==
The [[nirrosula]], an African musical instrument woven out of strips of plant leaves, is made in the form of a series of elongated bipyramids with non-equilateral triangles as the faces of their end caps.{{r|gerdes}}

==References==
{{Reflist|refs=

<ref name="berman">{{citation
 | 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=gerdes>{{citation|first=Paulus|last=Gerdes|author-link=Paulus Gerdes|title=Exploration of technologies, emerging from African cultural practices, in mathematics (teacher) education|journal= ZDM Mathematics Education |volume=42|issue=1|pages=11–17|doi=10.1007/s11858-009-0208-2|year=2009|s2cid=122791717 }}.</ref>

<ref name="johnson">{{citation
 | 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="pye">{{citation
 | last = Sapiña | first = R.
 | title = Area and volume of the Johnson solid <math> J_{8} </math>
 | url = https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J8/calculadora-area-volumen-formulas.html
 | issn = 2659-9899
 | access-date = 2020-09-09
 | language = es
 | journal = Problemas y Ecuaciones
}}.</ref>

<ref name="rajwade">{{citation
 | 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
 | publisher = Hindustan Book Agency
 | page = 84&ndash;89
 | isbn = 978-93-86279-06-4
 | doi = 10.1007/978-93-86279-06-4
}}.</ref>

<ref name="uehara">{{citation
 | 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 | urlname2 = JohnsonSolid | title2 = Johnson solid| urlname =ElongatedTriangularDipyramid| title =Elongated triangular bipyramid}}

{{Johnson solids navigator}}

[[Category:Johnson solids]]
[[Category:Bipyramids]]