Editing Truncated icosahedron

{{Short description|A polyhedron resembling a soccerball}}
{{CS1 config|mode=cs1}}
{{infobox polyhedron
 | name = Truncated icosahedron
 | image = File:Truncatedicosahedron.jpg
 | type = [[Archimedean solid]]<br>[[Uniform polyhedron]]<br>[[Goldberg polyhedron]]
 | faces = 32
 | edges = 90
 | vertices = 60
 | symmetry = [[Icosahedral symmetry]] <math> \mathrm{I}_{\mathrm{h}} </math>
 | vertex_figure = File:Polyhedron truncated 20 vertfig.svg
 | net = File:Polyhedron truncated 20 net compact.svg
 | dual = [[Pentakis dodecahedron]]
}}
[[File:Truncated icosahedron.stl|thumb|3D model of a truncated icosahedron]]

In [[geometry]], the '''truncated icosahedron''' is a polyhedron that can be constructed by [[Truncation (geometry)|truncating]] all of the [[regular icosahedron]]'s vertices. Intuitively, it may be regarded as [[Ball (association football)|footballs]] (or soccer balls) that are typically patterned with white hexagons and black pentagons. It can be found in the application of [[geodesic dome]] structures such as those whose architecture [[Buckminster Fuller]] pioneered are often based on this structure. It is an example of an [[Archimedean solid]], as well as a [[Goldberg polyhedron]].

== Construction ==
The truncated icosahedron can be constructed from a [[regular icosahedron]] by cutting off all of its vertices, known as [[Truncation (geometry)|truncation]]. Each of the 12 vertices at the one-third mark of each edge creates 12 pentagonal faces and transforms the original 20 triangle faces into regular hexagons.{{r|co}} Therefore, the resulting polyhedron has 32 faces, 90 edges, and 60 vertices.{{r|berman}} A [[Goldberg polyhedron]] is one whose faces are 12 pentagons and some multiple of 10 hexagons.  There are three classes of Goldberg polyhedron, one of them is constructed by truncating all vertices repeatedly, and the truncated icosahedron is one of them, denoted as <math> \operatorname{GP}(1,1) </math>.{{r|hart}}

== Properties ==
The surface area <math> A </math> and the volume <math> V </math> of the truncated icosahedron of edge length <math> a </math> are:{{r|berman}}
<math display=block>\begin{align}
A &= \left ( 20 \cdot \frac32\sqrt{3} + 12 \cdot \frac54\sqrt{ 1 + \frac{2}{\sqrt{5}}} \right) a^2 \approx 72.607a^2 \\
V &= \frac{125+43\sqrt{5}}{4} a^3 \approx 55.288a^3. 
\end{align}</math>
The [[sphericity]] of a polyhedron <math> \Psi </math> describes how closely a polyhedron resembles a [[sphere]]. It can be defined as the ratio of the surface area of a sphere with the same volume to the polyhedron's surface area, from which the value is between 0 and 1. In the case of a truncated icosahedron, it is:{{r|berman}}
<math display="block"> \Psi = \frac{6\pi^{1/2} V}{A^{3/2}} \approx 0.9504. </math>

The [[dihedral angle]] of a truncated icosahedron between adjacent hexagonal faces is approximately 138.18°, and that between pentagon-to-hexagon is approximately 142.6°.{{r|johnson}}

The truncated icosahedron is an [[Archimedean solid]], meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.{{r|diudea}} It has the same symmetry as the regular icosahedron, the [[icosahedral symmetry]], and it also has the property of [[vertex-transitivity]].{{r|kocakoca|cromwell}} The polygonal faces that meet for every vertex are one pentagon and two hexagons, and the [[vertex figure]] of a truncated icosahedron is <math> 5 \cdot 6^2 </math>. The truncated icosahedron's dual is [[pentakis dodecahedron]], a [[Catalan solid]],{{r|williams}} shares the same symmetry as the truncated icosahedron.{{r|holden}}

