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{{Semireg polyhedra db|Semireg polyhedron stat table|ID}}
== ROCHESTER Mbt Footwear Australia ==
[[Image:Hoberman Sphere.jpg|thumb|A [[Hoberman sphere]] as an icosidodecahedron]]
In [[geometry]], an '''icosidodecahedron''' is a [[polyhedron]] with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the [[Archimedean solid]]s and more particularly, a [[quasiregular polyhedron]].


An icosidodecahedron has icosahedral symmetry, and its first [[stellation]] is the compound of a [[dodecahedron]] and its dual [[icosahedron]], with the vertices of the icosahedron located at the midpoints of the edges of either.  
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Its [[dual polyhedron]] is the [[rhombic triacontahedron]]. An icosidodecahedron can be split along any of six planes to form a pair of [[pentagonal rotunda]]e, which belong among the [[Johnson solid]]s.
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The icosidodecahedron can be considered a [[#Pentagonal gyrobirotunda|''pentagonal gyrobirotunda'']], as a combination of two [[pentagonal rotunda|rotundae]] (compare [[pentagonal orthobirotunda]], one of the [[Johnson solid]]s). In this form its symmetry is [[Dihedral symmetry|D<sub>5d</sub>]], [10,2<sup>+</sup>], (2*5), order 20.
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The [[wire-frame figure]] of the icosidodecahedron consists of six [[decagon|flat regular decagons]], meeting in pairs at each of the 30 vertices.
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==Cartesian coordinates==
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Convenient [[Cartesian coordinates]] for the vertices of an icosidodecahedron with unit edges are given by:<ref>{{mathworld |title=Icosahedral group |urlname=IcosahedralGroup}}</ref>
 
*(0,0,±φ)
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*(0,±φ,0)
 
*(±φ,0,0)
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*(±1/2, ±φ/2, ±(1+φ)/2)
 
*(±φ/2, ±(1+φ)/2, ±1/2)
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*(±(1+φ)/2, ±1/2, ±φ/2)
 
where φ is the [[golden ratio]], (1+√5)/2.
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</ul>


==Orthogonal projections==
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The icosidodecahedron has four special [[orthogonal projection]]s, centered on a vertex, an edge, a triangular face, and a pentagonal face. The last two correspond to the A<sub>2</sub> and H<sub>2</sub> [[Coxeter plane]]s.
{|class=wikitable
|+ Orthogonal projections
|-
!Centered by
!Vertex
!Edge
!Face<br>Triangle
!Face<br>Pentagon
|-
!Image
|[[File:Dodecahedron_t1_v.png|120px]]
|[[File:Dodecahedron_t1_e.png|120px]]
|[[File:Dodecahedron_t1_A2.png|120px]]
|[[File:Dodecahedron_t1_H3.png|120px]]
|- align=center
!Projective<br>symmetry
|[2]
|[2]
|[6]
|[10]
|}


==Surface area and volume==
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The surface area ''A'' and the volume ''V'' of the icosidodecahedron of edge length ''a'' are:
 
:<math>A = \left(5\sqrt{3}+3\sqrt{25+10\sqrt{5}}\right) a^2 \approx 29.3059828a^2</math>
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:<math>V = \frac{1}{6} \left(45+17\sqrt{5}\right) a^3 \approx 13.8355259a^3.</math>
 
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==Related polyhedra==
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The icosidodecahedron is a [[Rectification (geometry)|rectified]] [[dodecahedron]] and also a rectified [[icosahedron]], existing as the full-edge truncation between these regular solids.


The icosidodecahedron contains 12 pentagons of the [[dodecahedron]] and 20 triangles of the [[icosahedron]]:
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{{Icosahedral truncations}}
 
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</ul>


The icosidodecahedron can be seen in a sequence of [[quasiregular polyhedron]]s and tilings:
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{{Quasiregular figure table}}


===Pentagonal gyrobirotunda===
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It is also related to the [[Johnson solid]] called a [[pentagonal orthobirotunda]] created by two [[pentagonal rotunda]] connected as mirror images.
 
