Triaugmented triangular prism: Difference between revisions

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{{Short description|Convex polyhedron with 14 triangle faces}}

{{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]]

}}

In [[geometry]], theThe '''triaugmented triangular prism'''{{r|trigg}}, 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,. aThe same shape is processalso called [[Augmentationthe (geometry)'''tetrakis triangular prism''',{{r|augmentation]].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]], [[composite polyhedron]], and [[Johnson solid]].

The same shape is also called the '''tetrakis triangular prism''',{{r|shdc}} '''tricapped trigonal prism''',{{r|kepert}} '''tetracaidecadeltahedron''',{{r|burgiel}} or '''tetrakaidecadeltahedron''';{{r|shdc}} these last names mean a polyhedron with 14 triangular faces. It is an example of a [[deltahedron]] and of a [[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|pughtimofeenko-2009|trigg}} These pyramids cover each square, replacing it with four [[equilateral triangle]]s, so that the resulting polyhedron compriseshas 14 equilateral triangles as its faces. A polyhedron with only equilateral triangles as faces, like this one, 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">\begin{align}

{{bi|left=1.6|<!-- avoid unnecessary interior scrollbar --><math>\displaystyle

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

\begin{align}

\left(0,-\frac2frac{1+\sqrt6}{\sqrt3},0 \pm1right),\qquad & \left(\pm\frac{1+\sqrt6}{2},\frac{1+\sqrt6}{2\sqrt3},0\right).\\

\end{align}</math>}}

(\pm1,-\frac1{\sqrt3},\pm1),\\

(0,-\sqrt2-\frac1{\sqrt3},0),\qquad &

(\pm\frac{1+\sqrt6}{2},\frac{1+\sqrt6}{2\sqrt3},0).\\

\end{align}</math>}}

==Properties==

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the area of 14 equilateral triangles. Its volume,{{r|berman}}

<math display=block>\frac{2\sqrt{2}+\sqrt{3}}{4}a^3\approx 1.140a^3,</math>

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 ==

[[File:Fritsch map.svg|thumb|left|The Fritsch graph and its dual map. For the partial 4-coloring shown, the red–green and blue–green [[Kempe chain]]s cross. It is not possible to free a color for the uncolored center region by swapping colors in a single chain, contradicting [[Alfred Kempe]]'s false proof of the four color theorem.]]

The graph of the triaugmented triangular prism has 9 vertices and 21 edges. It was used by {{harvtxt|Fritsch|Fritsch|1998}} as a small counterexample to [[Alfred Kempe]]'s false proof of the [[four color theorem]] using [[Kempe chain]]s, and its dual map was used as their book's cover illustration.{{r|ff98}} Therefore, this graph has subsequently been named the '''Fritsch graph'''.{{r|involve}} An even smaller counterexample, called the Soifer graph, is obtained by removing one edge from the Fritsch graph (the bottom edge in the illustration here).{{r|involve|soifer}}

The Fritsch graph is one of only six connected graphs in which the [[Neighbourhood (graph theory)|neighborhood]] of every vertex is a cycle of length four or five. More generally, when every vertex in a graph has a cycle of length at least four as its neighborhood, the triangles of the graph automatically link up to form a [[manifold|topological surface]] called a [[Triangulation (topology)|Whitney triangulation]]. These six graphs come from the six Whitney triangulations that, when their triangles are equilateral, have positive [[angular defect]] at every vertex. This makes them a combinatorial analogue of the positively curved smooth surfaces. They come from six of the eight deltahedra—excluding the two that have a vertex with a triangular neighborhood. As well as the Fritsch graph, the other five are the graphs of the [[regular octahedron]], [[regular icosahedron]], [[pentagonal bipyramid]], [[snub disphenoid]], and [[gyroelongated square bipyramid]].{{r|knill}}

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The [[dual polyhedron]] of the triaugmented triangular prism has a face for each vertex of the triaugmented triangular prism, and a vertex for each face. It is an [[enneahedron]] (that is, a nine-sided polyhedron){{r|fr07}} that can be realized with three non-adjacent [[Square (geometry)|square]] faces, and six more faces that are congruent irregular [[pentagon]]s.{{r|as18}} It is also known as an order-5 [[associahedron]], a polyhedron whose vertices represent the 14 triangulations of a [[regular hexagon]].{{r|fr07}} A less-symmetric form of this dual polyhedron, obtained by slicing a [[truncated octahedron]] into four congruent quarters by two planes that perpendicularly bisect two parallel families of its edges, is a [[space-filling polyhedron]].{{r|goldberg}}

More generally, when a polytope is the dual of an associahedron, its boundary (a [[simplicial complex]]) of triangles, tetrahedra, or higher-dimensional simplices) is called a "cluster complex". In the case of the triaugmented triangular prism, it is a cluster complex of type <math>A_3</math>, associated with the the <math>A_3</math> [[Dynkin diagram]] {{Dynkin|node|3|node|3|node}}, the <math>A_3</math> [[root system]], and the <math>A_3</math> [[cluster algebra]].{{r|bsw13}} The connection with the associahedron provides a correspondence between the nine vertices of the triaugmented triangular prism and the nine diagonals of a hexagon. The edges of the triaugmented triangular prism correspond to pairs of diagonals that do not cross, and the triangular faces of the triaugmented triangular prism correspond to the triangulations of the hexagon (consisting of three non-crossing diagonals). The triangulations of other regular polygons correspond to polytopes in the same way, with dimension equal to the number of sides of the polygon minus three.{{r|fr07}}

