Triaugmented triangular prism: Difference between revisions

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{{good article}}

{{Infobox polyhedron

| image = Triaugmented triangular prism (symmetric view).svg

| type = [[Deltahedron]],<br>[[Johnson solid|Johnson]]<br>{{math|[[biaugmented triangular prism|''J''{{sub|50}}]] – '''''J''{{sub|51}}''' – [[augmented pentagonal prism|''J''{{sub|52}}]]}}

| faces = 14 [[triangle]]s

| edges = 21

| vertices = 9

| symmetry = <math>D_{3\mathrm{h}}</math>

| vertex_config = <math>3\times 3^4+6\times 3^5</math>

| dual = [[Associahedron]] <|Associahedron {{math|''K''<sub>K_55</mathsub>}}]]

| angle = 109.5°<br>144.7°<br>169.5°

| properties = [[convex polytope|convex]],<br>[[composite polyhedron|composite]]

| net = [[File:Triaugmented triangular prism (symmetric net).svg|300px]]

}}

The '''triaugmented triangular prism''', in geometry, is a [[convex polyhedron]] with 14 [[equilateral triangle]]s as its faces. It can be constructed from a [[triangular prism]] by attaching [[equilateral square pyramid]]s to each of its three square faces. The same shape is also called the '''tetrakis triangular prism''',{{r|shdc}} '''tricapped trigonal prism''',{{r|kepert}} '''tetracaidecadeltahedron''',{{r|burgiel|pugh}} or '''tetrakaidecadeltahedron''';{{r|shdc}} these last names mean a polyhedron with 14 triangular faces. It is an example of a [[deltahedron]], and[[composite ofpolyhedron]], aand [[Johnson solid]].

The edges and vertices of the triaugmented triangular prism form a [[maximal planar graph]] with 9 vertices and 21 edges, called the '''Fritsch graph'''. It was used by Rudolf and Gerda Fritsch to show that [[Alfred Kempe]]'s attempted proof of the [[four color theorem]] was incorrect. The Fritsch graph is one of only six graphs in which every [[Neighbourhood (graph theory)|neighborhood]] is a 4- or 5-vertex cycle.

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==Construction==

[[File:J51 triaugmented triangular prism.stl|thumb|3D model of the triaugmented triangular prism]]

The triaugmented triangular prism is a [[composite polyhedron]], meaning it can be constructed by attaching [[equilateral square pyramid]]s to each of the three square faces of a [[triangular prism]], a process called [[Augmentation (geometry)|augmentation]].{{r|timofeenko-2009|trigg}} These pyramids cover each square, replacing it with four [[equilateral triangle]]s, so that the resulting polyhedron has 14 equilateral triangles as its faces. A polyhedron with only equilateral triangles as faces is called a [[deltahedron]]. There are only eight different [[Convex set|convex]] deltahedra, one of which is the triaugmented triangular prism.{{r|fw47|cundy}} More generally, the convex polyhedra in which all faces are [[regular polygon]]s are called the [[Johnson solid]]s, and every convex deltahedron is a Johnson solid. The triaugmented triangular prism is numbered among the Johnson solids {{nowrap|as <math>J_{51}</math>.{{r|francis}}}}

One possible system of [[Cartesian coordinates]] for the vertices of a triaugmented triangular prism, giving it edge length 2, is:{{r|shdc}}

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

\left(0,\frac2{\sqrt3},\pm1 \right),\qquad & \left(\pm1,-\frac1{\sqrt3},\pm1 \right),\\

\left(0,-\sqrt2-frac{1+\frac1sqrt6}{\sqrt3},0 \right),\qquad & \left(\pm\frac{1+\sqrt6}{2},\frac{1+\sqrt6}{2\sqrt3},0\right).\\

\end{align}</math>

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can be derived by slicing it into a central prism and three square pyramids, and adding their volumes.{{r|berman}}

[[File:Triaugmented triangular prism (geodesic nets).svg|thumb|upright=1.2|Two unfolded nets of the triaugmented triangular prism, showing its two types of closed geodesics. Prism faces are pink; pyramid faces are blue and yellow.]]

It has the same [[Point groups in three dimensions|three-dimensional symmetry group]] as the triangular prism, the [[dihedral group]] <math>D_{3\mathrm{h}}</math> of order twelve. Its [[dihedral angle]]s can be calculated by adding the angles of the component pyramids and prism. The prism itself has square-triangle dihedral angles <math>\pi/2</math> and square-square angles <math>\pi/3</math>. The triangle-triangle angles on the pyramid are the same as in the [[regular octahedron]], and the square-triangle angles are half that. Therefore, for the triaugmented triangular prism, the dihedral angles incident to the degree-four vertices, on the edges of the prism triangles, and on the square-to-square prism edges are, respectively,{{r|johnson}}

The triaugmented triangular prism has two types of [[closed geodesic]]s. These are paths on its surface that are locally straight: they avoid vertices of the polyhedron, follow line segments across the faces that they cross, and form [[complementary angles]] on the two incident faces of each edge that they cross. One of the two types of closed geodesic runs parallel to the square base of a pyramid, through the eight faces surrounding the pyramid. For a polyhedron with unit-length sides, this geodesic has length <math>4</math>. The other type of closed geodesic crosses ten faces, and has length <math>\sqrt{19}\approx 4.36</math>. For each type there is a continuous family of parallel geodesics, all of the same length.{{r|lptw}}

ItThe triaugmented triangular prism has the same [[Point groups in three dimensions|three-dimensional symmetry group]] as the triangular prism, the [[dihedral group]] <math>D_{3\mathrm{h}}</math> of order twelve. Its [[dihedral angle]]s can be calculated by adding the angles of the component pyramids and prism. The prism itself has square-triangle dihedral angles <math>\pi/2</math> and square-square angles <math>\pi/3</math>. The triangle-triangle angles on the pyramid are the same as in the [[regular octahedron]], and the square-triangle angles are half that. Therefore, for the triaugmented triangular prism, the dihedral angles incident to the degree-four vertices, on the edges of the prism triangles, and on the square-to-square prism edges are, respectively,{{r|johnson}}

<math display=block>

\begin{align}

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\frac{\pi}{3}+\arccos\left(-\frac13\right)&\approx 169.5^\circ.\\

\end{align}</math>

{{-}}

== Fritsch graph ==

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| title = Geometric combinatorics

| volume = 13

| year = 2007| s2cidisbn = 11435731978-0-8218-3736-8

| s2cid = 11435731

}}; see Definition 3.3, Figure 3.6, and related discussion</ref>

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| title = A simple sphere theorem for graphs

| year = 2019}}</ref>

<ref name=lptw>{{citation

| last1 = Lawson | first1 = Kyle A.

| last2 = Parish | first2 = James L.

| last3 = Traub | first3 = Cynthia M.

| last4 = Weyhaupt | first4 = Adam G.

| doi = 10.12732/ijpam.v89i2.1 | doi-access = free

| issue = 2

| journal = International Journal of Pure and Applied Mathematics

| pages = 123–139

| title = Coloring graphs to classify simple closed geodesics on convex deltahedra

| volume = 89

| year = 2013

| zbl = 1286.05048}}</ref>

<ref name=pugh>{{citation

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| title-link = The Mathematical Coloring Book

| year = 2008}}</ref>

<ref name="timofeenko-2009">{{citation

| 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>

<ref name=trigg>{{citation

| last = Trigg | first = Charles W. |author-link = Charles W. Trigg

| doi = 10.1080/0025570X.1978.11976675

| issue = 1

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[[Category:Composite polyhedron]]

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