Augmented triangular prism: Difference between revisions

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{{Short description|49th Johnson solid (8 faces)}}

{{Infobox polyhedron

| image =augmented Augmented triangular prism.png

| type = [[Johnson solid|Johnson]]<br>{{math|[[gyroelongated pentagonal birotunda|''J<''{{sub>|48</sub>}}]] -– '''''J<''{{sub>|49</sub>}}''' -– [[biaugmented triangular prism|''J<''{{sub>|50</sub>}}]]}}

| faces =3x2 6 [[triangle]]s<br>2 [[Square (geometry)|square]]s

| edges = 13

| vertices = 7

| symmetry =''C'' <submath>2v C_{2\mathrm{v}} </submath>

| vertex_config =2(3.4 <supmath>2</sup>)<br>1(3⁴)<br>4(3³.4) \begin{align}

&2 \times (3 \times 4^2) \, + \\

|dual=-

&1 \times (3^4) \, + \\

|properties=[[Convex set|convex]]

&4 \times (3^3 \times 4) \end{align} </math>

|net=Johnson solid 49 net.png

| properties = [[Convex polyhedron|convex]], [[composite polyhedron|composite]]

| net = Johnson solid 49 net.png

| angle = triangle-triangle: 109.5°, 169.4°<br>triangle-square: 90°, 114.7°<br>square-square: 60°

}}

[[File:J49 augmented triangular prism.stl|thumb|3D model of an augmented triangular prism]]

In [[geometry]], the '''augmented triangular prism''' is a polyhedron constructed by attaching an [[equilateral square pyramid]] onto the square face of a [[triangular prism]]. As a result, it is an example of [[Johnson solid]]. It can be visualized as the chemical compound, known as [[capped trigonal prismatic molecular geometry]].

== Construction ==

In [[geometry]], the '''augmented triangular prism''' is one of the [[Johnson solid]]s (''J''₄₉). As the name suggests, it can be constructed by augmenting a triangular [[prism (geometry)|prism]] by attaching a [[square pyramid]] (''J''₁) to one of its equatorial faces. The resulting solid bears a superficial resemblance to the [[gyrobifastigium]] (''J''₂₆), the difference being that the latter is constructed by attaching a second triangular prism, rather than a square pyramid.

The augmented triangular prism is [[composite polyhedron|composite]]: it can be constructed from a [[triangular prism]] by attaching an [[equilateral square pyramid]] to one of its square faces, a process known as [[Augmentation (geometry)|augmentation]].{{r|timofeenko-2009|rajwade}} This square pyramid covers the square face of the prism, so the resulting polyhedron has six [[equilateral triangle]]s and two [[Square (geometry)|square]]s as its faces.{{r|berman}} A [[Convex set|convex]] polyhedron in which all faces are [[Regular polygon|regular]] is [[Johnson solid]]. The augmented triangular prism is among them, enumerated as the forty-ninth Johnson solid <math> J_{49} </math>.{{r|francis}}

== Properties ==

An augmented triangular prism with edge length <math> a </math> has a surface area, calculated by adding six equilateral triangles and two squares' area:{{r|berman}}

<math display="block"> \frac{4 + 3\sqrt{3}}{2}a^2 \approx 4.598a^2. </math>

Its volume can be obtained by slicing it into a regular triangular prism and an equilateral square pyramid, and adding their volume subsequently:{{r|berman}}

<math display="block"> \frac{2\sqrt{2} + 3\sqrt{3}}{12}a^3 \approx 0.669a^3. </math>

It has [[Point groups in three dimensions|three-dimensional symmetry group]] of the cyclic group <math> C_{2\mathrm{v}} </math> of order four. Its [[dihedral angle]] can be calculated by adding the angle of an equilateral square pyramid and a regular triangular prism in the following:{{r|johnson}}

* The dihedral angle of an augmented triangular prism between two adjacent triangles is that of an equilateral square pyramid between two adjacent triangular faces, <math display="inline"> \arccos \left(-1/3 \right) \approx 109.5^\circ </math>

* The dihedral angle of an augmented triangular prism between two adjacent squares is that of a triangular prism between two lateral faces, the [[interior angle]] of a triangular prism <math> \pi/3 = 60^\circ </math>.

* The dihedral angle of an augmented triangular prism between square and triangle is the dihedral angle of a triangular prism between the base and its lateral face, <math display="inline"> \pi/2 = 90^\circ </math>

* The dihedral angle of an equilateral square pyramid between a triangular face and its base is <math display="inline"> \arctan \left(\sqrt{2}\right) \approx 54.7^\circ </math>. Therefore, the dihedral angle of an augmented triangular prism between a square (the lateral face of the triangular prism) and triangle (the lateral face of the equilateral square pyramid) on the edge where the equilateral square pyramid is attached to the square face of the triangular prism, and between two adjacent triangles (the lateral face of both equilateral square pyramids) on the edge where two equilateral square pyramids are attached adjacently to the triangular prism, are <math display="block"> \begin{align}

\arctan \left(\sqrt{2}\right) + \frac{\pi}{3} &\approx 114.7^\circ, \\

2 \arctan \left(\sqrt{2}\right) + \frac{\pi}{3} &\approx 169.4^\circ.

\end{align}

</math>

== Application ==

In the geometry of [[chemical compounds]], a polyhedron may commonly be visualized an [[atom cluster]] surrounding a central atom. The [[capped trigonal prismatic molecular geometry]] describes clusters for which this polyhedron is an augmented triangular prism.{{r|hbmr}} An example of such compound is the [[potassium heptafluorotantalate]].{{r|kaupp}}

== See also ==

* [[Biaugmented triangular prism]] &mdash; the 50th Johnson solid, constructed by attaching a triangular prism to two equilateral square pyramids.

* [[Triaugmented triangular prism]] &mdash; the 51st Johnson solid, constructed by augmenting each square face of a triangular prism with a square pyramid.

== 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="hbmr">{{cite journal

| last1 = Hoffmann | first1 = Roald

| last2 = Beier | first2 = Barbara F.

| last3 = Muetterties | first3 = Earl L.

| last4 = Rossi | first4 = Angelo R.

| year = 1977

| title = Seven-coordination. A molecular orbital exploration of structure, stereochemistry, and reaction dynamics

| journal = [[Inorganic Chemistry (journal)|Inorganic Chemistry]]

| volume = 16 | issue = 3 | pages = 511–522

| doi = 10.1021/ic50169a002

}}</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="kaupp">{{cite journal

| last = Kaupp | first = Martin

| year = 2001

| title = "Non-VSEPR" Structures and Bonding in d(0) Systems

| journal = Angew Chem Int Ed Engl

| volume = 40 | issue = 1 | pages = 3534–3565

| doi = 10.1002/1521-3773(20011001)40:19<3534::AID-ANIE3534>3.0.CO;2-#

| pmid = 11592184

}}</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="timofeenko-2009">{{cite journal

| last = Timofeenko | first = A. V.

| year = 2009

| title = Convex Polyhedra with Parquet Faces

| journal = Docklady Mathematics

| url = https://www.interocitors.com/tmp/papers/timo-parquet.pdf

| volume = 80 | issue = 2

| pages = 720–723

| doi = 10.1134/S1064562409050238

}}</ref>

}}

{{Johnson solid}}

==External links==

* {{MathWorld | urlname=JohnsonSolid | title=Johnson Solid}}

** {{MathWorld | urlname=AugmentedTriangularPrism | title=Augmented triangular prism }}

[[Category:{{Johnson solids]] navigator}}

[[Category:Johnson solids]]

{{Polyhedron-stub}}