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| {{Semireg polyhedra db|Semireg polyhedron stat table|tI}}
| | Not much to tell about me at all.<br>I enjoy of finally being a member of wmflabs.org.<br>I just wish I am useful at all<br><br>Feel free to visit my site ... Jose Luis [https://Www.vocabulary.com/dictionary/Pati%C3%B1o Patiño] Esquivel, [http://Www.Google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=47&ved=0CFcQFjAGOCg&url=http%3A%2F%2Fwww.boe.es%2Fboe%2Fdias%2F2013%2F03%2F04%2Fpdfs%2FBOE-A-2013-2350.pdf&ei=dsHGU7PhPMem8AXKyoLIAg&usg=AFQjCNGMnaLpjG5jJe30MO0DJLjKoGtZeg&sig2=GwW3oYks2OmXGeUDNh_4Uw http://Www.Google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=47&ved=0CFcQFjAGOCg&url=http%3A%2F%2Fwww.boe.es%2Fboe%2Fdias%2F2013%2F03%2F04%2Fpdfs%2FBOE-A-2013-2350.pdf&ei=dsHGU7PhPMem8AXKyoLIAg&usg=AFQjCNGMnaLpjG5jJe30MO0DJLjKoGtZeg&sig2=GwW3oYks2OmXGeUDNh_4Uw], |
| In [[geometry]], the '''truncated [[icosahedron]]''' is an [[Archimedean solid]], one of 13 convex [[Isogonal figure|isogonal]] nonprismatic solids whose [[Face (geometry)|face]]s are two or more types of [[regular polygons]].
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| It has 12 regular [[pentagon]]al faces, 20 regular [[hexagon]]al faces, 60 vertices and 90 edges.
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| It is the [[Goldberg polyhedron]] G<sub>V</sub>(1,1), containing pentagonal and hexagonal faces.
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| ==Construction==
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| This polyhedron can be constructed from an [[icosahedron]] with the 12 vertices [[Truncation (geometry)|truncated]] (cut off) such that one third of each edge is cut off at each of both ends. This creates 12 new pentagon faces, and leaves the original 20 triangle faces as regular hexagons. Thus the length of the edges is one third of that of the original edges.
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| {|class="wikitable"
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| |[[Image:Icosahedron.png|160px]]<br>[[Icosahedron]]
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| |}
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| ==Cartesian coordinates==
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| [[Cartesian coordinates]] for the vertices of a ''truncated icosahedron'' centered at the origin are all even permutations of:
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| :(0, ±1, ±3φ)
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| :(±2, ±(1+2φ), ±φ)
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| :(±1, ±(2+φ), ±2φ)
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| where φ = (1 + √5) / 2 is the [[golden ratio|golden mean]]. Using φ<sup>2</sup> = φ + 1 one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 9φ + 10. The edges have length 2.<ref>{{mathworld |title=Icosahedral group |urlname=IcosahedralGroup}}</ref>
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| ==Orthogonal projections==
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| The ''truncated icosahedron'' has five special [[orthogonal projection]]s, centered, on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A<sub>2</sub> and H<sub>2</sub> [[Coxeter plane]]s.
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| {|class=wikitable
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| |+ Orthogonal projections
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| |-
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| !Centered by
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| !Vertex
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| !Edge<br>5-6
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| !Edge<br>6-6
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| !Face<br>Hexagon
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| !Face<br>Pentagon
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| |-
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| !Image
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| |[[File:Dodecahedron t12 v.png|120px]]
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| |[[File:Dodecahedron t12 e56.png|120px]]
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| |[[File:Dodecahedron t12 e66.png|120px]]
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| |[[File:Icosahedron t01 A2.png|120px]]
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| |[[File:Icosahedron t01 H3.png|120px]]
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| |- align=center
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| !Projective<br>symmetry
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| |[2]
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| |[2]
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| |[2]
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| |[6]
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| |[10]
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| |}
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| ==Dimensions==
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| [[Image:Icosahedron-golden-rectangles.svg|120px|thumb|right|Mutually orthogonal golden rectangles drawn into the original icosahedron (before cut off)]]
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| If the edge length of a truncated icosahedron is ''a'', the [[radius]] of a [[circumscribed sphere]] (one that touches the truncated icosahedron at all vertices) is:
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| :<math>r_u = \frac{a}{2} \sqrt{1 + 9\varphi^2} = \frac{a}{4} \sqrt{58 +18\sqrt{5}} \approx 2.47801866 \cdot a</math>
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| where ''φ'' is the [[golden ratio]].
