Gyrate rhombicosidodecahedron

The gyrate rhombicosidodecahedron, or gyrid, is one of the 92 Johnson solids (J72). It consists of 5+5+5+5 triangles, 5+5+5+5+10 squares, and 1+1+5+5 pentagons. It can be constructed by rotating one of the pentagonal cupolaic caps of the small rhombicosidodecahedron by 36°.

Vertex coordinates
A gyrate rhombicosidodecahedron of edge length 1 has vertices given by:
 * $$\left(±\frac{5+\sqrt5}{4},\,0,\,±\frac{3+\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{3+\sqrt5}{4},\,-\frac{5+\sqrt5}{4}\right),$$
 * $$\left(0,\,-\frac{3+\sqrt5}{4},\,\frac{5+\sqrt5}{4}\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,0\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac{2+\sqrt5}{2}\right),$$
 * $$\left(±\frac{2+\sqrt5}{2},\,±\frac12,\,±\frac12\right),$$
 * $$\left(±\frac12,\,±\frac{2+\sqrt5}{2},\,-\frac12\right),$$
 * $$\left(±\frac12,\,-\frac{2+\sqrt5}{2},\,\frac12\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,-\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,-\frac{1+\sqrt5}{2},\,\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,\frac{5+4\sqrt5}{10},\,\frac{10+3\sqrt5}{10}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\frac{5+2\sqrt5}{5},\,\frac{15+\sqrt5}{20}\right),$$
 * $$\left(0,\,\frac{15+13\sqrt5}{20},\,\frac{5+\sqrt5}{20}\right).$$