Triaugmented triangular prism: Difference between revisions

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| 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]]

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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>

{{-}}

The [[closed geodesic]]s of a polyhedron mean the path on the surface avoiding the vertices and locally look like the shortest path. In other words, the path follows straight line segments across each face that intersect, and creates complementary angles on the two incident faces of the edge as it crosses. In the case of a triaugmented triangular prism, and with unit-length, it has two types of closed geodesics, the first geodesic crosses the edges of two equilateral square pyramids and a triangular prism, an [[equator]] of the solid, with length of <math> 4 </math>; the second geodesic crosses the edges of three equilateral square pyramids, with length of <math> \sqrt{19} </math>.{{r|lptw}}

== Fritsch graph ==

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| 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 | url = https:}}<//ijpam.eu/contents/2013-89-2/1/1.pdfref>

| doi-access = free

}}.</ref>

<ref name=pugh>{{citation