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 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>.

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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>