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| This result is easy to get by using one of the three orthogonal golden rectangles drawn into the original icosahedron (before cut off) as the starting point for our considerations. The angle between the segments joining the center and the vertices connected by shared edge (calculated on the basis of this construction) is approx. 23.281446°.
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| ==Area and volume== | |
| The area ''A'' and the volume ''V'' of the truncated icosahedron of edge length ''a'' are:
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| :<math>\begin{align}
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| A & = \left ( 20 \cdot \frac32\sqrt{3} + 12 \cdot \frac54\sqrt{ 1 + \frac{2}{\sqrt{5}}} \right ) a^2 \approx 72.607253a^2 \\
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| V & = \frac{1}{4} \left(125+43\sqrt{5}\right) a^3 \approx 55.2877308a^3. \\
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| \end{align}</math>
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| ==Geometric relations== | |
| The truncated icosahedron easily verifies the [[Euler characteristic]]:
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| :32 + 60 − 90 = 2.
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| With unit edges, the surface area is (rounded) 21 for the pentagons and 52 for the hexagons, together 73 (see [[Regular polygon#Area|areas of regular polygons]]).
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| ==Applications==
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| [[Image:Trunc-icosa.jpg|125px|thumb|left|The truncated icosahedron (left) compared to a [[football]].]]
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| [[File:Buckminsterfullerene Model in Red Beads.jpg|thumb|[[Buckminsterfullerene|Fullerene]] C<sub>60</sub> molecule]]
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| [[File:Peter Stehlik 2010.08.03 003.jpg|thumb|Truncated icosahedral [[radome]] on a [[weather station]]]]
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| The [[Football (association football)|balls used in 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. <ref>{{Cite journal |title=The Topology and Combinatorics of Soccer Balls |first1=Dieter |author1-link=Dieter Kotschick |last1=Kotschick |journal=American Scientist |year=2006 |volume=94 |issue=4 |pages=350–357 |postscript=<!--None-->}}</ref> The ball comprises the same pattern of regular pentagons and regular hexagons, but it is more spherical due to the pressure of the air inside and the elasticity of the ball. This ball type was introduced in 1970; starting with the [[2006 World Cup]], the design has been superseded by newer patterns.
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| A variation of the icosahedron was used as the basis of the honeycomb wheels (made from a polycast material) used by the Pontiac Motor Division between 1971 to 1976 on its Trans Am and Grand Prix. | |
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| This 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.<ref>{{cite book
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| |author=Rhodes, Richard
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| |title=Dark Sun: The Making of the Hydrogen Bomb
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| |isbn=0-684-82414-0
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| |publisher=Touchstone Books
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| |year=1996
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| |pages=195}}</ref>
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| The truncated icosahedron can also be described as a model of the [[Buckminsterfullerene]] (fullerene) (C<sub>60</sub>), or "buckyball," molecule, an [[allotrope]] of elemental carbon, discovered in 1985. The diameter of the football and the fullerene molecule are 22 cm and ca. 1 [[Nanometer|nm]], respectively, hence the size ratio is 220,000,000:1.
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| ==Truncated icosahedra in the arts== | |
| A truncated icosahedron with "solid edges" is a drawing by [[Luca Pacioli|Lucas Pacioli]] illustrating ''The Divine Proportion''.