== Truncated icosahedral graph ==
[[File:Truncated icosahedral graph.png|thumb|The truncated icosahedral graph]]
According to [[Steinitz's theorem]], the [[Skeleton (topology)|skeleton]] of a truncated icosahedron, like that of any [[convex polyhedron]], can be represented as a [[polyhedral graph]], meaning a [[planar graph]] (one that can be drawn without crossing edges) and [[k-vertex-connected graph|3-vertex-connected graph]] (remaining connected whenever two of its vertices are removed).{{r|negami}} The graph is known as '''truncated icosahedral graph''', and it has 60 [[Vertex (graph theory)|vertices]] and 90 edges. It is an [[Archimedean graph]] because it resembles one of the Archimedean solids. It is a [[cubic graph]], meaning that each vertex is incident to exactly three edges.{{r|rw|gr|kostant}}
{{-}}

== Appearance ==
[[File:Comparison of truncated icosahedron and soccer ball.png|thumb|upright=1.1|The truncated icosahedron (left) compared with an [[Ball (association football)|association football]]]]
The balls used in [[Football (association football)|association football]] and [[Team handball#Ball|team handball]] are perhaps the best-known example of a [[spherical polyhedron]] analog to the truncated icosahedron, found in everyday life.{{r|kotschick}} The ball comprises the same pattern of regular pentagons and regular hexagons, each of which is painted in black and white respectively; still, its shape is more spherical. It was introduced by [[Adidas]], which debuted the [[Adidas Telstar|Telstar ball]] during [[1970 FIFA World Cup|World Cup in 1970]].{{r|hh}} However, it was superseded in [[2006 World Cup|2006]].{{r|pmtsgsd}}

[[File:Buckminsterfullerene Model in Red Beads.jpg|thumb|left|upright=0.8|The [[buckminsterfullerene]] molecule]]
[[Geodesic dome]]s are typically based on triangular facetings of this geometry with example structures found across the world, popularized by [[Buckminster Fuller]]. An example can be found in the model of a [[buckminsterfullerene]], a truncated icosahedron-shaped geodesic dome [[allotrope]] of elemental carbon discovered in 1985.{{r|katz-2006}} In other engineering and science applications, its shape was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in both [[the gadget]] and [[Fat Man]] [[atomic bomb]]s.{{r|rhodes}} Its structure can also be found in the [[protein]] of [[clathrin]].{{r|kostant}}

[[File:Piero della Francesca - Libellus de quinque corporibus regularibus - p52b (cropped).jpg|thumb|upright=1.1|[[Piero della Francesca]]'s image of a truncated icosahedron from his book ''[[De quinque corporibus regularibus]]'']]
The truncated icosahedron was known to [[Archimedes]], who classified the 13 Archimedean solids in a lost work. All that is now known of his work on these shapes comes from [[Pappus of Alexandria]], who merely lists the numbers of faces for each: 12 pentagons and 20 hexagons, in the case of the truncated icosahedron. The first known image and complete description of a truncated icosahedron are from a rediscovery by [[Piero della Francesca]], in his 15th-century book ''[[De quinque corporibus regularibus]]'', which included five of the Archimedean solids (the five truncations of the regular polyhedra).{{r|katz-2011}} The same shape was depicted by [[Leonardo da Vinci]], in his illustrations for [[Luca Pacioli]]'s plagiarism of della Francesca's book in 1509. Although [[Albrecht Dürer]] omitted this shape from the other Archimedean solids listed in his 1525 book on polyhedra, ''Underweysung der Messung'', a description of it was found in his posthumous papers, published in 1538. [[Johannes Kepler]] later rediscovered the complete list of the 13 Archimedean solids, including the truncated icosahedron, and included them in his 1609 book, ''[[Harmonices Mundi]]''.{{r|field}}

== See also ==
* [[Chamfered dodecahedron]]
* [[Icosahedral twins]] - Nanoparticles which can have the shape of a truncated icosahedron

== References ==
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}}

==External links==
{{commons category}}
{{wiktionary}}
* {{mathworld2 |urlname=TruncatedIcosahedron |title=Truncated icosahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
** {{mathworld |urlname=TruncatedIcosahedralGraph |title=Truncated icosahedral graph}}
* {{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x5o - ti}}
* [http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html Editable printable net of a truncated icosahedron with interactive 3D view]
* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
* [http://www.georgehart.com/virtual-polyhedra/vp.html "Virtual Reality Polyhedra"]—''The Encyclopedia of Polyhedra''
* [http://visualoop.com/blog/22881/3d-world-cup-dataviz-ball-by-times-of-oman 3D paper data visualization World Cup ball]

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[[Category:Goldberg polyhedra]]
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