View source for Great rhombic triacontahedron

{{Short description|Polyhedron with 30 faces}}
{{Uniform polyhedra db|Uniform dual polyhedron stat table|gID}}
[[File:Great rhombic triacontahedron.stl|thumb|3D model of a great rhombic triacontahedron]]
In [[geometry]], the '''great rhombic triacontahedron''' is a nonconvex [[Isohedral figure|isohedral]], [[isotoxal]] [[polyhedron]]. It is the [[Dual polyhedron|dual]] of the [[great icosidodecahedron]] (U54). Like the convex [[rhombic triacontahedron]] it has 30 [[rhombus|rhombic]] faces, 60 edges and 32 vertices (also 20 on 3-fold and 12 on 5-fold axes).

It can be constructed from the convex solid by expanding the faces by factor of <math>\varphi^3 \approx 4.236</math>, where <math>\varphi\!</math> is the [[golden ratio]].

This solid is to the [[compound of great icosahedron and great stellated dodecahedron]] what the convex one is to the [[compound of dodecahedron and icosahedron]]:
The crossing edges in the [[dual compound]] are the diagonals of the rhombs.

What resembles an "excavated" rhombic triacontahedron (compare [[excavated dodecahedron]] and [[great dodecahedron|excavated icosahedron]]) can be seen within the middle of this compound. The rest of the polyhedron strikingly resembles a [[rhombic hexecontahedron]].

The rhombs have two angles of <math>\arccos(\frac{1}{5}\sqrt{5})\approx 63.434\,948\,822\,92^{\circ}</math>, and two of <math>\arccos(-\frac{1}{5}\sqrt{5})\approx 116.565\,051\,177\,08^{\circ}</math>. Its [[dihedral angles]] equal <math>\arccos(-\frac{1}{4}+\frac{1}{4}\sqrt{5})= 72^{\circ}</math>. Part of each rhomb lies inside the solid, hence is invisible in solid models. The ratio between the lengths of the long and short diagonal of the rhombs equals the golden ratio <math>\varphi</math>.

{|
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|rowspan="2"|
{{multiple image
 | align=left | perrow=2 | total_width=400
 | image1 = Skeleton pair 12-20, size s.png
 | image2 = Rhombic triacontahedron 1 (convex), size s, pyritohedral.png
 | image3 = Skeleton pair Gr12 and dual, size s.png
 | image4 = Rhombic triacontahedron 2 (medial), pyritohedral.png
 | image5 = Skeleton pair Gr20 and dual, size s.png
 | image6 = Rhombic triacontahedron 3 (great), pyritohedral.png
 | footer = Convex, [[medial rhombic triacontahedron|medial]] and great rhombic triacontahedron on the right (shown with [[pyritohedral symmetry]]) and the corresponding [[dual compound]]s of regular solids on the left
}}
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[[File:Rhombs of convex, medial and great rhombic triacontahedron.svg|thumb|280px|center|The face diagonal lengths of the three rhombic triacontahedra are powers of {{nowrap|<math>\varphi</math>.}}]]
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{{multiple image
 | align = left  |  total_width = 400
 | image1 = Rhombic triacontahedron 3 (great), size s, 2-fold.png
 | image2 = Rhombic triacontahedron 3 (great), size s, 3-fold.png
 | image3 = Rhombic triacontahedron 3 (great), size s, 5-fold.png
 | footer = Orthographic projections from 2-, 3- and 5-fold axes
}}
|}




==References==
*{{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=[[Cambridge University Press]] | isbn=978-0-521-54325-5 | mr= 730208| year=1983}}

==External links==
* {{mathworld | urlname = GreatRhombicTriacontahedron| title =Great rhombic triacontahedron}}
* David I. McCooey: [http://dmccooey.com/polyhedra/GreatRhombicTriacontahedron.html animation and measurements]
* [http://gratrix.net/polyhedra/uniform/summary Uniform polyhedra and duals]
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[[Category:Dual uniform polyhedra]]


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