Elongated triangular pyramid: Difference between revisions

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{{Short description|7th Johnson solid (7 faces)}}

{{Infobox polyhedron

| image = elongated_triangular_pyramid.png

| type = [[Johnson solid|Johnson]]<br>{{math|[[pentagonal rotunda|''J<''{{sub>|6</sub>}}]] -– ''''' J<''{{sub>|7</sub>}}''' -– [[elongated square pyramid|''J<''{{sub>|8</sub>}}]]}}

| faces = 1+34 [[triangle]]s<br>3 [[Square (geometry)|square]]s

| edges = 12

| vertices = 7

| symmetry = ''{{math|C''<{{sub>|3v</sub>}}, [3], (*33)}}

| rotation_group = ''{{math|C''<{{sub>|3</sub>}}, [3]<{{sup>|+</sup>}}, (33)}}

| vertex_config = {{math|1(3<{{sup>|3</sup>}})<br>3(3.4<{{sup>|2</sup>}})<br>3(3<{{sup>|2</sup>}}.4<{{sup>|2</sup>}})}}

| dual = [[self-dual]]{{r|draghicescu}}

| properties = [[convex set|convex]]

| net = Elongated Triangular Pyramid Net.svg

}}

[[File:Tetraedro elongado 3D.stl|thumb|Johnson solid {{math|''J<''{{sub>|7</sub>}}}}.]]

In [[geometry]], the '''elongated triangular pyramid''' is one of the [[Johnson solid]]s (''J''₇). As the name suggests, it can be constructed by elongating a [[tetrahedron]] by attaching a [[triangular prism]] to its base. Like any elongated [[pyramid]], the resulting solid is topologically (but not geometrically) self-[[dual polyhedron|dual]].

In [[geometry]], the '''elongated triangular pyramid''' is one of the [[Johnson solid]]s ({{math|''J''{{sub|7}}}}). As the name suggests, it can be constructed by elongating a [[tetrahedron]] by attaching a [[triangular prism]] to its base. Like any elongated [[Pyramid (geometry)|pyramid]], the resulting solid is topologically (but not geometrically) self-[[dual polyhedron|dual]].

{{Johnson solid}}

==Formulae Construction ==

The elongated triangular pyramid is constructed from a [[triangular prism]] by attaching [[regular tetrahedron]] onto one of its bases, a process known as [[Elongation (geometry)|elongation]].{{r|rajwade}} The tetrahedron covers an [[equilateral triangle]], replacing it with three other equilateral triangles, so that the resulting polyhedron has four equilateral triangles and three [[Square (geometry)|square]]s as its faces.{{r|berman}} A convex polyhedron in which all of the faces are [[regular polygon]]s is called the [[Johnson solid]], and the elongated triangular pyramid is among them, enumerated as the seventh Johnson solid <math> J_7 </math>.{{r|uehara}}

The following [[formula]]e for [[volume]] and [[surface area]] can be used if all [[face (geometry)|faces]] are [[regular polygon|regular]], with edge length ''a'':<ref>[[Stephen Wolfram]], "[http://www.wolframalpha.com/input/?i=Elongated+triangular+pyramid Elongated triangular pyramid]" from [[Wolfram Alpha]]. Retrieved July 21, 2010.</ref>

== Properties ==

:<math>V=\left(\frac{1}{12}\left(\sqrt{2}+3\sqrt{3}\right)\right)a^3\approx0.550864...a^3</math>

An elongated triangular pyramid with edge length <math> a </math> has a height, by adding the height of a regular tetrahedron and a triangular prism:{{r|pye}}

<math display="block"> \left( 1 + \frac{\sqrt{6}}{3}\right)a \approx 1.816a. </math>

Its surface area can be calculated by adding the area of all eight equilateral triangles and three squares:{{r|berman}}

<math display="block"> \left(3+\sqrt{3}\right)a^2 \approx 4.732a^2, </math>

and its volume can be calculated by slicing it into a regular tetrahedron and a prism, adding their volume up:{{r|berman}}:

<math display="block"> \left(\frac{1}{12}\left(\sqrt{2}+3\sqrt{3}\right)\right)a^3 \approx 0.551a^3. </math>

It has the [[Point groups in three dimensions|three-dimensional symmetry group]], the cyclic group <math> C_{3\mathrm{v}} </math> of order 6. Its [[dihedral angle]] can be calculated by adding the angle of the tetrahedron and the triangular prism:{{r|johnson}}

:<math>A=\left(3+\sqrt{3}\right)a^2\approx4.73205...a^2</math>

* 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>.

== References ==

The [[height]] is given by<ref name="pye">{{cite journal |last=Sapiña |first=R. |title=Area and volume of the Johnson solid J₇ |url=https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J7/calculadora-area-volumen-formulas.html |issn=2659-9899 |access-date= 2020-08-12 |language=es |journal = Problemas y Ecuaciones}}</ref>

{{reflist|refs=

:<math>H = a\cdot \left( 1 + \frac{\sqrt{6}}{3}\right) \approx a\cdot 1.816496581</math>

If the edges are not the same length, use the individual formulae for the tetrahedron and triangular prism separately, and add the results together.

<ref name="berman">{{cite journal

=== Dual polyhedron ===

| last = Berman | first = Martin

Topologically, the elongated triangular pyramid is its own dual. Geometrically, the dual has seven irregular faces: one equilateral triangle, three isosceles triangles and three isosceles trapezoids.

| year = 1971

{| class=wikitable width=320

| title = Regular-faced convex polyhedra

|- valign=top

| journal = Journal of the Franklin Institute

!Dual elongated triangular pyramid

| volume = 291

!Net of dual

|- valignissue =top 5

| pages = 329–352

|[[File:Dual elongated triangular pyramid.png|160px]]

| doi = 10.1016/0016-0032(71)90071-8

|[[File:Dual elongated triangular pyramid net.png|160px]]

| mr = 290245

|}

}}</ref>

<ref name="draghicescu">{{cite book

==Related polyhedra and honeycombs==

| last = Draghicescu | first = Mircea

| contribution = Dual Models: One Shape to Make Them All

| contribution-url = https://archive.bridgesmathart.org/2016/bridges2016

| editor-first1 = Eva | editor-last1 = Torrence

| editor-first2 = Bruce | editor-last2 = Torrence

| editor-first3 = Carlo H. | editor-last3 = Séquin

| editor-first4 = Douglas | editor-last4 = McKenna

| editor-first5 = Kristóf | editor-last5 = Fenyvesi

| editor-first6 = Reza | editor-last6 = Sarhangi

| title = Bridges Finland: Mathematics, Music, Art, Architecture, Education, Culture

| url = https://archive.bridgesmathart.org/2016/frontmatter.pdf

| pages = 635–640

}}</ref>

<ref name="johnson">{{cite journal

The elongated triangular pyramid can form a [[tessellation of space]] with [[square pyramid]]s and/or [[Octahedron|octahedra]].<ref>{{Cite web|url=http://woodenpolyhedra.web.fc2.com/J7.html|title=J7 honeycomb}}</ref>

| 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">{{cite journal

| 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">{{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

| 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">{{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>

}}

==References==

{{Reflist}}

==External links==

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[[Category:Johnson solids]]

[[Category:Self-dual polyhedra]]

[[Category:Pyramids and bipyramids(geometry)]]

{{Polyhedron-stub}}