DBFS

2014-10-02T18:46:30Z

207.8.234.82: /* Analog levels */

Hello from France. I'm glad to came here. My first name is Antonia. I live in a city called Vienne in east France. I was also born in Vienne 24 years ago. Married in March year 2009. I'm working at the post office. Look into my blog post; [http://www.wikihowto.in/qtoa/index.php/1236/how-to-get-free-fifa-15-coins FIFA 15 Coin Hack]

Flash evaporation

2014-01-10T14:42:48Z

207.8.148.65: /* Contrast with spray drying */

{{Semireg polyhedra db|Semireg polyhedron stat table|grID}}
In [[geometry]], the '''truncated icosidodecahedron''' is an [[Archimedean solid]], one of thirteen convex [[Isogonal figure|isogonal]] nonprismatic solids constructed by two or more types of [[regular polygon]] [[Face (geometry)|face]]s.

It has 30 [[square (geometry)|square]] faces, 20 regular [[hexagon]]al faces, 12 regular [[decagon]]al faces, 120 vertices and 180 edges – more than any other nonprismatic [[uniform polyhedron]]. Since each of its faces has point symmetry (equivalently, 180° [[rotation]]al symmetry), the truncated icosidodecahedron is a [[zonohedron]].

==Other names==
Alternate interchangeable names include:
*''Truncated icosidodecahedron'' ([[Johannes Kepler]])
*''Rhombitruncated icosidodecahedron'' ([[Magnus Wenninger]]<ref>Wenninger, (Model 16, p. 30)</ref>)
*''Great rhombicosidodecahedron'' ([[Robert Williams (geometer)|Robert Williams]],<ref>Williamson (Section 3-9, p. 94)</ref> Peter Cromwell<ref>Cromwell (p. 82)</ref>)
*''[[Omnitruncation (geometry)|Omnitruncated]] dodecahedron'' or ''icosahedron'' ([[Norman Johnson (mathematician)|Norman Johnson]])

The name ''truncated icosidodecahedron'', originally given by [[Johannes Kepler]], is somewhat misleading. If one [[truncation (geometry)|truncates]] an [[icosidodecahedron]] by cutting the corners off, one does ''not'' get this uniform figure: instead of [[Square (geometry)|square]]s the truncation has [[golden rectangle]]s. However, the resulting figure is [[topologically]] equivalent to this and can always be deformed until the faces are regular.
{|class="wikitable" width=360
|valign=top|[[Image:icosidodecahedron.png|180px]] [[Icosidodecahedron]]
|valign=top|[[Image:Nonuniform truncated icosidodecahedron.png|180px]] A literal geometric [[truncation (geometry)|truncation]] of the icosidodecahedron produces [[rectangular]] faces rather than [[Square (geometry)|squares]].
|}

The alternative name ''great rhombicosidodecahedron'' (as well as rhombitruncated icosidodecahedron) refers to the fact that the 30 square faces lie in the same planes as the 30 faces of the [[rhombic triacontahedron]] which is dual to the [[icosidodecahedron]]. Compare to [[small rhombicosidodecahedron]].

One unfortunate point of confusion is that there is a nonconvex uniform polyhedron of the same name. See [[nonconvex great rhombicosidodecahedron]].

=== Variations ===
Within [[Icosahedral symmetry]] there are unlimited geometric variations of the ''truncated icosidodecahedron'' with [[Isogonal_figure|isogonal]] faces. The [[truncated dodecahedron]], [[rhombicosidodecahedron]], and [[truncated icosahedron]] as degenerate limiting cases.
{|
|[[File:Truncated dodecahedron.png|120px]]
|[[File:Great truncated icosidodecahedron convex hull.png|120px]]
|[[File:Nonuniform_truncated_icosidodecahedron.png|120px]]
|[[File:Truncated_dodecadodecahedron_convex_hull.png|120px]]
|[[File:Icositruncated_dodecadodecahedron_convex_hull.png|120px]]
|}

==Area and volume==
The surface area ''A'' and the volume ''V'' of the truncated icosidodecahedron of edge length ''a'' are:
:<math>\begin{align}
A & = 30 \left [ 1 + \sqrt{ 2 \left ( 4 + \sqrt{5} + \sqrt{15+6\sqrt{6}} \right ) } \right ] a^2 \\
& \approx 175.031045a^2 \\
V & = ( 95 + 50\sqrt{5} ) a^3 \approx 206.803399a^3. \\
\end{align}</math>

If a set of all 13 [[Archimedean solid]]s were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest.

