Triaugmented triangular prism: Difference between revisions

Revision as of 20:51, 27 November 2022

Latest revision as of 13:21, 8 August 2025

(44 intermediate revisions by 16 users not shown)

Line 1:

{{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.

Line 22 ⟶ 23:

==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==

Line 38 ⟶ 36:

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}

Line 47 ⟶ 48:

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

\end{align}</math>

{{-}}

== Fritsch graph ==

Line 61 ⟶ 63:

==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==

Line 71 ⟶ 73:

*{{Annotated link|Császár polyhedron}}

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

{{-Clear}}

==References==

Line 100 ⟶ 102:

| title = Regular-faced convex polyhedra

| volume = 291

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

<ref name=bsw13>{{citation

Line 114 ⟶ 117:

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

| volume = 23

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

<ref name=burgiel>{{citation

Line 141 ⟶ 144:

| pages = 263–266

| title = Deltahedra

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

<ref name=ff98>{{citation

Line 153 ⟶ 156:

| 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

Line 181 ⟶ 185:

| 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>

Line 194 ⟶ 200:

| title = On the space-filling enneahedra

| volume = 12

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

<ref name=involve>{{citation

Line 210 ⟶ 217:

| 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

Line 224 ⟶ 232:

| 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

Line 233 ⟶ 242:

| publisher = Springer

| title = Inorganic Chemistry Concepts

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

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

}}</ref>

<ref name=knill>{{citation

Line 240 ⟶ 251:

| 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

Line 261 ⟶ 286:

| title = Minimal-energy clusters of hard spheres

| volume = 14

| year = 1995}}</ref>| s2cid = 26955765 | doi-access = free

}}</ref>

<ref name=soifer>{{citation

Line 271 ⟶ 297:

| 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

Line 293 ⟶ 330:

| title = Unique arrangements of points on a sphere

| volume = 59

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

}}

{{Johnson solids navigator}}

[[Category:Composite polyhedron]]

[[Category:Johnson solids]]

[[Category:Deltahedra]]

{{Johnson solids navigator}}