Rhombic triacontahedron
In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirtyfaced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.
A face of the rhombic triacontahedron. The lengths of the diagonals are in the golden ratio. 
Rhombic triacontahedron  

(Click here for rotating model)  
Type  Catalan solid 
Coxeter diagram  
Conway notation  jD 
Face type  V3.5.3.5 rhombus 
Faces  30 
Edges  60 
Vertices  32 
Vertices by type  20{3}+12{5} 
Symmetry group  I_{h}, H_{3}, [5,3], (*532) 
Rotation group  I, [5,3]^{+}, (532) 
Dihedral angle  144° 
Properties  convex, facetransitive isohedral, isotoxal, zonohedron 
Icosidodecahedron (dual polyhedron) 
Net 
The ratio of the long diagonal to the short diagonal of each face is exactly equal to the golden ratio, φ, so that the acute angles on each face measure 2 tan^{−1}(1/φ) = tan^{−1}(2), or approximately 63.43°. A rhombus so obtained is called a golden rhombus.
Being the dual of an Archimedean solid, the rhombic triacontahedron is facetransitive, meaning the symmetry group of the solid acts transitively on the set of faces. This means that for any two faces, A and B, there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B.
The rhombic triacontahedron is somewhat special in being one of the nine edgetransitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron, and the rhombic dodecahedron.
The rhombic triacontahedron is also interesting in that its vertices include the arrangement of four Platonic solids. It contains ten tetrahedra, five cubes, an icosahedron and a dodecahedron. The centers of the faces contain five octahedra.
It can be made from a truncated octahedron by dividing the hexagonal faces into 3 rhombi:
Dimensions
If the edge length of a rhombic triacontahedron is a, surface area, volume, the radius of an inscribed sphere (tangent to each of the rhombic triacontahedron's faces) and midradius, which touches the middle of each edge are:[1]
where φ is the golden ratio.
The insphere is tangent to the faces at their face centroids. Short diagonals belong only to the edges of the inscribed regular dodecahedron, while long diagonals are included only in edges of the inscribed icosahedron.
Dissection
The rhombic triacontahedron can be dissected into 20 golden rhombohedra: 10 acute ones and 10 obtuse ones.[2][3]
10  10 

Acute form 
Obtuse form 
Orthogonal projections
The rhombic triacontahedron has four symmetry positions, two centered on vertices, one midface, and one midedge. Embedded in projection "10" are the "fat" rhombus and "skinny" rhombus which tile together to produce the nonperiodic tessellation often referred to as Penrose tiling.
Projective symmetry 
[2]  [2]  [6]  [10] 

Image  
Dual image 
Related polyhedra
Family of uniform icosahedral polyhedra  

Symmetry: [5,3], (*532)  [5,3]^{+}, (532)  
{5,3}  t{5,3}  r{5,3}  t{3,5}  {3,5}  rr{5,3}  tr{5,3}  sr{5,3} 
Duals to uniform polyhedra  
V5.5.5  V3.10.10  V3.5.3.5  V5.6.6  V3.3.3.3.3  V3.4.5.4  V4.6.10  V3.3.3.3.5 
This polyhedron is a part of a sequence of rhombic polyhedra and tilings with [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are also rectangles.
Symmetry mutations of dual quasiregular tilings: V(3.n)^{2}  

*n32  Spherical  Euclidean  Hyperbolic  
*332  *432  *532  *632  *732  *832...  *∞32  
Tiling  
Conf.  V(3.3)^{2}  V(3.4)^{2}  V(3.5)^{2}  V(3.6)^{2}  V(3.7)^{2}  V(3.8)^{2}  V(3.∞)^{2} 
6cube
The rhombic triacontahedron forms a 32 vertex convex hull of one projection of a 6cube to three dimensions.
The 3D basis vectors [u,v,w] are:

Shown with inner edges hidden 20 of 32 interior vertices form a dodecahedron, and the remaining 12 form a icosahedron. 
Uses
Danish designer Holger Strøm used the rhombic triacontahedron as a basis for the design of his buildable lamp IQlight (IQ for "Interlocking Quadrilaterals").
Woodworker Jane Kostick builds boxes in the shape of a rhombic triacontahedron.[6] The simple construction is based on the less than obvious relationship between the rhombic triacontahedron and the cube.
Roger von Oech's "Ball of Whacks" comes in the shape of a rhombic triacontahedron.
The rhombic triacontahedron is used as the "d30" thirtysided die, sometimes useful in some roleplaying games or other places.
Christopher Bird, coauthor of The Secret Life of Plants wrote an article for New Age Journal in May, 1975, popularizing the dual icosahedron and dodecahedron as "the crystalline structure of the Earth," a model of the "Earth (telluric) energy Grid." The EarthStar Globe by Bill Becker and Bethe A. Hagens purports to show "the natural geometry of the Earth, and the geometric relationship between sacred places such as the Great Pyramid, the Bermuda Triangle, and Easter Island." It is printed as a rhombic triacontahedron, on 30 diamonds, and folds up into a globe.[7]
References
 Stephen Wolfram, "" from Wolfram Alpha. Retrieved January 7, 2013.
 Dissection of the rhombic triacontahedron
 Pawley, G. S. (1975). "The 227 triacontahedra". Geometriae Dedicata. Kluwer Academic Publishers. 4 (2–4): 221–232. doi:10.1007/BF00148756. ISSN 15729168.
 Messer, P. W. (1995). "Stellations of the Rhombic Triacontahedron and Beyond". Structural Topology. 21: 25–46.
 triacontahedron box  KO Sticks LLC
 http://www.vortexmaps.com/gridhistory.php
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X. (Section 39)
 Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 9780521543255, MR 0730208 (The thirteen semiregular convex polyhedra and their duals, Page 22, Rhombic triacontahedron)
 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, ISBN 9781568812205 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, Rhombic triacontahedron )
External links
 Eric W. Weisstein, Rhombic triacontahedron (Catalan solid) at MathWorld.
 Rhombic Triacontrahedron – Interactive Polyhedron Model
 Virtual Reality Polyhedra – The Encyclopedia of Polyhedra
 Stellations of Rhombic Triacontahedron
 EarthStar globe – Rhombic Triacontahedral map projection
 IQlight—Danish designer Holger Strøm's lamp
 Make your own
 a wooden construction of a rhombic triacontahedron box – by woodworker Jane Kostick
 120 Rhombic Triacontahedra, 30+12 Rhombic Triacontahedra, and 12 Rhombic Triacontahedra by Sándor Kabai, The Wolfram Demonstrations Project
 A viper drawn on a rhombic triacontahedron.