==Cartesian coordinates==
[[Cartesian coordinates]] for the vertices of a truncated icosidodecahedron with edge length 2τ − 2, centered at the origin, are all the [[even permutation]]s of:<ref>{{mathworld|title=Icosahedral group|urlname=IcosahedralGroup}}</ref>
:(±1/τ, ±1/τ, ±(3+τ)),
:(±2/τ, ±τ, ±(1+2τ)),
:(±1/τ, ±τ2, ±(−1+3τ)),
:(±(-1+2τ), ±2, ±(2+τ)) and
:(±τ, ±3, ±2τ),
where τ = (1 + √5)/2 is the [[golden ratio]].

==Orthogonal projections==
The truncated icosidodecahedron has seven special [[orthogonal projection]]s, centered on a vertex, on three types of edges, and three types of faces: square, hexagonal and decagonal. The last two correspond to the A2 and H2 [[Coxeter plane]]s.
{|class=wikitable
|+ Orthogonal projections
|-
!Centered by
!Vertex
!Edge 4-6
!Edge 4-10
!Edge 6-10
!Face square
!Face hexagon
!Face decagon
|-
!Image
|[[File:Dodecahedron_t012_v.png|100px]]
|[[File:Dodecahedron_t012_e46.png|100px]]
|[[File:Dodecahedron_t012_e4x.png|100px]]
|[[File:Dodecahedron_t012_e6x.png|100px]]
|[[File:Dodecahedron_t012_f4.png|100px]]
|[[File:Dodecahedron_t012_A2.png|100px]]
|[[File:Dodecahedron_t012_H3.png|100px]]
|- align=center
!Projective symmetry
|[2]+
|[2]
|[2]
|[2]
|[2]
|[6]
|[10]
|}

==Spherical tiling==
The truncated icosidodecahedron can also be represented as a [[spherical tiling]], and projected onto the plane via a [[stereographic projection]]. This projection is [[Conformal map|conformal]], preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
{|class=wikitable width=640
|[[Image:Uniform tiling 532-t012.png|160px]]
|[[Image:Truncated icosidodecahedron stereographic projection decagon.png|160px]] [[Decagon]]-centered
|[[Image:Truncated icosidodecahedron stereographic projection hexagon.png|160px]] [[Hexagon]]-centered
|[[Image:Truncated icosidodecahedron stereographic projection square.png|160px]] [[square (geometry)|square]]-centered
|-
!Spherical tiling
!colspan=3|Stereographic projections (face-centered)
|}
== Related polyhedra and tilings==
{{Icosahedral truncations}}

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and [[Coxeter-Dynkin diagram]] {{CDD|node_1|p|node_1|3|node_1}}. For ''p'' < 6, the members of the sequence are [[Omnitruncation (geometry)|omnitruncated]] polyhedra ([[zonohedron]]s), shown below as spherical tilings. For ''p'' > 6, they are tilings of the hyperbolic plane, starting with the [[truncated triheptagonal tiling]].

{{Omnitruncated table}}

==See also==
*[[:Image:Truncatedicosidodecahedron.gif|Spinning great rhombicosidodecahedron]]
*[[Dodecahedron]]
*[[Great truncated icosidodecahedron]]
*[[Icosahedron]]
*[[Truncated cuboctahedron]]

==Notes==
{{reflist}}

==References==
*{{Citation |last1=Wenninger |first1=Magnus |author1-link=Magnus Wenninger |title=Polyhedron Models |publisher=[[Cambridge University Press]] |isbn=978-0-521-09859-5 |id={{MathSciNet |id=0467493}} |year=1974}}
*{{cite book|author=Cromwell, P.|year=1997|title=Polyhedra|location=United Kingdom|publisher=Cambridge|pages=79-86 ''Archimedean solids''|isbn=0-521-55432-2}}
*{{The Geometrical Foundation of Natural Structure (book)}}
*Cromwell, P.; [http://books.google.com/books?id=OJowej1QWpoC&lpg=PP1&pg=PA82#v=onepage&q=&f=false ''Polyhedra''], CUP hbk (1997), pbk. (1999).
*{{mathworld2 |urlname=GreatRhombicosidodecahedron |title=GreatRhombicosidodecahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x5x - grid}}

==External links==
*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=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&name=Truncated+Icosidodecahedron#applet Editable printable net of a truncated icosidodecahedron 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

{{Archimedean solids}}
{{Polyhedron navigator}}

{{DEFAULTSORT:Truncated Icosidodecahedron}}
[[Category:Uniform polyhedra]]
[[Category:Archimedean solids]]
[[Category:Zonohedra]]

formulasearchengine - User contributions [en]

DBFS

Flash evaporation