{|class="wikitable"
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|align=center|[[Image:Dissected icosidodecahedron.png|320px]]<br>(Dissection)
 
|
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{|
 
|align=center valign=bottom|[[Image:Icosidodecahedron.png|100px]]<br>Icosidodecahedron<br>(''pentagonal gyrobirotunda'')
  <li>[http://www.ovnprod.com/spip.php?article24/ http://www.ovnprod.com/spip.php?article24/]</li>
|-
 
|align=center valign=bottom|[[Image:Pentagonal orthobirotunda solid.png|100px]]<br>Pentagonal orthobirotunda
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|-
 
|align=center valign=bottom|[[Image:pentagonal rotunda.png|100px]]<br>Pentagonal rotunda
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|}
 
|}
</ul>


Eight [[uniform star polyhedron|uniform star polyhedra]] share the same [[vertex arrangement]]. Of these, two also share the same [[edge arrangement]]: the [[small icosihemidodecahedron]] (having the triangular faces in common), and the [[small dodecahemidodecahedron]] (having the pentagonal faces in common). The vertex arrangement is also shared with the [[uniform polyhedron compound|compounds]] of [[compound of five octahedra|five octahedra]] and of [[compound of five tetrahemihexahedra|five tetrahemihexahedra]].
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{|class="wikitable" width="400" style="vertical-align:top;text-align:center"
|align=center|[[Image:Icosidodecahedron.png|100px]]<br>Icosidodecahedron
|align=center|[[Image:Small icosihemidodecahedron.png|100px]]<br>[[Small icosihemidodecahedron]]
|align=center|[[Image:Small dodecahemidodecahedron.png|100px]]<br>[[Small dodecahemidodecahedron]]
|-
|align=center|[[Image:Great icosidodecahedron.png|100px]]<br>[[Great icosidodecahedron]]
|align=center|[[Image:Great dodecahemidodecahedron.png|100px]]<br>[[Great dodecahemidodecahedron]]
|align=center|[[Image:Great icosihemidodecahedron.png|100px]]<br>[[Great icosihemidodecahedron]]
|-
|align=center|[[Image:Dodecadodecahedron.png|100px]]<br>[[Dodecadodecahedron]]
|align=center|[[Image:Small dodecahemicosahedron.png|100px]]<br>[[Small dodecahemicosahedron]]
|align=center|[[Image:Great dodecahemicosahedron.png|100px]]<br>[[Great dodecahemicosahedron]]
|-
|align=center|[[Image:Compound of five octahedra.png|100px]]<br>[[Compound of five octahedra]]
|align=center|[[Image:UC18-5_tetrahemihexahedron.png|100px]]<br>[[Compound of five tetrahemihexahedra]]
|}
=== Related polytopes===


In four-dimensional geometry the '''icosidodecahedron''' appears in the [[Regular polytopes|regular]] [[600-cell]] as the equatorial slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words: the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed [[hypersphere]] from a pair of opposite vertices, are the vertices of an icosidodecahedron. The wire frame figure of the 600-cell consists of 72 flat regular decagons. Six of these are the equatorial decagons to a pair of opposite vertices. They are precisely the six decagons which form the wire frame figure of the icosidodecahedron.
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</ul>


==See also==
== see studies with "Stockner" Shox Nz Eu ==
*[[Cuboctahedron]]
*[[Great truncated icosidodecahedron]]
*[[Icosahedron]]
*[[Rhombicosidodecahedron]]
*[[Truncated icosidodecahedron]]


