 in geometry, the rhombic triacontahedron 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. {|style="float: right; border: 1px solid #bbb; margin: .5em 0 0 .2em;" |- style="font-size: 88%;" |valign="top" |240px a face of the rhombic triacontahedron. the lengths of the diagonals are in the golden ratio. |} the ratio of the long diagonal to the short diagonal of each face is exactly equal to the golden ratio, Ï, so that the acute angles on each face measure , or approximately 63.43Â°. a rhombus so obtained is called a golden rhombus. being the dual of an archimedean solid, the rhombic triacontahedron is face-transitive, 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 edge-transitive 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 includes the arrangement of all the platonic solids. it contains ten tetrahedrons, five hexahedrons, five octahedrons, an icosahedron and a dodecahedron. ==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:stephen wolfram, "" from wolfram alpha. retrieved january 7, 2013. s = a^2 \cdot 12\sqrt{5} \approx 26.8328 \cdot a^2 v = a^3 \cdot 4\sqrt{5+2\sqrt{5}} \approx 12.3107 \cdot a^3 r_i = a \cdot \frac{\varphi^2}{\sqrt{1 + \varphi^2}} = a \cdot \sqrt{1 + \frac{2}{\sqrt{5}}} \approx 1.37638 \cdot a r_m = a \cdot \left(1+\frac{1}{\sqrt5{}}\right) \approx 1.44721 \cdot a where Ï is the golden ratio. the plane of each face is perpendicular to the center of the rhombic triacontahedron, and is located at the same distance (short diagonals belong only to the edges of the inscribed regular dodecahedron, while long diagonals are included only in edges of the inscribed icosahedron). using one of the three orthogonal golden rectangles drawn into the inscribed [
Catalan_solids Quasiregular_polyhedra Zonohedra