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| ==Related polyhedra==
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| {{Icosahedral truncations}}
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| {{Truncated figure2 table}}
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| These [[uniform star-polyhedra]], and one icosahedral stellation have nonuniform truncated icosahedra [[convex hull]]s:
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| {|class=wikitable
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| ![[File:Nonuniform truncated icosahedron.png|64px]]<br>Nonuniform<br>truncated icosahedron<br>2 5 | 3
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| |[[Image:Great truncated dodecahedron.png|64px]]<br>[[Truncated great dodecahedron|U37]]<br> 2 5/2 | 5
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| |[[Image:Great dodecicosidodecahedron.png|64px]]<br>[[Great dodecicosidodecahedron|U61]]<br> 5/2 3 | 5/3
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| |[[Image:Uniform great rhombicosidodecahedron.png|64px]]<br>[[Nonconvex great rhombicosidodecahedron|U67]]<br> 5/3 3 | 2
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| |[[Image:Great rhombidodecahedron.png|64px]]<br>[[Great rhombidodecahedron|U73]]<br>2 5/3 (3/2 5/4)
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| |[[File:Complete icosahedron ortho stella.png|64px]]<br>[[Complete stellation of the icosahedron|Complete stellation]]
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| |-
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| ![[File:Rhombidodecadodecahedron convex hull.png|64px]]<br>Nonuniform<br>truncated icosahedron<br>2 5 | 3
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| |[[Image:Rhombidodecadodecahedron.png|64px]]<br>[[Rhombidodecadodecahedron|U38]]<br>5/2 5 | 2
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| |[[Image:Icosidodecadodecahedron.png|64px]]<br>[[Icosidodecadodecahedron|U44]]<br>5/3 5 | 3
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| |[[Image:Rhombicosahedron.png|64px]]<br>[[Rhombicosahedron|U56]]<br>2 3 (5/4 5/2) |
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| |-
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| ![[File:Small snub icosicosidodecahedron convex hull.png|64px]]<br>Nonuniform<br>truncated icosahedron<br>2 5 | 3
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| |[[Image:Small snub icosicosidodecahedron.png|64px]]<br>[[Small snub icosicosidodecahedron|U32]]<br> | 5/2 3 3
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| |}
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| ==See also== | |
| * [[Dodecahedron]]
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| * [[Icosidodecahedron]]
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| * [[Truncated dodecahedron]]
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| * [[truncated rhombic triacontahedron]]
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| * [[hyperbolic soccerball]]
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| * [[fullerene]]
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| ==Notes== | |
| <references/>
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| ==References==
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| * {{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
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| *{{cite book|author=Cromwell, P.|year=1997|title=Polyhedra|location=United Kingdom|publisher=Cambridge|pages=79-86 ''Archimedean solids''|isbn=0-521-55432-2}}
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| ==External links==
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| {{wiktionary}}
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| * {{mathworld2 |urlname=TruncatedIcosahedron |title=Truncated icosahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
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| * {{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x5o - ti}}
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| * [http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=FcWqcgCFwJy8hdr7Tpr0E5bZt5x3cWEH6dwwKHCmiAYTfaZwqWF0uEVMInwLYVXfN8PDwpUyauaLHhwJXg40Gas6xdXAY3rsGHjcD6k2QDzGzKkuLMI31aQ26rI3uD7eErma5wI9FxPhfb5HKttbwy3zq25bAOIRYmGDfrCf7sgmOwZShNBxNitreNFUEgEVyUztA9NGYxLLw6iOI3DGr1wAFohIvfYJ9v6CVDw7VE0C0NFaLQlslQ6L5acUJqQrPut6CxdTdxW1jRbZLXocrlCgQRpfanYeiaOBkYeHTpTQ1A5A8vkLNMrMDBl02ot2TMhJrhkI5ignSVOJ5X4S7GyUQMhdnriBZb447C6DLmqsGccUdj3C87AInlwfzYEQZvAXy0RR5QIt7WdIlLlcQldm&name=Truncated+Icosahedron#applet Editable printable net of a truncated icosahedron with interactive 3D view]
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| * [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
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| * [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
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| {{Archimedean solids}}
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| {{Polyhedron navigator}}
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| [[Category:Archimedean solids]]
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| [[Category:Uniform polyhedra]]
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