==Applications==

In the geometry of [[chemical compound]]s, it is common to visualize an [[atom cluster]] surrounding a central atom as a polyhedron—the [[convex hull]] of the surrounding atoms' locations. The [[tricapped trigonal prismatic molecular geometry]] describes clusters for which this polyhedron is a triaugmented triangular prism, although not necessarily one with equilateral triangle faces.{{r|kepert}} For example, the [[lanthanide]]s from [[lanthanum]] to [[dysprosium]] dissolve in water to form [[cation]]s surrounded by nine water molecules arranged as a triaugmented triangular prism.<ref name=Persson2022>{{citation |last1=Persson |first1=Ingmar |date=2022 |title=Structures of Hydrated Metal Ions in Solid State and Aqueous Solution |journal=Liquids |volume=2 |issue=3 |pages=210–242 |doi=10.3390/liquids2030014 |doi-access=free }}</ref>

In the [[Thomson problem]], concerning the minimum-energy configuration of <math>n</math> charged particles on a sphere, and for the [[Tammes problem]] of constructing a [[spherical code]] maximizing the smallest distance among the points, the minimum solution known for <math>n=9</math> places the points at the vertices of a triaugmented triangular prism with non-equilateral faces, [[Circumscribed sphere|inscribed in a sphere]]. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is not known.{{r|whyte}}

{{-Clear}}

==See also==

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*{{Annotated link|Császár polyhedron}}

*{{Annotated link|Steffen's polyhedron}}

{{-Clear}}

==References==

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| title = Regular-faced convex polyhedra

| volume = 291

| year = 1971| issue = 5
}}; see Table IV, line 71, p. 338</ref>

<ref name=bsw13>{{citation

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| title = The discrete fundamental group of the associahedron, and the exchange module

| volume = 23

| year = 2013| s2cid = 14722555 }}</ref>

<ref name=burgiel>{{citation

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| pages = 263–266

| title = Deltahedra

| volume = 36| s2cid = 250435684 }}</ref>

<ref name=ff98>{{citation

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| publisher = Springer-Verlag

| title = The Four-Color Theorem: History, Topological Foundations, and Idea of Proof

| year = 1998}}</ref>| doi-access = free

}}</ref>

<ref name=fw47>{{citation

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| title = Geometric combinatorics

| volume = 13

| year = 2007| isbn = 978-0-8218-3736-8
| s2cid = 11435731
}}; see Definition 3.3, Figure 3.6, and related discussion</ref>

<ref name=francis>{{citation|first=Darryl|last=Francis|title=Johnson solids & their acronyms|journal=Word Ways|date=August 2013|volume=46|issue=3|page=177|url=https://go.gale.com/ps/i.do?id=GALE%7CA340298118}}</ref>

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| title = On the space-filling enneahedra

| volume = 12

| year = 1982| s2cid = 120914105
}}; see polyhedron 9-IV, p. 301</ref>

<ref name=involve>{{citation

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| doi = 10.2140/involve.2009.2.249

| issue = 3

| journal = Involve,: aA Journal of Mathematics

| pages = 249–265

| publisher = Mathematical Sciences Publishers

| title = How false is Kempe's proof of the Four Color Theorem? Part II

| volume = 2}}</ref>| doi-access = free

}}</ref>

<ref name=johnson>{{citation

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| title = Convex polyhedra with regular faces

| volume = 18

| year = 1966}};| sees2cid Table= III,122006114 line| 51</ref>doi-access = free

}}; see Table III, line 51</ref>

<ref name=kepert>{{citation

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| publisher = Springer

| title = Inorganic Chemistry Concepts

| year = 1982}}</ref>| volume = 6

| isbn = 978-3-642-68048-9

}}</ref>

<ref name=knill>{{citation

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| title = A simple sphere theorem for graphs

| year = 2019}}</ref>

<ref name=lptw>{{citation

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| last2 = Parish | first2 = James L.

| last3 = Traub | first3 = Cynthia M.

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

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| journal = International Journal of Pure and Applied Mathematics

| pages = 123–139

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

| volume = 89

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<ref name=pugh>{{citation

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| title = Minimal-energy clusters of hard spheres

| volume = 14

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<ref name=soifer>{{citation

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| title-link = The Mathematical Coloring Book

| year = 2008}}</ref>

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

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| year = 2009

| title = Convex Polyhedra with Parquet Faces

| journal = Docklady Mathematics

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| volume = 80 | issue = 2

| pages = 720–723

| doi = 10.1134/S1064562409050238

}}</ref>

<ref name=trigg>{{citation

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| doi = 10.1080/0025570X.1978.11976675

| issue = 1

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| title = Unique arrangements of points on a sphere

| volume = 59

| year = 1952| issue = 9
}}</ref>

}}

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