Augmented triangular prism: Difference between revisions

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

{{Infobox polyhedron

| image = Augmented triangular prism.png

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

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

| properties = [[Convex polytopepolyhedron|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 ==

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 6six [[equilateral triangle]]s and 2two [[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]],. and theThe augmented triangular prism is among them, enumerated as 49ththe forty-ninth Johnson solid <math> J_{49} </math>.{{r|francis}}

== Properties ==

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<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 4four. Its [[dihedral angle]] can be calculated by adding the angle of an equilateral square pyramid and a regular triangular prism. The dihedral angle of an equilateral square pyramid between two adjacent triangular faces is <math display="inline"> \arccos \left(-1/3 \right) \approx 109.5^\circ </math>, and that between a triangular face and its base is <math display="inline"> \arctan \left(\sqrt{2}\right) \approx 54.7^\circ </math>. The dihedral angle of a triangular prism between two adjacent square faces isin the [[internal angle]] of an equilateral triangle <math display="inline"> \pi/3 = 60^\circ </math>, and that between square-to-triangle is <math display="inline"> \pi/2 = 90^\circ </math>. Therefore, the dihedral angle of the augmented triangular prism between square-to-triangle and triangle-to-triangle on the edge where both square pyramid and triangular prism are attached is, respectivelyfollowing:{{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>

<math display="block"> \begin{align}

* 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>.

\frac{\pi}{3} + \arccos \left(-\frac{1}{3}\right) &\approx 104.5^\circ, \\

* 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>

\frac{\pi}{2} + \arccos \left(-\frac{1}{3}\right) &\approx 144.5^\circ.

* 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}

\end{align} </math>

\frac{arctan \pi}left(\sqrt{32}\right) + \arccos \left(-\frac{1\pi}{3}\right) &\approx 104114.57^\circ, \\

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

\end{align} </math>

</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 ==

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| 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>

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| 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>

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| 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>