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 in the following:{{r|johnson}}

* The dihedral angle of aan biaugmentedaugmented 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>, and is that of the interior angle of a triangular prism <math> \pi/3 = 60^\circ </math>.

* The dihedral angle of aan biaugmentedaugmented triangular prism between squaretwo andadjacent trianglesquares is the dihedral anglethat of a triangular prism between two lateral faces, the base[[interior andangle]] itsof laterala face,triangular prism <math display="inline"> \pi/23 = 9060^\circ </math>.

* The dihedral angle of an equilateralaugmented squaretriangular pyramidprism between square and triangle is the dihedral angle of a triangular faceprism between the base and its baselateral isface, <math display="inline"> \arctan \left(\sqrt{pi/2}\right) \approx= 54.790^\circ </math>.
* The dihedral angle of aan triangular prism between two adjacentequilateral square facespyramid isbetween thea [[internaltriangular angle]]face ofand anits equilateralbase triangleis <math display="inline"> \pi/3arctan =\left(\sqrt{2}\right) 60\approx 54.7^\circ </math>. Therefore, the dihedral angle of aan biaugmentedaugmented 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.

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== Application ==

In the geometry of [[chemical compounds]], a polyhedron may commonly visualizebe 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 withto two equilateral square pyramids.

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

== References ==

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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>