==Notes==
AbstractCapturing dependence among a large number of high dimensional random vectors is an extremely important and challenging problem. By arranging n random vectors of length p in the form of a matrix, we develop a linear spectral statistic from the constructed matrix to test whether the n random vectors are [http://www.ashgrovedance.com.au/Content/frame.asp?nikeshox=52 Shox Nz Eu] independent or otherwise. Specifically, the proposed statistic can also be applied to n random vectors, each of whose elements can be written as whether linear stationary process or perhaps a linear combination of a random vector with independent elements. The asymptotic distribution of the proposed test statistic is made in the case where both p and [http://www.stonecottage.com.au/cp/Scripts/ASP/Guestbook/file.asp?a=23-Nike-Air-Force-Ones-White Nike Air Force Ones White] n go to infinity at the same order. To prevent estimating the spectrum of each random vector, a modified test statistic, which is based on splitting the original n vectors into two equal parts and eliminating the word that contains the inner structure of every random vector or time series, is constructed. The facts that the limiting distribution is really a normal distribution and there is no need to know the inner structure of each investigated random vector lead to simple implementation of the constructed test statistic. Simulation results show the proposed test is powerful against many common dependent cases. See general here is how to correct material in RePEc. This allows to link your profile for this item. It also allows you to accept potential citations for this item that we are uncertain about.<br><br>Since like charges repel, Szilard thought, no matter how hard you fire them in, the majority would simply be deflected away. That morning he was one of the first to realise that the recently discovered neutron might be what was needed. The neutron, a subatomic particle just like a proton but with no electric charge is discovered in 1932. With no charge, Szilard believed the neutron would simply put on the heart of the atom undeflected.<br><br>The small markets that really need a new CBA are losing less cash by not playing than playing at this time. And the large markets realize that if [http://www.communitypharmacy.com.au/enews/December/info.asp?a=46 Abercrombie And Fitch Perth Australia] the league gets what it really wants, the financial deficit of losing a season could be more than made up over the course of the next agreement. <br><br>Why should I have to do Google searches watching a CBC documentary to attempt to learn more?I am not against this type of work completed, but it needs to be done transparently and properly. Fertilization of [http://www.paceappointments.com.au/schedule/section.asp?l=48 Longchamps Handbags] water bodies isn't a new concept as it has been done on large nursery lakes in the BC Interior and Southwestern part of the province (see studies with "Stockner").<ul>
{{reflist}}
 
 
  <li>[http://www.cert-centers.com/forum.php?mod=viewthread&tid=1247848&extra= http://www.cert-centers.com/forum.php?mod=viewthread&tid=1247848&extra=]</li>
==References==
 
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
  <li>[http://www.sorinchiriac.ro/forum/profile/tfmfetjm http://www.sorinchiriac.ro/forum/profile/tfmfetjm]</li>
*{{cite book|author=Cromwell, P.|year=1997|title=Polyhedra|location=United Kingdom|publisher=Cambridge|pages=79-86 ''Archimedean solids''|isbn=0-521-55432-2}}
 
 
  <li>[http://verdamilio.net/tonio/spip.php?article1739/ http://verdamilio.net/tonio/spip.php?article1739/]</li>
==External links==
 
*{{mathworld2 |urlname=Icosidodecahedron |title=Icosidodecahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
  <li>[http://discuz.zhiqian.wang/viewthread.php?tid=117950&extra= http://discuz.zhiqian.wang/viewthread.php?tid=117950&extra=]</li>
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|o3x5o - id}}
 
*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=1D1NiLVqdeGtmgRYVoqDtRVdULYX8yuwCc3QmkefHkBBQFfG061UUcmoeyIbYKETw4p9pLg3WecHMPxDRy8JGaGKHlYWQJCqLbA3yu66NU9ZKOdvejvHBGtMuZp4O89LBszvnEgWNFbU7I2JwhnVp8hQBn8wSw1ifNCAzgGapWzCkvOVoJPQgcrWxuNXlSA1ESwiCSBVdqj7pKquEaYsKQyb&name=Icosidodecahedron#applet Editable printable net of an icosidodecahedron  with interactive 3D view]
  <li>[http://matchsouls.com/tweet/?module=yafaomyb http://matchsouls.com/tweet/?module=yafaomyb]</li>
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
 
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
</ul>
 
{{Archimedean solids}}
{{Polyhedron navigator}}
 
[[Category:Archimedean solids]]
[[Category:Quasiregular polyhedra]]

Latest revision as of 14:24, 12 January 2015

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