Rhombic triacontahedron

The rhombic triacontahedron, or rhote, is one of the 13 Catalan solids. It has 30 rhombi as faces, with 12 order-5 and 20 order-3 vertices. It is the dual of the uniform icosidodecahedron.

It can also be obtained as the convex hull of a dodecahedron and an icosahedron scaled so that their edges are orthogonal. For this to happen, the icosahedron's edge length must be $$\frac{1+\sqrt5}{2}$$ times that of the dodecahedron's edge length. Each edge of the dodecahedron or icosahedron corresponds to one of the diagonals of the faces.

Each face of this polyhedron is a rhombus with longer diagonal $$\frac{1+\sqrt5}{2}$$ times the shorter diagonal, with acute angle $$\arccos\left(\frac{\sqrt5}{5}\right) ≈ 63.43495°$$ and obtuse angle $$\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$$.

Vertex coordinates
A rhombic triacontahedron of edge length 1 has vertex coordinates given by all permutations of: Plus all even permutations of:
 * $$\left(±\sqrt{\frac{5+\sqrt5}{8}},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\sqrt{\frac{5+\sqrt5}{8}}\right),$$
 * $$\left(±\sqrt{\frac{5+2\sqrt5}{5}},\,±\sqrt{\frac{5-\sqrt5}{10}},\,0\right),$$
 * $$\left(±\sqrt{\frac{5+2\sqrt5}{5}},\,±\sqrt{\frac{5+\sqrt5}{10}},\,0\right).$$